Colloquia for the Spring 2023 Semester

Matthew Rosenzweig, MIT

(Special Colloquium)

Monday, January 9, 2023, KAP 414, 3:30 PM – 4:30 PM

Recent progress on mean-field limits for systems with Riesz interactions

Abstract: In statistical physics, many particle models are described by an interaction energy determined by the Coulomb potential, or more generally an inverse power law called a Riesz potential. To this energy, one can associate a dynamics, either conservative or dissipative, which takes the form of a coupled system of nonlinear differential equations. In principle, one could solve this system of differential equations directly and perfectly describe the behavior of every particle in the system. But in practice, the number of particles (e.g., 1023 in a gas) is too large for this to be feasible. Instead, one can focus on the “average” behavior of a particle, which is encoded by the empirical measure of the system. Formally, this measure converges to a solution of a certain nonlinear PDE, called the mean-field limit, as the number of particles tends to infinity; but proving this convergence is a highly nontrivial matter. We will review results over the past few years on mean-field limits for Riesz systems, including important questions such as how fast this limit occurs and how it deteriorates with time, and discuss open questions that still remain.


Tomasz Mrowka, MIT

Double Header – Distinguished Lectures from MIT

Tuesday, January 17, 2023, KAP 414, 2:00 PM – 3:00 PM

Floer Homology for three manifolds and its applications

Abstract: Floer homology theories for 3-manifolds come from many sources Instantons, Seiberg-Witten Monopoles, Heegaard Floer and Embedded Contact Floer theories. They have proven to be a powerful tools in low dimensional topology. I’ll try to outline some of their applications and give some prospects for some future directions. This is meant to be a fly over without (m)any details hopefully accessible to a rather general mathematics audience.


Gigliola Staffilani, MIT

Double Header – Distinguished Lectures from MIT

Tuesday, January 17, 2023, KAP 414, 3:30 PM – 4:30 PM

On the wave turbulence theory for a stochastic KdV type equation

Abstract: This talk is a summary of a recent work completed with Binh Tran. Starting from the stochastic Zakharov-Kuznetsov (ZK) equation, a multidimensional KdV type equation on a hypercubic lattice, we provide a rigorous derivation of the 3-wave kinetic equation. We show that the two point correlation function can be asymptotically expressed as the solution of the 3-wave kinetic equation at the kinetic limit under very general assumptions: the initial condition is out of equilibrium, the dimension is d>1, the smallness of the nonlinearity is allowed to be independent of the size of the lattice, the weak noise is chosen not to compete with the weak nonlinearity and not to inject energy into the equation. Unlike the cubic nonlinear Schrödinger equation, for which such a general result is commonly expected without the noise, the kinetic description of the deterministic lattice ZK equation is unlikely to happen. One of the key reasons is that the dispersion relation of the lattice ZK equation leads to a singular manifold, on which not only 3-wave interactions but also all m-wave interactions are allowed to happen. To the best of our knowledge, this work provides the first rigorous derivation of nonlinear 3-wave kinetic equations. Also, this is the first derivation for wave kinetic equations in the lattice setting and out-of-equilibrium.


Steven Heilman, USC

(Special Colloquium)

Monday, January 23, 2023, KAP 414, 3:30 PM – 4:30 PM

Gaussian Isoperimetry with Discrete Applications

Abstract: Given votes for candidates, what is the best way to determine the winner of the election, if some of the votes have been corrupted or miscounted? As we saw in Florida in 2000, where a difference of 537 votes determined the president of the United States, the electoral college system does not seem to be the best voting method. We will survey some recent progress on the above question along with some open problems. Our results use tools from the calculus of variations, probability, discrete and continuous Fourier analysis, and from the geometry of the Gaussian measure on Euclidean space. Answering the above voting question reveals unexpected connections to Khot’s Unique Games Conjecture in theoretical computer science, the MAX-CUT problem, and mean curvature flows. We will discuss these connections and present recent results and open problems.


Roman Shvydkoy, UIC

Monday, February 6, 2023, KAP 414, 3:30 PM – 4:30 PM

On the problem of emergence arising in hydrodynamic systems of collective behavior

Abstract: Emergence is a phenomenon of formation of collective outcomes in systems where communications between agents has local range. In dynamics of swarms such outcomes often represent a globally aligned flock or congregation of aligned clusters. The classical result of Cucker and Smale states that alignment is unconditional in flocks that have global communication with non-integrable radial tails. Proving a similar statement for purely local interactions presents a major mathematical challenge. In this talk we will overview three programs of research directed on understanding the emergent phenomena: hydrodynamic topological interactions, kinetic approach based on hypocoercivity, and spectral energy method. We present a novel framework based on the concept of environmental averaging which allows us to obtain coercivity estimates leading to new flocking results.


Sameer Iyer, UC Davis

Monday, February 13, 2023, KAP 414, 3:30 PM – 4:30 PM

Reversal in the Stationary Prandtl Equations

Abstract: We discuss a recent result in which we investigate reversal and recirculation for the stationary Prandtl equations. Reversal describes the solution after the Goldstein singularity, and is characterized by spatio-temporal regions in which $u > 0$ and $u < 0$. The classical point of view of regarding the Prandtl equations as an evolution $x$ completely breaks down. Instead, we view the problem as a quasilinear, mixed-type, free-boundary problem. Joint work with Nader Masmoudi.

We discuss a recent result in which we investigate reversal and recirculation for the stationary Prandtl equations. Reversal describes the solution after the Goldstein singularity, and is characterized by spatio-temporal regions in which $u > 0$ and $u < 0$. The classical point of view of regarding the Prandtl equations as an evolution $x$ completely breaks down. Instead, we view the problem as a quasilinear, mixed-type, free-boundary problem. Joint work with Nader Masmoudi.


Stan Palasek, UCLA

Monday, February 27, 2023, KAP 414, 3:30 PM – 4:30 PM

Non-uniqueness and convex integration for the forced Euler equations

Abstract: This talk is concerned with the uniqueness and flexibility of C^α weak solutions of the incompressible Euler equations. A great deal of recent effort has led to the conclusion that the space of 3D Euler flows is flexible when α is below 1/3, the well-known Onsager regularity. We introduce an alternating convex integration framework for the forced Euler equations that is effective above the Onsager regularity, for all α<1/2. This leads to a new non-uniqueness theorem for any initial data. This work is joint with Aynur Bulut and Manh Khang Huynh.


Wojciech Ozanski, Florida State University

Monday, March 6, 2023, KAP 414, 3:30 PM – 4:30 PM

Instantaneous gap loss of Sobolev regularity of solutions to the 2D incompressible Euler equations

Abstract: We will discuss classical well-posedness results of the incompressible Euler equations, and recent results concerning ill-posedness. We will then discuss, in the 2D case, the first result of instantaneous gap loss of Sobolev regularity. Namely we will describe a construction of initial vorticity in the Sobolev space H^β, β ∈ (0,1) which gives rise to a unique global-in-time solution of the 2D Euler equations that instantaneously leaves H^β’ for every β’ > (2 – β)β/(2 – β^2)$. This is joint work with Diego Córdoba and Luis Martínez-Zoroa.


Mimi Dai, University of Illinois at Chicago

Monday, March 20, 2023, KAP 414, 3:30 PM – 4:30 PM

Singularity formation for models of fluids

Abstract: Finite time singularity formation for fluid equations will be discussed. Built on extensive study of approximating models, breakthroughs on this topic have emerged recently for Euler equation. Inspired by the progress for pure fluids, we attempt to understand this challenging issue for magnetohydrodynamics (MHD). Finite time singularity scenarios are discovered for some reduced models of MHD. The investigation also reveals connections of MHD with Euler equation and surface quasi-geostrophic equation.


Alexey Cheskidov, University of Illinois at Chicago

Monday, March 27, 2023, KAP 414, 3:30 PM – 4:30 PM

Turbulent solutions of fluid equations

Abstract: In the past couple of decades, mathematical fluid dynamics has been highlighted by numerous constructions of solutions to fluid equations that exhibit pathological or wild behavior. These include the loss of the energy balance, non-uniqueness, singularity formation, and dissipation anomaly. Interesting from the mathematical point of view, providing counterexamples to various well-posedness results in supercritical spaces, such constructions are becoming more and more relevant from the physical point of view as well. Indeed, a fundamental physical property of turbulent flows is the existence of the energy cascade. Conjectured by Kolmogorov, it has been observed both experimentally and numerically, but had been difficult to produce analytically. In this talk I will overview new developments in discovering not only pathological mathematically, but also physically realistic solutions of fluid equations.


Svetlana Jitomirskaya, UC Irvine and Georgia Tech

Distinguished Lecture from UC Irvine and Georgia Tech

Monday, April 24, 2023, KAP 414, 3:30 PM – 4:30 PM

Fractal properties of the Hofstadter butterfly, eigenvalues of the almost Mathieu operator, and topological phase transitions

Abstract: Harper’s operator – the 2D discrete magnetic Laplacian – is the model behind the Hofstadter’s butterfly and Thouless theory of the Quantum Hall Effect. It reduces to the critical almost Mathieu family, indexed by the phase. We will present a complete proof of singular continuous spectrum for the critical family, for all phases, finishing a program with a long history. The proof is based on a simple Fourier analysis and a new Aubry duality-type transform. We will also explain how these ideas provide for a very simple proof of zero measure of the spectrum of Harper’s operator, a problem previously solved by sophisticated dynamical systems techniques, as well as progress on some other outstanding conjectures.

Colloquia for the Fall 2022 Semester

Nathan Glatt-Holtz, Tulane University

(CAMS Colloquium Joint with Probability and Statistics Seminar)

Monday, Monday, September 19, 2022, KAP 414, 3:30 PM – 4:30 PM

A unified framework for Metropolis-Hastings Type Monte Carlo methods (Video)

Abstract: The efficient generation of random samples is a central task within many of the quantitative sciences. The workhorse of Bayesian statistics and statistical physics, Markov chain Monte Carlo (MCMC) comprises a large class of algorithms for sampling from arbitrarily complex and/or high-dimensional probability distributions. The Metropolis-Hastings method (MH) stands as the seminal MCMC algorithm, and its basic operation underlies most of the MCMC techniques developed to this day.
We provide an all-encompassing, measure theoretic mathematical formalism that describes essentially any Metropolis-Hastings algorithm using three ingredients: a random proposal, an involution on an extended phase space and an accept-reject mechanism. This unified framework illuminates under-appreciated relationships between a variety of known algorithms while yielding a means for deriving new methods.
As an immediate application we identify several novel algorithms including a multiproposal version of the popular preconditioned Crank- Nicolson (pCN) sampler suitable for infinite-dimensional target measures which are absolutely continuous with respect to a Gaussian base measure. We also develop a new class of ‘extended phase space’ methods, based on Hamiltonian mechanics. These methods provide a versatile approach to bypass expensive gradient computations through skillful reduced order modeling and/or data driven approaches. A selection of case studies will be presented that use our multiproposal pCN algorithm (mpCN) to resolve a selection of problems in Bayesian statistical inversion for partial differential equations motivated by fluid flow measurement.

This is joint work with Andrew J. Holbrook (UCLA), Justin Krometis (Virginia Tech) and Cecilia Mondaini (Drexel).

Career Advice Panel

Monday, September 26, 2022, KAP 414, 3:30 PM – 4:30 PM

Panel discussion: “Applying for Grants and Jobs: Information and Advice”

Panelists: Evgeni Dimitrov, Aaron Lauda and Trevor Leslie
Moderator: Susan Friedlander

All postdocs and graduate students are strongly encouraged to come and ask questions about applying for grants from organizations such as the NSF, the NSA and the Simons Foundation. We will also discuss applying for academic positions.

Natasa Pavlovic, University of Texas, Austin

Monday, October 10, 2022, KAP 414, 3:30 PM – 4:30 PM

A tale of two generalizations of Boltzmann equation (Video)

Abstract: In the first part of the talk we shall discuss dynamics of systems of particles that allow interactions beyond binary, and their behavior as the number of particles goes to infinity. This part of the talk is based on the joint work with Ioakeim Ampatzoglou on a derivation of a binary-ternary Boltzmann equation describing the kinetic properties of a dense hard spheres gas, where particles undergo either binary or ternary instantaneous interactions, while preserving momentum and energy. An important challenge we overcome in deriving this equation is related to providing a mathematical framework that allows us to detect both binary and ternary interactions. In the second part of the talk we will discuss a rigorous derivation of a Boltzmann equation for mixtures of gases, which is a joint work with Ioakeim Ampatzoglou and Joseph Miller. We prove that the microscopic dynamics of two gases with different masses and diameters is well defined, and introduce the concept of a two parameter BBGKY hierarchy to handle the non-symmetric interaction of these gases.

John Schotland, Yale University

Monday, October 31, 2022, KAP 414, 3:30 PM – 4:30 PM

Nonlocal PDEs and Quantum Optics

Abstract: Quantum optics is the quantum theory of the interaction of light and matter. In this talk, I will describe recent work on a real-space formulation of quantum electrodynamics for single photons interacting with two-level atoms. It is shown that the probability amplitude of a photon obeys a nonlocal partial differential equation. Applications to quantum optics in random media will be described, where there is a close relation to kinetic equations for PDEs with random coefficients.


Weiwei Hu, University of Georgia

Monday, November 7, 2022, KAP 414, 3:30 PM – 4:30 PM

Optimal control for suppression of singularity in chemotaxis (Video)

Abstract: In this talk, we discuss the problem of optimal control design for suppression of singularity via flow advection in chemotaxis modeled by the Patlak-Keller-Segel (PKS) equations. It is well-known that for the system without advection, singularity of the solution may develop at finite time. Specifically, if the initial condition is above certain critical threshold, the solution may blow up at finite time by concentrating positive mass at a single point. In this talk, we will first address the global regularity and stability of the PKS system in the presence of flow advection in a bounded domain, by using a semigroup approach. Then we focus on the design of an optimal flow field for suppressing such singularities. Rigorous theoretical framework and numerical experiments will be presented to demonstrate the ideas.


Adam Larios, University of Nebraska

Monday, November 14, 2022, KAP 414, 3:30 PM – 4:30 PM

A Song of Water and Fire: The Navier-Stokes and Kuramoto-Sivashinsky Equations (Video)

Abstract: The flame equation, also known as the Kuramoto-Sivashinsky equation (KSE) is a highly chaotic dynamical system that arises in flame fronts, plasmas, crystal growth, and many other phenomena. Due to its lack of a maximum principle, the KSE is often studied as an analogue to the 3D Navier-Stokes equations (NSE) of fluids. We will discuss some of the relationships between these equations of fire and water. Much progress has been made on the 1D KSE since roughly 1984, but for the 2D KSE, even global well-posedness remains a major open question. In analogy with regularizations of the 3D NSE, we present modifications of the 2D KSE which allow for global well-posedness, while still retaining many important features of the 2D KSE. However, as has been demonstrated recently by Kostianko, Titi, and Zelik, standard regularizations, which work well for Navier-Stokes, destabilize the system when applied to even the 1D KSE. Thus, we present entirely new types of modifications for the 2D KSE. This talk will describe key ideas of the analysis, and also show many colorful movies of solutions.


Thomas Hou, Caltech

Monday, November 21, 2022, KAP 414, 3:30 PM – 4:30 PM

Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data

Abstract: Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this talk, we will present a new exciting result with Dr. Jiajie Chen in which we prove finite time blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary. There are several essential difficulties in establishing such blowup results. We overcome these difficulties by first constructing an approximate self-similar blowup profile using the dynamic rescaling formulation. To establish the stability of the approximate blowup profile, we decompose the linearized operator into a leading order operator plus a finite rank perturbation operator. We use sharp functional inequalities and optimal transport to establish the stability of the leading order operator. To estimate the finite rank operator, we use energy estimates and space-time numerical solutions with rigorous error control. This enables us to establish nonlinear stability of the approximate self-similar profile and prove stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data. This provides the first rigorous justification of the Hou-Luo blowup scenario.

Marco Sammartino, University of Palermo

Monday, December 5, 2022,


Colloquia for the Spring 2022 Semester

Robert Ghrist, University of Pennsylvania

Monday, January 31, 2022, (Zoom), 3:30 PM – 4:30 PM

Opinion Dynamics on Sheaves (Video)

Abstract: There is a long history of networked dynamical systems that models the spread of opinions over social networks, with the graph Laplacian playing a lead role.  One of the difficulties in modeling opinion dynamics is the presence of polarization: not everyone comes to consensus. This talk will describe joint work with Jakob Hansen introducing a new model for opinion dynamics using sheaves of vector spaces over social networks. The graph Laplacian is enriched to a Hodge Laplacian, and the resulting dynamics on discourse sheaves can lead to some very interesting and perhaps more realistic outcomes. Additional work with Hans Riess extending the theory will be hinted at. The talk requires no background in sheaf theory and is suitable for graduate students in the mathematical sciences.


Alexis Vasseur, University of Texas Austin

Monday, February 7, 2022, (Zoom), 3:30 PM – 4:30 PM

Boundary vorticity estimate for the Navier-Stokes equation and  control of layer separation in the inviscid limit (Video)

Abstract: Consider the steady solution to the incompressible Euler equation $Ae_1$ in the periodic tunnel $\Omega=[0,1]\times \mathbb T^2$. Consider now the family of solutions $U_\nu$ to the associated Navier-Stokes equation with no-slip condition on the flat boundaries, for small viscosities $\nu=1/ Re$, and initial values close in $L^2$ to $Ae_1$. Under a conditional assumption on the energy dissipation close to the boundary,  Kato showed in 1984 that  $U_\nu$ converges to $Ae_1$ when the viscosity converges to 0 and the initial value converge to $A e_1$. It is still unknown whether this inviscid  is unconditionally true. Actually, the convex integration method predicts the possibility of a layer separation. It produces solutions to the Euler equation with initial values $Ae_1 $, but   with layer separation energy at time T up to:

$$\|U(T)-Ae_1\|^2_{L^2}\equiv A^3T.$$

In this work we prove that at the double limit for the  inviscid asymptotic $\bar{U}$, where both the Reynolds number $Re$ converges to infinity and the initial value $U_{\nu}$ converges to $Ae_1$ in $L^2$, the energy of layer separation cannot be more than:

$$\| \bar{U}(T)-Ae_1\|^2_{L^2}\lesssim A^3T.$$

Especially, it shows that, even if if the limit is not unique, the shear flow pattern is observable up to time $1/A$. This provides a notion of stability despite the possible non-uniqueness of the limit predicted by the convex integration theory. The result relies on a new boundary vorticity estimate for the Navier-Stokes equation. This new estimate, inspired by previous work on higher regularity estimates for Navier-Stokes, provides a non-linear control scalable through the inviscid limit.


Jacob Bedrossian, University of Maryland

Monday, February 14, 2022, (Zoom), 3:30 PM – 4:30 PM

Positive Lyapunov exponents for 2d Galerkin-Navier-Stokes with stochastic forcing (Video)

Abstract: In this talk we discuss our recently introduced methods for obtaining strictly positive lower bounds on the top Lyapunov exponent of high-dimensional, stochastic differential equations such as the weakly-damped Lorenz-96 (L96) model or Galerkin truncations of the 2d Navier-Stokes equations (joint with Alex Blumenthal and Sam Punshon-Smith). This hallmark of chaos has long been observed in these models, however, no mathematical proof had previously been made for any type of deterministic or stochastic forcing. We propose a new method which has the ability to obtain quantitative estimates on the top Lyapunov exponents of high-dimensional, weakly dissipative SDEs. It combines (A) a new identity connecting the Lyapunov exponents to a Fisher information of the Markov process tracking tangent directions; and (B) an quantitative hypoelliptic estimate to show that this (degenerate) Fisher information is an upper bound on some fractional regularity. For L96 and GNSE, we then further reduce the lower bound of the top Lyapunov exponent to proving a certain Lie algebraic condition on the nonlinearity. I will also discuss the recent work of Sam Punshon-Smith and I on verifying this condition for the 2d Galerkin-Navier-Stokes equations in a rectangular, periodic box of any aspect ratio using some special structure of matrix Lie algebras and ideas from computational algebraic geometry (for L96 it can be verified “by hand”).

Tarek Elgindi, Duke University

Monday, February 28, 2022, (Zoom), 3:30 PM – 4:30 PM

Singularity formation in incompressible fluids (Video)

Abstract: I will give a review of recent progress on finite-time singularity formation in incompressible fluid models. In the prorcess, I will discuss different joint works with Tej Ghoul, In-Jee Jeong, Nader Masmoudi, and Federico Pasqualotto.

Hyung Ju Hwang, POSTECH, Korea

Monday, March 7, 2022, (Zoom), 3:30 PM – 4:30 PM

Deep Neural Network Solutions of PDEs and Applications to COVID-19 spread model (Video)

Abstract:  Mathematics is closely related to the theory and algorithms of AI and machine learning. In this talk, we investigate how deep neural networks (DNNs) can be used in the forward-inverse problems of PDEs. We introduce a loss function that guides neural networks to find solutions of PDEs more efficiently. Next, we look into real-world implications of a rapidly-responsive COVID-19 spread model via deep learning. The methodology could also be employed for a short-term prediction of COVID-19, which could help the government prepare for a new outbreak.

Eitan Tadmor, University of Maryland

Monday, March 21, 2022, (Zoom), 3:30 PM – 4:30 PM

Hierarchical decomposition of images and the problem of Bourgain-Brezis (Video)

Abstract: Edges are noticeable features in images which can be extracted from noisy data using different variational models.

The analysis of such models leads to the question of expressing general L^2-data, f, as the divergence of uniformly bounded vector fields, div(U). We present a multi-scale approach to construct uniformly bounded solutions of div(U)=f for general f’s in the critical regularity space L^d(T^d). The study of this equation and related problems was motivated by results of Bourgain & Brezis.

The intriguing critical aspect here is that although the problems are linear, construction of their solution is not. Our constructive solution for such problems is a special case of a rather general framework for solving linear equations, formulated as inverse problems in critical regularity spaces. The solutions are realized in terms of nonlinear hierarchical decomposition, U=∑_ju_j, which we introduced earlier in the context of image processing, and yield a multi-scale decomposition of “objects” U.

Mihaela Ignatova, Temple University

Monday, March 28, 2022, (Zoom), 3:30 PM – 4:30 PM

Electroconvection in Fluids (Video)

Abstract: We present two electroconvection models describing the interaction between a surface charge density and a fluid in a two-dimensional situation. We compare these models with the surface quasi-geostrophic equation in bounded domains and recall some recent results. We discuss the global existence and long time behavior of solutions of the electroconvection models. For the first model, the global existence results can be obtained for bounded domains and for the torus. In the latter case, in joint work with graduate student E. Abdo, we proved that the long time asymptotic state of the system is finite dimensional, if body forces are applied to the fluid, and a singleton solution in the absence of fluid body forces. For the more challenging second model, corresponding to electroconvection through porous media, we proved global existence for subcritical and for small data cases.

Diego Cordoba, IAS and Madrid

Monday, April 4, 2022, (Zoom), 3:30 PM – 4:30 PM

Instant blow-up for the Surface Quasi-geostrophic equation (Video)

Abstract: The Surface Quasi-geostrophic equation (SQG) arises from a geophysical context and its mathematical interest is motivated by the number of traits it shares with the 3D incompressible Euler equations. Local existence of solutions for SQG has been established in $C^{k,\gamma}$ ($k\geq 1$ and $0<\gamma<1$) and in Sobolev spaces $H^s$ (s>2). In this talk we will introduce a method to construct solutions in $\mathds{R}^2$ with finite energy of the surface quasi-geostrophic equations (SQG) that initially are in $C^k$ ($k\geq 2$) but that are not in $C^{k}$ for $t>0$. We prove a similar result also for $H^{s}$ in the range $s\in(\frac32,2)$.

Jonathan Mattingly, Duke University

Monday, April 11, 2022, (Zoom), 3:30 PM – 4:30 PM

A random splitting model for the 2D Euler and Navier Stokes equations (Video)

Abstract: I will introduce a newly developed model of fluid motion which introduces randomness via a random splitting mechanism. After explaining the model in the galerkin approximation setting I will give some basic properties of the system around its long time behavior.

Paolo Galdi, University of Pittsburgh

Monday, April 18, 2022, (Zoom), 3:30 PM – 4:30 PM

Navier-Stokes Equations around a Rigid Body: Three Remarkable Open Problems (Video)

Abstract: The motion of a (finite) rigid body, $\mathfrak{B}$, in a viscous liquid is a fundamental and widely investigated problem of mathematical fluid mechanics, in both cases when the motion of $\mathfrak{B}$ is prescribed or it becomes part of the problem. However, in spite of the many outstanding contributions tracing back to the works of Leray, Ladyzhenskaya and Finn, there is still a plethora of fundamental questions that remain still unanswered and call for the attention of the interested mathematician. Objective of this talk is to present and discuss three of them, which I believe to be among the most remarkable and intriguing ones.

László Székelyhidi, IAS and Leipzig

Monday, April 25, 2022, (Zoom), 3:30 PM – 4:30 PM

Magnetohydrodynamic Turbulence: weak solutions and conserved quantities (Video)

Abstract: The ideal MHD system in three space dimensions consists of the incompressible Euler equations coupled to the Faraday system via Ohm’s law. This system has a wealth of interesting structure, including three conserved quantities: the total energy, cross-helicity and magnetic helicity. Whilst the former two are analogous (and analytically comparable) to the total kinetic energy for the Euler system, magnetic helicity is known to be more robust and of a different nature. In particular, when studying weak solutions, Onsager-type conditions for all three quantities are known, and are basically on the same level of 1/3-differentiability as the kinetic energy in the ideal hydrodynamic case for the former two. In contrast, magnetic helicity does not require any differentiability, only L^3 integrability. From the physical point of view this difference lies at the heart of the Taylor-Woltjer relaxation theory. From the mathematical point of view it turns out to be closely related to the div-curl structure of the Faraday system. In the talk we present and compare some recent constructions of weak solutions and, along the way highlight some of the hidden structures in the ideal MHD system. Joint work with Daniel Faraco and Sauli Lindberg.

Colloquia for the Fall 2021 Semester

Trevor Leslie, USC

Monday, August 30, 2021, KAP 414, 3:30 PM – 4:30 PM

Sticky Particle Methods for the 1D Euler Alignment System

Abstract: The Euler Alignment system is a hydrodynamic version of the celebrated Cucker–Smale ODE’s of collective behavior.  It can have a hyperbolic or parabolic character, depending on the specified nonlocal interaction protocol; this talk concerns the hyperbolic case in 1D.  It is well-established that solutions may lose regularity in finite time, but it has been unknown until recently how to continue to evolve the dynamics after a blowup.  After brief orientation on the special structure of these equations, I will describe a recent joint work with Changhui Tan (University of South Carolina), where we developed a theory of weak solutions.  Inspired by Brenier and Grenier’s work on the pressureless Euler equations, we show that the dynamics of our system are captured by a nonlocal scalar balance law.  We generate the unique entropy solution of a discretization of this balance law by introducing the ‘sticky particle Cucker–Smale’ system to track the shock locations.  Our approximation scheme for the density converges in the Wasserstein metric; it does so with a quantifiable rate as long as the initial velocity is at least Hölder continuous.

Thomas Hou, Caltech

Monday, September 13, 2021, (Zoom), 3:30 PM – 4:30 PM

Potential singularity of 3D incompressible Euler equations and nearly singular solutions of 3D Navier-Stokes equations

Abstract: Whether the 3D incompressible Euler and Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In an effort to provide a rigorous proof of the potential Euler singularity revealed by Luo-Hou’s computation, we develop a novel method of analysis and prove that the original De Gregorio model and the Hou-Lou model develop a finite time singularity from smooth initial data. Using this framework and some techniques from Elgindi’s recent work on the Euler singularity, we prove the finite time blowup of the 2D Boussinesq and 3D Euler equations with $C^{1,\alpha}$ initial velocity and boundary. Further, we present some new numerical evidence that the 3D incompressible Euler equations with smooth initial data develop a potential finite time singularity at the origin, which is quite different from the Luo-Hou scenario.  Our study also shows that the 3D Navier-Stokes equations develop nearly singular solutions with maximum vorticity increasing by a factor of $10^7$. However, the viscous effect eventually dominates vortex stretching and the 3D Navier-Stokes equations narrowly escape finite time blowup.  Finally, we present strong numerical evidence that the 3D Navier-Stokes equations with slowly decaying viscosity develop a finite time singularity.

Sijue Wu, University of Michigan

Monday, September 20, 2021, (Zoom), 3:30 PM – 4:30 PM

The quartic integrability and long time existence of steep water waves in 2d

Abstract: It is known since the work of Dyachenko & Zakharov in 1994 that for weakly nonlinear 2d infinite depth water waves,  there are no 3-wave interactions and all of the 4-wave interaction coefficients vanish on the non-trivial resonant manifold. In this talk I will present a recent result that proves this partial integrability from a different angle.  We construct a sequence of energy functionals $E_j(t)$,  directly in the physical space,  which are explicit in the Riemann mapping variable and involve material derivatives of order $j$ of the solutions for the 2d water wave equation,  so that  $\frac d{dt}  E_j(t)$ is quintic or higher order.  We show that if some scaling invariant norm, and a norm involving one  spacial derivative above the scaling of the initial data are of size no more than $\epsilon$, then the lifespan of the solution for the 2d water wave equation is at least of order $O(\epsilon^{-3})$, and the solution remains as regular as the initial data during this time. If only the scaling invariant norm of the data is of size $\epsilon$, then the lifespan of the solution is at least of order $O(\epsilon^{-5/2})$. Our long time existence results do not impose size restrictions on the slope bof the initial interface and the magnitude of the initial velocity, they allow the interface to have arbitrary large steepnesses  and initial velocities to have arbitrary large magnitudes.

Marta Lewicka, University of Pittsburgh

Monday, September 27, 2021, (Zoom), 3:30 PM – 4:30 PM

Geometry, analysis and morphogenesis: problems and prospects

Abstract: The remarkable range of biological forms in and around us, raise a number of questions: how might these shapes be predicted, and how can they eventually be designed and controlled for function? We review our current understanding of this problem, that brings together analysis, geometry and mechanics in the description of the morphogenesis of low-dimensional objects. Starting from the view that shape is the consequence of metric frustration in an ambient space, we revisit known rigorous results on curvature-driven patterning of thin elastic films, especially the asymptotic behaviors of the solutions as the thickness becomes vanishingly small and the local curvature can become large. Along the way, we discus open problems that include those in mathematical modeling, analysis and applications in science and engineering.

Career Advice Panel

Monday, October 4, 2021, KAP 414, 3:30 PM – 4:30 PM

Panel discussion: “Applying for Grants and Jobs: Information and Advice”

Panelists: Anne Dranowski, Aaron Lauda and Cris Negron

Moderator: Susan Friedlander

All postdocs and graduate students are strongly encouraged to come and ask questions about applying for grants from organizations such as the NSF, the NSA and the Simons Foundation. We will also discuss applying for academic positions.


Lin Lin, UC Berkeley

Monday, October 11, 2021, (Zoom), 3:30 PM – 4:30 PM

Quantum numerical linear algebra (Video)

Abstract: The two “quantum supremacy” experiments (by Google in 2019 and by USTC in 2020, respectively) have brought quantum computation to the public’s attention.  In this talk, I will discuss how to use a quantum computer to solve linear algebra problems. I will start with a toy linear system Ax=b, where A is merely a 2 x 2 matrix. I will then propose a quantum LINPACK benchmark, and quantum Hamiltonian simulation benchmark for early fault-tolerant quantum computers. I will also discuss some recent works of solving general quantum linear systems with near-optimal complexity, and quantum preconditioning techniques  for ill-conditioned linear systems

Vincent Martinez, CUNY

Monday, October 18, 2021, (Zoom), 3:30 PM – 4:30 PM

On well-posedness at critical regularity for a family of active scalar equations arising in hydrodynamics (Video)

Abstract: This talk discusses a family of active scalar transport equations characterized by increasingly singular constitutive laws. This family includes the 2D Euler and surface quasi-geostrophic (SQG) equations as members, and extrapolates beyond them. Although local well-posedness of the initial value problem in sufficiently regular settings are classical results for both the Euler and SQG equations, ill-posedness at critical regularity has only recently been established. For this talk, we consider various regularizations of this family in its most singular range that recover well-posedness results at the threshold regularity level, in spite of the apparent strongly quasilinear structure of the equations in this regime. This is joint work with M.S. Jolly and A. Kumar.

Sung-Jin Oh, UC Berkeley

Monday, October 25, 2021, (Zoom), 3:30 PM – 4:30 PM

A tale of two tails 

Abstract: In this talk, I will introduce a general method for understanding the late-time tail for solutions to wave equations on asymptotically flat spacetimes with odd spatial dimensions. A particular consequence of the method is a re-proof of Price’s law-type results, which concern the sharp decay rate of the late-time tails on stationary spacetimes. Moreover, the method also applies to dynamical spacetimes. In this case, I will explain how the late-time tails are in general different(!) from the stationary case in the presence of dynamical perturbations of spacetime. This is joint work with Jonathan Luk (Stanford).

Theodore Drivas, SUNY Stonybrook

Monday, November 15, 2021, (Zoom), 3:30 PM – 4:30 PM

Simultaneous Development of Shocks and Weak Discontinuities from Smooth Data (Video)

Abstract: I will discuss some recent work on shock formation and propagation of singularities for compressible Euler. The aim is to describe precisely the structure of an entropy producing shock in its early phase (starting from smooth initial conditions), and also describe with some detail a collection of weaker singularities that are born with it. These singularities are in the derivatives of the solution fields and travel along different characteristics than the shock.  This reports on joint work with T. Buckmaster, S. Shkoller, and V. Vicol.

Nathan Glatt Holtz, Tulane

Monday, November 22, 2021, (Zoom), 3:30 PM – 4:30 PM

Some Recent Developments in the Bayesian Approach to PDE Inverse Problems: Statistical Sampling and Consistency (Video)

Abstract: This talk concerns some of our recent work on a statistical methodology to quantify an unknown, infinite dimensional, parameter $\mathbf{u}$ specifying a class of partial differential equations whose solutions we observe in a limited fashion and which is subject to measurement error. As a paradigmatic model problem we consider the estimation of a divergence free flow field $\mathbf{u}$ from the partial and noisy observation of a scalar $\theta$ which is advected by $\mathbf{u}$ and which diffuses  passively in the fluid medium.  Thus we suppose that $\theta$ solves
\partial_t \theta + \mathbf{u} \cdot \nabla \theta = \kappa \Delta \theta, \quad  \theta(0) = \theta_0,
up to the unknown $\mathbf{u}$ and where $\kappa > 0$ is the diffusivity parameter. For this problem \eqref{eq:AD}, as in a variety of other PDE inverse problems of interest, our task is thus to recover $\mathbf{u}$ from a data set  $Y \in \mathbb{R}^n$ which obeys a statistical measurement model of the form
Y = \mathcal{G}(\mathbf{u}) + \eta.
Here $\mathcal{G}$ is nonlinear and defined as a composition of a parameter to PDE solution map $\mathcal{S}$ and an observation operator $\mathcal{O}$. The term $\eta$ represents an additive observational error.

A Bayesian approach  pioneered recently by Andrew Stuart and others allows for the effective treatment of infinite-dimensional unknowns $\mathbf{u}$ via a suitable regularization at small scales through the consideration of certain classes of Gaussian priors. In this framing, by positing such prior distributions $\mu_0$ on $\mathbf{u}$, and assuming $\eta$ has the pdf $p_\eta$ we obtain a posterior distribution of the form
\mu(d \mathbf{u}) \propto p_\eta(Y – \mathcal{G}(\mathbf{u})) \mu_0(d\mathbf{u})
This measure $\mu$ therefore provides a comprehensive model of the uncertainty in our unknown $\mathbf{u}$.

In this talk we will survey some recent results which analyzes such PDE constrained posterior measures $\mu$ both analytically and numerically. We will discuss the issue of posterior contraction (consistency) in the large observation limit for the inverse problem defined by \eqref{eq:AD}.  We also describe some infinite dimensional Markov Chain Monte Carlo (MCMC) algorithms which we have developed, refined and rigorously analyzed to effectively sample from $\mu$.

This is joint work with Jeff Borggaard (Virginia Tech), Justin Krometis (Virginia Tech) and Cecilia Mondaini (Drexel).

Colloquia for the Spring 2021 Semester

Virtual USC Film Screening: SECRETS OF THE SURFACE

Friday, March 12, 2021, (Zoom), 3:30 PM

The Mathematical Vision of Maryam Mirzakhani
A documentary film by GEORGE CSICSERY

The film examines the life and mathematical work of Maryam Mirzakhani, an Iranian immigrant to the United States who became a superstar in her field. In 2014, she was both the first woman and the first Iranian to be honored by mathematics’ highest prize, the Fields Medal.

Co-Sponsored by
Center for Applied Mathematical Sciences
Women in Science and Engineering
USC Math Department, Charlotte’s Web and Probability & Statistics Seminar,
Women in Physics

Colloquia for the Fall 2020 Semester

Virtual Career Advice Panel

Wednesday, September 2, 2020, (Zoom), 3:30 PM – 4:30 PM

Panel discussion: “Applying for Grants and Jobs: Information and Advice”

Panelists: Aaron Lauda and Harold Williams

Moderator: Susan Friedlander

Will discuss some basic information concerning applying for jobs, fellowships, grants etc and urge you to ask questions.


Katherine Bouman, Caltech

(Joint with Probability & Statistics Seminar)

Monday, September 11, 2020, (Zoom), 3:30 PM – 4:30 PM

Capturing the First Image of a Black Hole & Designing the Future of Black Hole Imaging

Abstract: This talk will present the methods and procedures used to produce the first image of a black hole from the Event Horizon Telescope, as well as discuss future developments for black hole imaging. It had been theorized for decades that a black hole would leave a “shadow” on a background of hot gas. Taking a picture of this black hole shadow would help to address a number of important scientific questions, both on the nature of black holes and the validity of general relativity. Unfortunately, due to its small size, traditional imaging approaches require an Earth-sized radio telescope. In this talk, I discuss techniques the Event Horizon Telescope Collaboration has developed to photograph a black hole using the Event Horizon Telescope, a network of telescopes scattered across the globe. Imaging a black hole’s structure with this computational telescope required us to reconstruct images from sparse measurements, heavily corrupted by atmospheric error. The talk will also discuss future developments, including new imaging techniques and how we are developing machine learning methods to help design future telescope arrays.

Colloquia for the Spring 2020 Semester

Charles Collot, Courant Institute

Monday, January 27, 2020, KAP 414, 3:30 PM – 4:30 PM

Singular solutions to evolution nonlinear PDEs: key notions and recent results for certain semilinear/quasilinear equations

Abstract: Some partial differential equations of evolution possess singular solutions. At some time T > 0 something happens that prevents the solution to be continued smoothly beyond that point. The natural questions regarding this “blow-up” are that of the existence, the description, and the classification. This talk will be mostly concerned by the problem of the construction and description of particular solutions, for the semilinear heat equation and the Prandtl’s system. First, certain key notions will be detailed on simple examples (Ricatti and Burgers): self-similarity, renormalisation, and modulation. Then, the construction and stability of type I self-similar solutions of the supercritical heat equation will be explained as a canonical example. This will allow us to consider next some more recent results on this problem for compressible and incompressible fluids, and some degenerate anisotropic blow-ups for the semilinear heat equation. We will conclude with the problem of the possible classification of the dynamics, with the example of the inviscid Prandtl’s system. The main two works that will be presented are collaborations with P. Raphaël and J. Szeftel, and with T.-E. Ghoul. and N. Masmoudi.


Oleg Lazarev, Columbia University

(NOTE: Special Colloquium)

Wednesday, February 12, 2020, KAP 414, 3:30 PM – 4:30 PM

Flexibility and rigidity in symplectic topology

Abstract: While symplectic manifolds have no local invariants, they do have powerful global invariants coming from Gromov’s theory of pseudoholomorphic curves. These curves have been packaged into rich algebraic structures like the Fukaya category, which plays a central role in homological mirror symmetry. On the other hand, a recent development is the discovery of “flexible” symplectic structures, which are essentially determined by the underlying smooth topology and have trivial pseudoholomorphic curve invariants. In this talk, I will explain how flexibility provides insights into the rigid world of Fukaya categories and the general structure of symplectic manifolds. In particular, I will discuss an h-cobordism type theorem for symplectic Morse functions and relate this flexibility to a classical result of Thomason in algebraic K-theory in the context of Fukaya categories.


Qingtang Su, USC

Monday, February 24, 2020, KAP 414, 3:30 PM – 4:30 PM

Long time behavior of rotational water waves

Abstract: The study of the water waves has been one of the central problems in applied mathematics for centuries, yet the rigorous mathematical analysis for the full water waves was quite recently. Indeed, the local wellposedness for large initial data in Sobolev spaces was open until the breakthrough works of S. Wu in the late 1990s, and the global wellposedness for small localized initial data were proved in the last decade. The aforementioned global wellposedness result assumes irrotionality. However, most fluids feature the vorticity.  Although the local wellposedness of rotational water waves is well understood, the study of its long time behavior is still largely open.  In this talk I will focus on my work on the long time dynamics of 2d water waves with concentrated vorticity. First, I will survey the results on the long time behavior of irrotational water waves,  in particular, I’ll discuss the idea of proving local and long time existence. Then I’ll formulate the rotational water waves as the coupling of the evolution of the free surface and the transport of the vorticity. In particular, if the vorticity is given by point vortices, then the problem is reduced to the evolution of the free boundary and the motion of the point vortices. In general, we don’t expect  such a system to remain smooth and small for a long time. However, for some special cases, such as for the case with a pair of couter-rotating point vortices  traveling away from the free boundary, we are able to obtain a long time existence and describe its long time behavior.


Hao Jia, University of Minnesota

Monday, March 9, 2020, KAP 414, 3:30 PM – 4:30 PM

Nonlinear asymptotic stability in two dimensional incompressible Euler equations

Abstract: Stability of coherent structures in two dimensional Euler equations, such as shear flows and vortices, is an important problem and a classical topic in fluid dynamics. Full nonlinear asymptotic stability results are difficult to obtain since the rate of stabilization is slow, the convergence of vorticity occurs only in weak, distributional sense, and the nonlinearity is strong. In a breakthrough work, Bedrossian and Masmoudi proved the first nonlinear asymptotic stability result, near the Couette flow (linear shear). In this talk, we will explain the physical relevance of the problem, survey recent progresses and in particular discuss our results on the nonlinear asymptotic stability of general monotonic shear flows. If time permits, further open problems in the area will also be mentioned. This is joint work with Alexandru Ionescu.


Vincent Martinez, CUNY


Monday, March 30, 2020, KAP 414, 3:30 PM – 4:30 PM

Title: TBA
Abstract: TBA


USC Film Screening


Monday, April 3, 2020, Irani Hall 101, 3:30 PM – 4:30 PM

SECRETS OF THE SURFACE: The Mathematical Vision of Maryam Mirzakhani

A documentary film by


The film examines the life and mathematical work of Maryam Mirzakhani, an Iranian immigrant to the United States who became a superstar in her field. In 2014, she was both the first woman and the first Iranian to be honored by mathematics’ highest prize, the Fields Medal.

Co-Sponsored by
Center for Applied Mathematical Sciences
Women in Science and Engineering
Charlotte’s Web
Women in Physics

CAMS Director:
Susan Friedlander


Shouhong Wang, Indiana University


Monday, April 6, 2020, KAP 414, 3:30 PM – 4:30 PM

Title: TBA
Abstract: TBA


Eleanor Rieffel, NASA


(Joint with the Department Colloquium)

Wednesday, April 8, 2020, KAP 414, 3:30 PM – 4:30 PM

Title: TBA
Abstract: TBA


Steve Shkoller, UC Davis

Monday, April 13, 2020, (Zoom), 3:30 PM – 4:30 PM

Shock formation for the 3d Euler equations

Abstract: The Euler equations are the fundamental model of gas dynamics and it has been known for some time (both numerically and analytically) that smooth solutions of these equations experience a  breakdown in finite time.  For solutions to the Euler equations in 1d,  Lax proved in 1964 that when the initial data has a negative slope, a shock singularity will form in a finite-time.   A shock occurs when the gradient of the velocity and density becomes infinite.

In multiple space dimensions, new ideas were needed to understand how singularities form from smooth solutions.  In 2007, Christodoulou developed a novel geometric framework which allowed him to prove shock formation in the irrotational setting for the 3d relativistic Euler equations, and using this same methodology, Christodoulou & Miao in 2014 proved the same result for irrotational solutions to the non-relativistic 3d Euler equations.   More recently, again using this geometric method, Luk & Speck in 2018 proved shock formation for the 2d Euler equations in the presence of small vorticity.    The presence of vorticity complicates matters greatly.

In this talk, I will show a number of numerical simulations of the Euler equations that will demonstrate the types of shocks that can form, and then I will describe a new method developed in recent work with Tristan Buckmaster (Princeton) and Vlad Vicol (NYU) which establishes the first  finite-time shock formation result for the 3d Euler equations with vorticity.   In particular, it will be shown that from a large open set of smooth (Sobolev-class) data, smooth solutions form a “generic” and stable shock.   Unlike the previous methods (including even those in 1d), the exact blow up time and location can be explicitly computed, and solutions at the blow up time are smooth except for a single point, where they are of cusp-type with Holder-1/3 regularity. Our proof is based on the use of modulated self-similar variables that are used to enforce a number of constraints on the blow up profile, necessary to establish the stability in self-similar variables of the generic shock profile.


Sung-Jin Oh, UC Berkeley


Monday, April 20, 2020, KAP 414, 3:30 PM – 4:30 PM

Title: TBA
Abstract: TBA


Sijue Wu, University of Michigan


Monday, April 27, 2020, KAP 414, 3:30 PM – 4:30 PM

Title: TBA
Abstract: TBA


Simon Tavare, Columbia University

(CAMS Distinguished Lecturer)


Monday, May 1, 2020, KAP 414, 3:30 PM – 4:30 PM

Title: TBA
Abstract: TBA

Colloquia for the Fall 2019 Semester

Tadashi Tokieda, Stanford University

Monday, September 9, 2019, Irani Hall 101 (Note Location), 3:30 PM – 4:30 PM

Toy models

Abstract: Would you like to come see some toys?

‘Toys’ here have a special sense: objects from daily life which you can find or make in minutes, yet which, if played with imaginatively, reveal behaviors that puzzle seasoned scientists for a while.  We’ll see table-top demos of a series of such toys.  The theme that emerges is singularity.

Suncica Canic, UC Berkeley

Friday, September 13, 2019, KAP 414, 3:30 PM – 4:30 PM

Weak solutions to fluid-mesh-shell interaction problem

Abstract: We give an overview of the recent developments in the design of constructive existence proofs for nonlinear moving boundary problems involving 3D incompressible, viscous fluids and various elastic structures. A special attention will be paid to the interaction involving elastic mesh-supported shells. Real life examples of such problems are many, including the interaction between blood flow and vascular walls treated with mesh-like devices called stents. Examples of applications to vascular procedures will be shown.

Terence Tao, UCLA

(CAMS Distinguished Lecturer)

Monday, September 16, 2019, Irani Hall 101 (Note Location), 3:30 PM – 4:30 PM

The global regularity problem for Navier-Stokes

Abstract: We survey some recent developments towards the infamous global regularity problem for the Navier-Stokes equations for incompressible viscous fluids.

Career Advice Panel

Monday, September 23, 2019, KAP 414, 3:30 PM – 4:30 PM

Panel discussion: “Applying for Grants and Jobs: Information and Advice”

Panelists: Steven Heilman, Aaron Lauda and Gary Rosen

Moderator: Susan Friedlander

All postdocs and graduate students are strongly encouraged to come and ask questions about applying for grants from organizations such as the NSF, the NSA and the Simons Foundation. We will also discuss applying for academic positions.


Jonathan Luk, Stanford University

Monday, October 7, 2019, KAP 414, 3:30 PM – 4:30 PM

Stability of vacuum for the Landau equation with moderately soft potentials

Abstract: Consider the Landau equation with moderately soft potentials in the whole space. We prove that sufficiently small and localized regular initial data give rise to unique global-in-time smooth solutions. Moreover, the solutions approach that of the free transport equation as $t\to +\infty$. This is the first stability of vacuum result for a binary collisional kinetic model featuring a long-range interaction.

Zaher Hani, University of Michigan

Monday, October 14, 2019, KAP 414, 3:30 PM – 4:30 PM

On the kinetic description of the long-time behavior of dispersive PDE

Abstract: Wave turbulence theory claims that at very long timescales, and in appropriate limiting regimes, the effective behavior of a nonlinear dispersive PDE on a large domain can be described by a kinetic equation called the “wave kinetic equation” (WKE). This is the wave-analog of Boltzmann’s equation for particle collisions. A fundamental scientific question to resolve here is to provide a rigorous derivation of this kinetic equation, in a way that allows to justify its significance in describing the long-time dynamics of the Hamiltonian dispersive PDE we started with. In this talk, we shall consider the nonlinear Schrodinger equation on a large box with periodic boundary conditions, and provide a rigorous derivation of its kinetic equation on timescales that are significantly shorter than the conjectured kinetic time scale, but still long enough to exhibit the onset of the kinetic behavior. (This is joint work with Tristan Buckmaster, Pierre Germain, and Jalal Shatah).

Wojciech Ozanski, USC

Monday, October 28, 2019, KAP 414, 3:30 PM – 4:30 PM

Partial regularity results of solutions to the 3D incompressible Navier–Stokes equations and other models

Abstract: We discuss some modern developments of the partial regularity theory for the Navier–Stokes equations, as well as other models of fluid mechanics, since the ground-breaking work of Scheffer (1976-1980) and Caffarelli, Kohn & Nirenberg (1982).

Huy Nguyen, Brown University

Monday, November 4, 2019, KAP 414, 3:30 PM – 4:30 PM

On regularity for the Muskat problem

Abstract: The Muskat problem models the evolution of two immiscible fluids of varying density in a porous medium. The free interface between the two fluids obeys a quasilinear parabolic equation, which to leading order has a natural scaling. We will discuss our recent results on local and global well-posedness respectively in all subcritical Sobolev spaces and certain critical spaces, allowing for curvature singularities of the initial interface.  We employ a paradifferential calculus approach which is robust enough to incorporate various features in the problem such as varying viscosity and  physical boundaries. We also obtain results on the vanishing surface tension limit and the infinite depth limit.

Anna Mazzucato, Penn State University

Monday, November 8, 2019, KAP 414, 3:30 PM – 4:30 PM

Irregular transport and mixing in fluids

Abstract: I will discuss the effect of irregular transport on mixing properties in incompressible fluids, in particular describing measures of mixing and examples of optimal mixers. I will also discuss how mixing and transport can suppress singularity formation or lead to complete loss of regularity.

Boris Khesin, University of Toronto

Monday, December 2, 2019, KAP 414, 3:30 PM – 4:30 PM

Beyond Arnold’s geodesic framework of an ideal hydrodynamics

Abstract: We discuss ramifications of Arnold’s group-theoretic approach to ideal hydrodynamics as the geodesic flow for a right-invariant metric on the group of volume-preserving diffeomorphisms. We show that problems of optimal mass transport are in a sense dual to the Euler hydrodynamics. Moreover, many equations of mathematical physics, such as the motion of vortex sheets or fluids with moving boundary, have Lie groupoid, rather than Lie group, symmetries (this is a joint work with Anton Izosimov).

Vladimir Sverak, University of Minnesota

Monday, December 9, 2019, KAP 414, 3:30 PM – 4:30 PM

Regularity and long-time behavior for certain 1d model equations related to fluid flows

Abstract: In the 1980s, P. Constantin, P. Lax, and A. Majda introduced a simple 1d model equation sharing certain features with the 3d incompressible Euler equation. In the 1990s, S. De Gregorio suggested a modification of the original Constantin-Lax-Majda model which has some intriguing features.  Since then, other interesting model equations appeared. In spite of the apparent simplicity of these models, various natural questions about them remain open. I will discuss some of the known results, as well as the relation of these models to the full 3d equations.

Colloquia for the Spring 2019 Semester

Vlad Vicol, Courant Institute

Friday, January 11, 2019, KAP 414, 3:30 PM – 4:30 PM

Convex integration on thin sets

I will discuss the construction of wild weak solutions to the Navier-Stokes equation which are smooth on the complement of a thin set of times (with Hausdorff dimension strictly less than 1). This is based on joint work with T. Buckmaster and M. Colombo.


Misha Vishik, University of Texas at Austin

Monday, January 14, 2019, KAP 414, 3:30 PM – 4:30 PM

Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid

Abstract: We prove non-uniqueness of the solution to  Cauchy problem of the Euler equations of an ideal incompressible fluid in plane with vorticity in some L^Q(R^2) space. The lack of uniqueness is of the symmetry breaking type, with the radially symmetric external force locally integrable in time with values in the same Lebesgue space.

We prove linear instability of a certain class of incompressible flows in the “linear” part of the paper.


Mimi Dai, University of Illinois at Chicago

Monday, January 28, 2019, KAP 414, 3:30 PM – 4:30 PM

Non-uniqueness of Leray-Hopf weak solutions for the 3D Hall-MHD system

Abstract: We will talk about the non-uniqueness of weak solutions in Leray-Hopf space for the three dimensional magneto-hydrodynamics with Hall effect. We adapt the widely appreciated convex integration framework developed in a recent work of Buckmaster and Vicol  for the Navier-Stokes equation, and with deep roots in a sequence of breakthrough papers for the Euler equation.


Lenya Ryzhik, Stanford University

Monday, February 4, 2019, KAP 414, 3:30 PM – 4:30 PM

The stochastic heat equation and KPZ in dimensions three and higher

Abstract. I will give an introduction to the behavior of the solutions to the heat equation with a space-time stationary random potential in d\ge 3. In these “high” dimensions, when the potential is sufficiently weak, this equation admits a space-time stationary solution that serves as an analog of the principal eigenfunction in a periodic problem. As a consequence, one obtains an effective diffusion equation, and an Edwards-Wilkinson limit for fluctuations in the long time limit. Similar results hold for the KPZ equation in d\ge 3. I will also try to explain the connection to the stochastic Burgers’ equation and standard elliptic homogenization. This is a joint work with A. Dunlap, Y. Gu and O. Zeitouni.


Susan Holmes, Stanford University

(CAMS Distinguished Lecturer)

Monday, February 25, 2019, IRANI HALL 101 (PLEASE NOTE ROOM CHANGE), 3:30 PM – 4:30 PM

Hidden variables: finding latent variables in bacterial communities

Abstract: The analyses of complex biological systems often results in output that may seem just as complex, with little useful knowledge extracted as a result of the multiple layers of information. Although distances are an important component of effective data science, we will show examples where distances taken in isolation of probability measure information give spurious results. In bioinformatics for instance standard methods for identifying taxa used fixed radii at 97% similarity regardless of sequence prevalence leading to spurious results. The standard base rate neglect fallacy (Kahneman and Tversky, 1974) still prevails even in mathematics where methods such as topological data analyses still ignore relevant changes in measure.

The use of multi-scale strategies is providing useful predictions of preterm birth and a deeper understanding of resilience of the human microbiome after antibiotic perturbations.


Kavita Ramanan, Brown University

Monday, March 4, 2019, KAP 414, 3:30 PM – 4:30 PM

Hydrodynamic limits for randomized load balancing networks

Abstract: Randomized load balancing algorithms allow for the efficient use of resources and are of particular importance in large-scale networks.   Since these networks are too complex to be amenable to an exact analysis, an established framework is instead to obtain tractable approximations that provide qualitative insight into the dynamics, and whose accuracy can be rigorously justified via limit theorems in a suitable (asymptotic) regime.    However, load balancing networks with jobs having general service distributions fall outside the purview of existing methods.  We introduce a novel interacting particle representation for these networks, describe their hydrodynamic scaling limits, and show how they can be used to provide insight into both transient and equilibrium performance measures of the network.


Ian Tice, Carnegie Mellon University

Monday, March 18, 2019, KAP 414, 3:30 PM – 4:30 PM

Trace operators for homogeneous Sobolev spaces in infinite strip-like domains

Abstract: Sobolev spaces are an indispensable tool in the modern theory of partial differential equations.  Trace embeddings show that functions in Sobolev spaces, which are a priori defined as elements of $L^p$ and hence are only defined almost everywhere, can actually be restricted to sufficiently regular hypersurfaces in a bounded way.  Characterizing the resulting trace spaces and constructing bounded right inverses (lifting results) then plays an essential role in using Sobolev spaces to study boundary-value problems in PDE.  The use of Sobolev spaces to study equations in unbounded, infinite-measure sets often requires employing homogeneous Sobolev seminorms, in which only the highest-order derivatives are controlled in $L^p$.  In this setting, the classical trace results may fail for certain choices of sets that appear naturally in PDE applications, such as infinite strip-like sets $\mathbb{R}^{n-1}\times (0,b) \subset \mathbb{R}^n$.  In this talk we will survey the classical theory and then turn to recent developments in the homogeneous trace theory and applications.  In particular, we will show that in strip-like sets the homogeneous trace spaces are characterized by a new type of fractional homogeneous Sobolev regularity and an interaction between the traces on the different connected components of the boundary.


Haitian Yue, USC

Monday, March 25, 2019, KAP 414, 3:30 PM – 4:30 PM

Well-posedness for the periodic cubic NLS

Abstract: The cubic nonlinear Schrödinger equation (NLS) is energy-critical with respect to the scaling symmetry in the dimensions four. The initial value problem (IVP) of cubic NLS is scaling invariant in the Sobolev norm H^1.

First this talk introduce the deterministic global well-posedness result of the periodic cubic NLS in four dimensional space in the critical regime (with H^1 initial data).

Second we consider the periodic cubic NLS in the super-critical regime (with H^s data, s<1). A probabilistic approach is applied to obtain an “almost sure” well-posedness result for the periodic cubic NLS in the super-critical regime.


James Kelliher, UC Riverside

Monday, April 1, 2019, KAP 414, 3:30 PM – 4:30 PM

The strong vanishing viscosity limit with Dirichlet boundary conditions: facts, speculations, and conjectures

Abstract: We say that the strong vanishing viscosity limit holds if solutions to the Navier-Stokes equations converge in the energy norm to a solution to the Euler equations uniformly over finite time as the viscosity is taken to zero. Starting with the seminal 1983 paper of Tosio Kato, we show how to establish necessary and sufficient conditions for such convergence to hold in the presence of a boundary. We extend various conditions developed by subsequent authors for no-slip boundary conditions to allow non-homogeneous Dirichlet boundary conditions, establishing a few new conditions along the way. Finally, we make a few speculations and conjectures on the strong vanishing viscosity limit.


Yan Guo, Brown University

Monday, April 15, 2019, KAP 414, 3:30 PM – 4:30 PM

Validity of Steady Prandtl Expansion

Abstract: In a joint work with Sameer Iyer, the validity of steady Prandtl layer expansion is established. Our result covers the celebrated Blasius boundary layer profile, which is based on uniform quotient estimates for the derivative Navier-Stokes equations, as well as a positivity estimate at the flow entrance.


Daniel Tataru, UC Berkeley

Monday, April 22, 2019, KAP 414, 3:30 PM – 4:30 PM

Long time dynamics in two dimensional water waves

Abstract: The water wave equations describe the motion of the free surface of a fluid (e.g. water) under the action of various physical forces. Understanding the long time properties of water wave flows is a very interesting yet also very challenging class of problems. The talk will provide an overview of recent and ongoing work in this direction. This is joint work with Mihaela Ifrim.

Colloquia for the Fall 2018 Semester

Angkana Rueland, University of Leipzig

Monday, August 20, 2018, KAP 414, 3:30 PM – 4:30 PM

Microstructures in Shape-Memory Alloys: Rigidity, Flexibility and Some Numerical Experiments

Abstract: In this talk I will discuss a striking dichotomy which occurs in the mathematical analysis of microstructures in shape-memory alloys: On the one hand, a number of interesting models display strong rigidity with only very specific microstructures emerging, if one assumes that surface energies are penalised. On the other hand, without this penalisation, for the same models a plethora of very “wild” solutions exists. Motivated by this observation, we seek to further understand and analyse the underlying mechanisms. By discussing a two-dimensional toy model and by constructing explicit solutions, we show that adding only little regularity to the model does not suffice to exclude the wild solutions. We illustrate these constructions by presenting numerical simulations of them. The talk is based on joint work with J. M. Taylor, Ch. Zillinger and B. Zwicknagl


Career Advice Panel

Monday, August 27, 2018, KAP 414, 3:30 PM – 4:30 PM

Panel discussion: “Applying for Grants and Jobs: Information and Advice”

Panelists: Aravind Asok, Greta Panova and Aaron Lauda

Moderator: Susan Friedlander

All postdocs and graduate students are strongly encouraged to come and ask questions about applying for grants from organizations such as the NSF, the NSA and the Simons Foundation. We will also discuss applying for academic positions.


Monica Visan, UCLA

Monday, September 10, 2018, KAP 414, 3:30 PM – 4:30 PM

KdV is well posed on H-1

Abstract: While initial introduced as a model for water waves, the Korteweg — de Vries equation has grown to be one of the most-studied partial differential equations.  It attracts interest as a model exhibiting solitons, as a representative dispersive PDE, and as an integrable system.  In this talk, I will present a proof of well-posedness for KdV for initial data which is merely in H-1. The argument applies equally well for KdV posed on the line or the circle.  This is based on joint work with Rowan Killip.


Philip Isett, Caltech

Monday, September 17, 2018, KAP 414, 3:30 PM – 4:30 PM

Nonuniqueness and Dissipation of Energy in the Incompressible Euler Equations

Abstract: Ideas concerning energy dissipation in turbulence have inspired the notion that solutions to the incompressible Euler equations that dissipate kinetic energy may be appropriate for describing fluid behavior in the vanishing viscosity limit.  Such solutions that do not satisfy the conservation of energy must have limited regularity, and may therefore in principle be nonunique.  The maximal regularity for the existence of an energy non-conserving solution was conjectured in 1949 by Lars Onsager to be the Hölder exponent 1/3.

I will discuss the proof of this conjecture up to the endpoint regularity, as well as a new result on the existence and nonuniqueness of continuous solutions that exhibit local dissipation of energy.  The local dissipation of energy is a key criterion for isolating the most physically relevant weak solutions to the equations.  A new approach has been required to overcome the limitations faced by previous results on the dissipation of total kinetic energy or on local energy dissipation for bounded, measurable solutions.  We prove that continuous weak solutions are nonunique even if local conservation of energy is imposed.  Our proof of nonuniqueness represents the first application of the probabilistic method within the method of convex integration, which we employ to obtain a family of such solutions that has positive Hausdorff dimension in the energy space emanating from a single initial datum.


Jack Xin, UC Irvine

Monday, September 23, 2018, KAP 414, 3:30 PM – 4:30 PM

Enhanced Diffusivity in Chaotic Flows

Abstract: Transport phenomena in fluid flows are observed ubiquitously in nature such as smoke rings in the air, pollutants in the aquifers, plankton blooms in the ocean,  and stirring drops of cream in a cup of coffee. We begin with examples of two dimensional Hamiltonian systems modeling incompressible planar flows, and illustrate the transition from ordered to chaotic flows as the Hamiltonian function becomes more time dependent. We introduce effective diffusivity and show how its enhancement in the flows relate to the existence of periodic orbits (order) and chaotic behavior of flow trajectories (disorder). Then we discuss enhanced diffusivity in three dimensional chaotic flows, viz., the Arnold-Beltrami-Childress flow and the Kolmogorov flow, through recent analytical and computational findings.


Marcelo Disconzi, Vanderbilt

Monday, October 8, 2018, KAP 414, 3:30 PM – 4:30 PM

A new formulation of the relativistic Euler equations

Abstract: In this talk, we will discuss some recent results concerning the problem of regularity and shock formation for relativistic fluids, with focus on the relativistic Euler equations. Highlighting some “hidden” geometric aspects of the problem, we will present a new formulation of the relativistic Euler equations that exhibits remarkable properties. This new formulation is well-suited for various applications, in particular for the study of stable shock formation, as we will discuss. Furthermore, using the new formulation, we establish a local well-posedness result showing that the vorticity and the entropy of the fluid are one degree more regular than one might naively expect. This is a joint work with Jared Speck.


Christian Zillinger, USC

Monday, October 15, 2018, KAP 414, 3:30 PM – 4:30 PM

Stabilization by mixing: On linear damping for the 2D Euler equations

Abstract: In recent years, following the seminal works of Villani and Mouhot on Landau damping, phase-mixing as a damping mechanism and, in particular, inviscid damping in fluids have attracted much interest. In this talk, I will provide an introduction to the underlying mechanism and discuss new linear stability and damping results near Taylor-Couette flow between concentric cylinders. This is based on joint work with Michele Coti Zelati at Imperial College, London.


Connor Mooney, UC Irvine

Monday, October 22, 2018, KAP 414, 3:30 PM – 4:30 PM

Singular Solutions to Parabolic Systems

Abstract: Regularity results for linear elliptic and parabolic systems with measurable coefficients play an important role in the calculus of variations. Morrey showed that in two dimensions, solutions to linear elliptic systems are continuous. We will discuss some surprising recent examples of discontinuity formation in the plane for the parabolic problem.


Michael Wolf, University of Zurich

Monday, November 5, 2018, KAP 414, 3:30 PM – 4:30 PM

Analytical Nonlinear Shrinkage of Large Covariance Matrices

Abstract: This paper gives the first analytical formula for optimal nonlinear shrinkage of large-dimensional covariance matrices. We achieve this by identifying and mathematically exploiting a deep connection between nonparametric estimation of the Hilbert transform of the sample spectral density and nonlinear shrinkage.  Previous nonlinear shrinkage methods were numerical: QuEST requires numerical inversion of a complex equation from random matrix theory, and NERCOME is a cross-validation scheme. Analytical is more elegant and has more potential to accommodate future variations or extensions. Immediate benefits are that it is 1,000 times faster with same accuracy, and accommodates covariance matrices of dimension up to 10,000. The difficult case where the matrix dimension exceeds the sample size is also covered.


Maciej Zworski, UC Berkeley

Monday, November 12, 2018, KAP 414, 3:30 PM – 4:30 PM

Microlocal methods in chaotic dynamics

Abstract: Microlocal analysis exploits mathematical manifestations of the classical/quantum (particle/wave) correspondence and has been a successful tool in spectral theory and partial differential equations.  Recently, microlocal methods have been applied to the study of classical dynamical problems, in particular of chaotic (Anosov, Axiom A) flows. I will survey results obtained with Dyatlov and present some more recent results of, among others, Guillarmou, Dang–Riviere, Shen, Bonthonneau–Weich.


Richard Stanley, MIT

(Distinguished Lecture,  Joint with the Whiteman Lecture)

Monday, Nov 26, 2018, Irani Hall 101, 4:00 PM – 5:00 PM

Increasing and decreasing subsequences

Abstract: An increasing subsequence of a permutation a_1, a_2, … , a_n of 1,2, … , n is a subsequence b_1,b_2, … , b_k satisfying b_1 < b_2 <  … < b_k, and similarly for decreasing subsequence. The earliest result in this area is due to Erdös and Szekeres in 1935: any permutation of 1,2, … , pq + 1 has an increasing subsequence of length p+1 or a decreasing subsequence of length q+1. This result turns out to be closely connected to the RSK algorithm from the representation theory of the symmetric group. A lot of work has been devoted to the length k of the longest increasing subsequence of a permutation 1,2, … , n, beginning with Ulam’s question of determining the average value of this number over all such permutations. There are many interesting analogues of longest increasing subsequences, such as longest alternating subsequences, i.e., subsequences b_1,b_2, … , b_k of a permutation a_1, a_2, …, an satisfying b_1>b_2<b_3>b_4< … . We will survey these highlights of the remarkable theory of increasing and decreasing subsequences


Richard Stanley, MIT

(Distinguished Lecture,  Joint with the Whiteman Lecture)

Tuesday, Nov 27, 2018, KAP 414, 3:30 PM – 4:30 PM

A survey of parking functions

Abstract: A parking function of length n is a sequence a_1, a_2, … , an of positive integers whose increasing rearrangement b_1 \le b_2 \le … \le b_n satisfies b_i \le i. Parking functions go back to Pyke in 1959; the term “parking function” and the connection with the parking of cars is due to Konheim and Weiss (1966).  Pollak gave an elegant proof that the number of parking functions of length n is (n + 1)^{n-1}. There are close connections between parking functions and other combinatorial objects such as trees, noncrossing partitions, and the Shi hyperplane arrangement. Parking functions arise in several unexpected algebraic areas, such as representations of the symmetric group and Haiman’s theory of diagonal harmonic. Parking functions also have a number of natural generalizations which fit together in a nice way. We will survey these aspects of the theory of parking functions.

Colloquia for the Spring 2018 Semester

Marcelo Viana, IMPA

(CAMS Distinguished Lecturer, Joint with the Mathematics Department Colloquium)

Monday, January 17, 2018, KAP 414, 3:30 PM – 4:30 PM

Random products of matrices

Abstract: By an old theorem of H. Furstenberg and H. Kesten, the norm of a random product of d-by-d invertible matrices grows at a well-defined (i.e. almost certain) exponential rate, that we call the Lyapunov  exponent.A recent result of A. Avila, A. Eskin and myself asserts that this number depends continuously on the data, that is, on the matrix coefficients and their probability weights. For d=2 this was proven before, in my student C. Bocker´s thesis.This behavior is in sharp contrast with some classical results of R. Mane and J. Bochi about the Lyapunov exponents of continuous linear cocycles. I´ll also discuss some moduli of continuity for the Lyapunov exponent of random matrices.


Tarek Elgindi, UC San Diego

Monday, January 29, 2018, KAP 414, 3:30 PM – 4:30 PM

Singularity formation in incompressible fluids

The rapid formation of small scale structures is a ubiquitous feature of incompressible fluids. Despite this, actually proving small scale formation analytically is highly non-trivial. In this regard, there are two major problems in the field: whether smooth solutions of the 3D Euler equations become singular in finite time and whether generic smooth solutions of the 2D Euler equations become singular in infinite time. I will discuss recent advances on both of these problems as well as our recent proof of finite-time blow-up for strong solutions to the 3D Euler equations. This is a joint work with I. Jeong.


Weiwei Hu, Oklahoma State University

Monday, February 5, 2018, KAP 414, 3:30 PM – 4:30 PM

Boundary Control of Optimal Mixing via Stokes and Navier-Stokes Flows

Abstract: We discuss the problem of optimal mixing of an inhomogeneous distribution of a scalar field via an active control of the flow velocity, governed by  Stokes or Navier-Stokes equations, in a two dimensional open bounded and connected domain.  We consider  the velocity field steered by a control input that acts tangentially on the boundary of the domain through the  Navier slip boundary conditions. This is motivated by the problem of mixing  within a cavity or vessel  by moving the walls or stirring at the boundaries. Our main objective is to design an optimal Navier slip  boundary  control  that optimizes mixing at a given final time. Non-dissipative scalars, both passive and active, governed by the transport equation will be discussed.  In the absence of diffusion, transport and mixing occur due to pure advection.  This essentially leads to a nonlinear control problem of a semi-dissipative system.   A rigorous  proof of the existence of an  optimal controller  and the first-order necessary conditions for optimality will be presented.


Theodore Drivas, Princeton University

Monday, February 12, 2018, KAP 414, 3:30 PM – 4:30 PM

Remarks on Onsager’s Conjecture and a Lagrangian formula for anomalous dissipation

Abstract: We discuss the inviscid limit of the global energy dissipation of Leray solutions of incompressible Navier-Stokes on the torus, assuming that the solutions have norms for Besov space Bps,∞ with s ∈ (0,1]  that are bounded in the L3-sense in time, uniformly in viscosity. We establish an upper bound on energy dissipation of the form O(ν(3s-1)/(s+1)). A consequence is that Onsager type “quasi-singularities” are required in the Leray solutions, even if the total energy dissipation vanishes in the limit ν → 0, as long as it does so sufficiently slowly.  We then discuss a novel Lagrangian expression of the dissipative anomaly under the same assumptions.  These formulae give insight into Lagrangian time irreversibility and its connection to the direction of the turbulent cascade. In 3d turbulence, the cascade is downscale and tracer particles initially disperse faster backward-in-time than forward while in 2d, the cascade is upscale and the particles initially disperse faster forward in time. The first part of the talk concerns joint work with G. Eyink.


Jonathan Aurnou, UCLA

Monday, February 26, 2018, KAP 414, 3:30 PM – 4:30 PM

Planetary Magnetohydrodynamics: The View from the Lab

Abstract: Stars and planets are broadly capable of generating their own large-scale magnetic fields via magnetohydrodynamic (MHD) dynamo processes.  These dynamos are likely the end-product of turbulent MHD cascades. Presently, numerical models can generate beautiful facsimiles of geophysical and astrophysical dynamo fields.  However, it is unclear that they accurately model the turbulence that exists in planetary cores and stellar convection zones.  To better access MHD planetary core-style turbulence, we have built a number of laboratory experiments.  After discussing the state of planetary dynamo modeling, I will present our experiments and experimental findings. I will close with a brief discussion of future aims and the need for continual collusion with theorists.


Andrej Zlatos, UC San Diego

Monday, March 5, 2018, KAP 414, 3:30 PM – 4:30 PM

Stochastic homogenization for reaction-diffusion equations

Abstract: We study spreading of reactions in random media and prove that homogenization takes place under suitable hypotheses.  That is, the medium becomes effectively homogeneous in the large-scale limit of the dynamics of solutions to the PDE.  In contrast to the original (second-order) reaction-diffusion equations, the limiting “homogenized” PDE for this model are (first-order) Hamilton-Jacobi equations, and the limiting solutions are discontinuous functions that solve these in a weak sense.  A key ingredient is a new relationship between spreading speeds and front speeds for these models (as well as a new method to prove existence of these speeds).  This can be thought of as the inverse of the classical Freidlin-Gartner formula in the case of periodic media, but we are able to establish it even for more general stationary ergodic reactions.


Franca Hoffmann, Caltech

Monday, March 19, 2018, KAP 414, 3:30 PM – 4:30 PM

Equilibria in energy landscapes with nonlinear diffusion and nonlocal interaction

Abstract: We study interacting particles behaving according to a reaction-diffusion equation with nonlinear diffusion and nonlocal attractive interaction. This class of partial differential equations has a very nice gradient flow structure that allows us to make links to variations of well-known functional inequalities. Depending on the nonlinearity of the diffusion, the choice of interaction potential and the space dimensionality, we obtain different regimes. Our goal is to understand better the asymptotic behavior of solutions in each of these regimes, starting with the fair-competition regime where attractive and repulsive forces are in balance. This is joint work with José A. Carrillo and Vincent Calvez.


Yuri Bakhtin, Courant Institute

Monday, March 26, 2018, KAP 414, 3:30 PM – 4:30 PM

Burgers equation with random forcing

Abstract: The Burgers equation is a basic nonlinear evolution PDE of Hamilton–Jacobi type related to fluid dynamics and growth models.  I will talk about the ergodic theory of randomly forced Burgers equation in noncompact setting. The basic objects are one-sided infinite minimizers of random action (in the inviscid case) and polymer measures on one-sided infinite trajectories (in the positive viscosity case). This is joint work with Eric Cator, Kostya Khanin, and Liying Li.


Tam Do, USC

Monday, April 2, 2018, KAP 414, 3:30 PM – 4:30 PM

Vorticity Growth in Axial Symmetric Euler Flows

Abstract: For two-dimensional Euler flows, it is known that the L^\infty norm of the gradient of vorticity can grow at most double exponentially in time. In recent years, this bound has been proven to be sharp by Kiselev and Sverak on the unit disc. Drawing inspiration from their results, we examine the possibility of similar growth in the 3D axisymmetric setting. For flows with no swirl, the 3D axisymmetric Euler equations are globally well posed and bear similarities with the 2D Euler Equations. However, we will show that for these flows, one cannot sustain double exponential growth up to the axis of symmetry.


Benoit Pausader, Brown University

Monday, April 16, 2018, KAP 414, 3:30 PM – 4:30 PM

Global regularity for the Einstein equation with a massive scalar field

Abstract: We consider the global in time dynamics of perturbations of Minkowski space for the Einstein equation with a massive scalar field. We show that, if the perturbation is small enough, the solution will be global in time and we derive their asymptotic behavior. The difficulty when a massive scalar field is present is that some of the modes of propagation travel at a speed smaller than the speed of light, so that one cannot simply treat this model as a system of wave equations. Thus it is a good setup to develop robust methods to handle the presence of matter in Einstein’s equations.


Yu Deng, The Courant Institute

(PDE Seminar)

Friday, April 20, 2018, KAP 414, 2:00 PM – 3:00 PM

On 3D gravity-capillary water waves

Abstract: The global stability of water wave equation under small perturbations has been an active direction of research in recent years. When either gravity or surface tension is neglected, stability results have been obtained by several groups of authors. In this talk we present the recent work where we establish global stability when both gravity and surface tension are present. The main new ingredients are a specially designed energy estimate, and a refinement of the spacetime resonance method of Germain-Masmoudi-Shatah. This is joint work with A. Ionescu, B.Pausader and F. Pusateri.


John Grace, Earth Sciences Associates

Monday, April 23, 2018, KAP 414, 3:30 PM – 4:30 PM

Automated Salt Recognition in 2D Seismic and Mapping Basin-Wide Salt in the Gulf of Mexico

Abstract: The most powerful and widely utilized tool for mapping subsurface geologic structure is reflection seismic technology, based on inducing sound waves at the earth’s surface and measuring the energy reflected back by geologic features underground and the time it takes for waves to return. A seismic cross-section is typically 20-40 miles long, 4 -10 miles deep and collects several gigabytes of data. A key goal in seismic interpretation is the discrimination rock types and the thickness and areal extent of strata. Of particular importance in exploring for oil and gas in the Gulf of Mexico is identifying and mapping salt accumulations, which can form subsurface domes, or mountain-like structures, 5 to 8 miles high underground. We have developed a suite of unsupervised machine-learning algorithms to automatically discriminate salt on seismic cross-sections, map its depth and assess the confidence in the results. We have applied them to a set of approximately 10,000 seismic cross-sections covering the Gulf of Mexico (an area about the size of California). The approach combines analysis of the “texture” of raster images derived from the raw seismic data and vector representation of individual reflectors extracted from the rasters. Mapping several dimensions of image texture (e.g., contrast, entropy) to two exhaustive rock categories is the main goal of raster analysis; vector information contributes identifying where salt is not and separately to find the boundaries. Collectively, these dimensions are reduced to a common salt “score” at each point, which is then subject to a threshold analysis to produce two output classifications: salt and non-salt. A map of top of salt is estimated in time and then converted to depth by a separately estimated velocity field for rocks in the Gulf of Mexico. Certainty assessment of the estimated salt/non-salt boundary derives from the strength of the texture differences across the boundary.

Joint work with Scott Morris, Shuang Li and Tony Dupont.


Colloquia for the Fall 2017 Semester

Career Advice Panel

Monday, August 28, 2017, KAP 414, 3:30 PM – 4:30 PM

Panel discussion: “Applying for Grants and Jobs: Information and Advice”

Panelists: Jay Bartroff, Juhi Jang and Aaron Lauda

Moderator: Susan Friedlander

All postdocs and graduate students are strongly encouraged to come and ask questions about applying for grants from organizations such as the NSF, the NSA and the Simons Foundation. We will also discuss applying for academic positions.


Thomas Sideris, UC Santa Barbara

Monday, September 18, 2017, KAP 414, 3:30 PM – 4:30 PM

The affine motion of 3d compressible fluids surrounded by vacuum

Abstract: In continuum mechanics, the motion of a body $B\subset\mathbb R^3$ is described by a family of diffeomorphisms $y\mapsto x(t,y)$ assigning all points $y\in B$ a position $x$ in a moving domain $\Omega_t$ at time $t$. Affine motion is one which is linear in $y$.  That is, $x(t,y)=\mathbf A(t)y$, where for each time $t$, $\mathbf A(t)\in GL^+(3,\mathbb R)$, the group of invertible, orientation preserving  linear transformations of $\mathbb R^3$. In the affine case, we will show that the system of partial differential equations describing the motion of a compressible fluid surrounded by vacuum reduces to a globally solvable Hamiltonian system of ordinary differential equations in $GL^+(3,\mathbb R)$.  For each time $t$, the fluid domain $\Omega_t$ is  an ellipsoid whose diameter grows at a rate proportional to time, as $|t|\to\infty$.  We shall investigate the asymptotic behavior of these domains, and in particular, we will show that for a certain range of the adiabatic index there a scattering theory.


Colette Guillope, University of Paris

Monday, September 25, 2017, KAP 414, 3:30 PM – 4:30 PM

Propagation of long-crested water waves

Abstract: In this talk we will present some results about the  propagation of waves such as those sometimes observed in canals and in near-shore zones of large bodies of water.  A special interest will also be on waves arising in bore propagation, when a surge of water invades an otherwise constantly flowing river. The results are developed in the context of Boussinesq models, so they are applicable to waves that have small amplitude and long wave length when compared with the undisturbed depth. We will discuss the theory of well-posednses results on the long, Boussinesq time scale. In the case of bore propagation, where the mass of water has an infinite energy a priori, we will show how to use suitable approximations with which to compare the full solution. This work is in collaboration with Jerry Bona and Thierry Colin.


Ivan Corwin, Columbia University

Monday, October 2, 2017, KAP 414, 3:30 PM – 4:30 PM

Beyond the Gaussian universality class: traffic, growth, matrices and their universal fluctuations

Abstract: The Gaussian distribution is ubiquitous across science and society. Yet, there are many complex random systems which fail to be well describe by Gaussian processes. In this talk, we will consider certain models of traffic flow, interface growth and random matrices. In their large scale limits, they surprisingly all display the same limit behaviors describe by the so called Kardar-Parisi-Zhang universality class. This represents a rich universality class beyond that of the Gaussian which is widely applicable to many other types of spatial systems.


Arieh Warshel, USC

Monday, October 9, 2017, KAP 414, 3:30 PM – 4:30 PM

How to Simulate the Action of Complex Biological Systems on a Molecular Level?

Abstract: Despite the enormous advances in structural studies of biological systems we are frequently left without a clear structure function correlation and cannot fully describe how different systems actually work. This introduces a major challenge for computer modeling approaches that are aimed at a realistic simulation of biological functions. The unresolved questions range from the elucidation of the basis for enzyme action to the understanding of the directional motion of complex molecular motors. Here we review the progress in simulating biological functions, starting with the early stages of the field and the development of QM/MM approaches for simulations of enzymatic reactions (1). We provide overwhelming support to the idea that enzyme catalysis is due to electrostatic preorganization and then move to the renormalization approaches aimed at modeling long time processes, demonstrating that dynamical effects cannot change the rate of the chemical steps in enzymes (2). Next we describe the use our electrostatic augmented coarse grained (CG) model (2) and the renormalization method to simulate the action of different challenging complex systems. It is shown that our CG model produces, for the first time, realistic landscapes for vectorial process such as the actions of F1 ATPase (3,4), F0 ATPase (5) and myosinV (6,7). It is also shown that such machines are working by exploiting free energy gradients and cannot just use Brownian motions as the vectorial driving force. Significantly, at present, to the best of our knowledge, theses studies are the only studies that reproduced consistently (rather than assumed) a structure based vectorial action of molecular motors. We also describe a breakthrough in CG modeling of voltage activated ion channels (8). We also outline a simulation of the tag of war between staled elongated peptide in the ribosome and the translocon as an illustration of the power of our CG approach (9). The emerging finding from all of our simulations is that electrostatic effects are the key to generating functional free energy landscapes.

1  Electrostatic Basis for Enzyme Catalysis, A. Warshel, P. K. Sharma, M. Kato, Y. Xiang, H. Liu and M. H. M. Olsson, Chem. Rev., 106, 3210 (2006).
2  Coarse-Grained (Multiscale) Simulations in Studies of Biophysical and Chemical Systems, S. C.   L.Kamerlin, S. Vicatos, A. Dryga and A. Warshel, Ann. Rev. Phys. Chem. 62,41 (2011).
3  Electrostatic Origin of The Mechanochemical Rotary Mechanism And The Catalytic Dwell of F1-ATPase, S. Mukherjee and A.Warshel, Proc. Natl. Acad. Sci. USA, 108, 20550 (2011).
4  Torque, chemistry and efficiency in molecular motors: a study of the rotary–chemical coupling in F1-ATPase, S. Mukherjee, R. B. Prasad and A. Warshel, QRB, Discovery, 48, 395–403 (2015).
5  Realistic simulations of the coupling between the protomotive force and the mechanical rotation of the F0-ATPase, Proc. Natl. Acad. Sci. USA, 109,14876 (2012).
6  Electrostatic origin of the unidirectionality of walking myosin V motors, S. Mukherjee and  A. Warshel, Proc. Natl. Acad. Sci. USA, 110, 17326-17331 (2013).
7  Simulating The Dynamics of The Mechanochemical Cycle of Myosin-V, S. Mukherjee, R. Alhadeff, and A. Warshel, Proc. Natl. Acad. Sci.USA,114,2259-64 (2017).
8  Coarse-Grained Simulation of the Gating Current in the Voltage-Activation Kv1.2 Channel, I. Kim and A. Warshel, Proc. Natl. Acad. Sci. USA 111, 2128-2133 (2013).
9  Simulating the pulling of stalled elongated peptide from the ribosome by the translocon, A. Rychkova, S. Mukherjee, R. P. Bora, and A.  Warshel, Proc. Natl. Acad. Sci. USA, 110, 10195-10200 (2013).


Thomas Banks, N.C. State University

(CAMS Distinguished Lecturer)

Monday, October 23, 2017, KAP 414, 3:30 PM – 4:30 PM

Modeling Bumble Bee Population Dynamics with Delay Differential Equations or What’s the Buzz about Global Bumblebee Decline?

Abstract: We report on our continuing efforts between our group at NCSU and ecologists at California State University, Monterey Bay and the Swedish University of Agricultural Sciences, Uppsala. To provide a tool for projecting and testing sensitivity of growth and death of populations under contrasting and combined pressures, we developed a non-linear, non-autonomous delay differential equation (DDE) model of bumblebee colonies and resources model that describes multi-colony bumble bee population dynamics.  We explain the usefulness of delay differential equations as a natural modeling formulation, particularly for bumble bee modeling.  We then introduce a specific spline-based numerical method that approximates the solution of the delay model.  We demonstrate that the model satisfies sufficient conditions to assure the subsequent theoretical developments therein in order to attain convergent approximate solutions. We report on our recent efforts on studies of response to toxic substances, in particular our simulations related to growth, death and sublethal responses to neonicotinoid exposure.


Thomas Hou, California Institute of Technology

Monday, October 30, 2017, KAP 414, 3:30 PM – 4:30 PM

Blowup or no blowup? The interplay between analysis and computation in the study of 3D Euler equations

Abstract: Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. This question is closely related to the Clay Millennium Problem on 3D Navier-Stokes Equations. A potential singularity in the 3D Euler equations is significant because it may be responsible for the onset of energy cascade in turbulent flows. We first review some recent theoretical and computational studies of the 3D Euler equations. Our study suggests that the convection term could have a nonlinear stabilizing effect for certain flow geometry. We then present strong numerical evidence that the 3D Euler equations develop finite time singularities.  The singularity is a ring like singularity that occurs at a stagnation point in the symmetry plane located at the boundary of the cylinder.  A careful local analysis also suggests that the blowing-up solution is highly anisotropic and is not of Leray type. A 1D model is proposed to study the mechanism of the finite time singularity. We have recently proved rigorously that the 1D model develops finite time singularity. Finally, we present some recent progress in developing an integrated analysis and computation strategy to analyze the finite time singularity of the original 3D Euler equations.


Claire Vishik, Intel Corporation

Monday, November 6, 2017, KAP 414, 3:30 PM – 4:30 PM

Trust and cybersecurity: In search of a multi-disciplinary theory and solutions for real world problems

Abstract: Cybersecurity has become a global priority, but it encompasses a set of problems that are difficult to solve. The broad definition of cybersecurity encompasses a range of subfields, from computer and network security to cryptography, psychology, human behavior, economics, policy, and international collaboration.  The interplay of these subjects is important for the theory and practice in cybersecurity, but the underlying relationships are difficult to define, thus impeding the understanding of real life problems.

Trust is a foundational concept in cybersecurity that can provide a common thread linking its many components. But this link is not straightforward. For a device, an application, or a system, trust is based on the premise that the other party behaves in an expected manner under the same conditions. Trust complements security requirements, enabling various security models.  But trust is defined somewhat differently in social sciences or in international relations, all parts of the broad definition of cybernetics. The talk will examine trust in different contexts. From the human side of trust, to cryptographically-supported trusted systems, to policies that maintain trust, a broad view of the topic will be presented. What are the foundations of the individual’s trust in technology and what are the consequences of the lack of trust?  How can trust between systems depend on subtle differences in the integrated circuits in their hardware?  Can technical trust be nuanced, allowing a system to trust another system a little or a lot, depending on the circumstances? How is trust misused by cyber criminals, and how can technology and cyber norms stop them? How is trust connected to privacy? What are the real world problems that the concept of trust can solve?  We will touch upon many of these questions. Trust is complex and multi-faceted, but it is a concept that can explain many successes and failures in cybersecurity.

Bio:  Claire Vishik  is a Sinior Director for Global Cybersecuity at Intel. Her work focuses on hardware security, Trusted Computing, privacy enhancing technologies, applied cryptography and related technology areas. Claire is a member of the Permanent Stakeholders Group (Advisory Board) of the European Network and Information Security Agency (ENISA). She is on the Board of Directors of the Trusted Computing Group (TCG) and a Council Member of the Information Security Forum. a Board member for Trust in Digital Life (TDL),  co-chair of NIST CPS (Cyber-physical Systems) Public Working Group, and  member of the Cybersecurity Steering Group for the UK Royal Society. She serves on advisory  and review boards of a number of organizations and research initiatives in security and privacy in Europe, Asia,  and the US. Claire is the author of a large number of peer reviewed papers and an inventor on 30+ pending and granted U.S. patents. Prior to joining Intel, Claire worked at Schlumberger Laboratory for Computer Science and AT&T Laboratories. She received her Ph.D from the University of Texas at Austin.


Jared Whitehead, Brigham Young University

Monday, November 13, 2017, KAP 414, 3:30 PM – 4:30 PM

The impact of laminar boundary layers on the search for the ultimate regime of turbulent convection

Abstract: For several decades, careful experiments and detailed numerical simulations have tried to observe the ever elusive ‘ultimate’ regime of turbulent Rayleigh-Bénard convection as predicted by Kraichnan in 1962, but to date no decisive evidence has been brought to light.  At the same time, rigorous upper bound analysis has been used to show that this ultimate state does not occur for certain types of boundary conditions and/or asymptotic limits of the underlying Boussinesq equations.  Even so, in the most general setting upper bound analysis fails to eliminate the possibility of this ultimate regime from occurring, but does not confirm its existence either.

We approach this problem from a different perspective than upper bound analysis.  The prevailing current theory motivating the existence of the ultimate regime is that the thermal boundary layers become turbulent at a theoretically predicted level of the driving force.  We show that for Navier-slip and fixed flux boundaries this is not possible, as the fluid velocity and temperature fields satisfy a linear equation in the boundary layer.  Coupled with very recent results from upper bound analysis this implies that either the rigorous upper bounds so far derived are not saturated, i.e. there are stronger results than those currently known, or the ultimate regime is not a result of turbulent boundary layers.


Gigliola Staffilani, MIT

Monday, December 4, 2017, KAP 414, 3:30 PM – 4:30 PM

The many faces of dispersive and wave equations

Abstract: In recent years great progress has been made in the study of dispersive and wave equations.  Over the years the toolbox used in order to attack  highly nontrivial problems related to these equations has developed to include a variety of techniques from Fourier and harmonic analysis, analytic number theory, math physics, dynamical systems, probability and symplectic geometry. In this talk I will introduce a variety of problems connected with dispersive and wave equations, such as the derivation of a certain  nonlinear Schrodinger equations from a quantum many-particles system, periodic Strichartz estimates, the concept of energy transfer, the invariance of a Gibbs measure associated to an infinite dimension Hamiltonian system and non-squeezing theorems for such systems when they  also enjoy a symplectic structure.

Colloquia for the Spring 2017 Semester

Amjad Tuffaha, American University of Sharjah

Monday, January 9, 2017, KAP 414, 3:30 PM – 4:30 PM

Free Boundary Problems in Fluid Flow and Fluid-Structure Interactions

Abstract: We consider some mathematical  problems involving the Navier-Stokes and the Euler Equations on an evolving domain and other systems of fluid-structure interaction involving the Euler or the Navier-Stokes equations  coupled with elasticity or plate equations. We examine historical and recent developments in studying the well-posedness of these systems.


Vlad Vicol, Princeton University

Wednesday, January 11, 2017, KAP 414, 3:30 PM – 4:30 PM

Turbulent weak solutions to hydrodynamic equations

Abstract: Motivated by Kolmogorov’s theory of turbulence, we prove the existence of weak solutions to formally conservative hydrodynamic models, which do not conserve the associated Hamiltonian. In particular, this shows that up to a certain regularity threshold weak solutions are not unique. For the 2D surface quasi-geostrophic equations this answers an open problem posed by De Lellis and Szekelyhidi Jr.


Messoud Efendiyev, Helmholtz Center, Munich

Monday, February 6, 2017, KAP 414, 3:30 PM – 4:30 PM

Mathematical modelling of biofilms

Abstract: In this talk I will discuss spatio-temporal mesoscale  biofilm models. On the mesoscale level, mathematical  biofilm models must incorporate population and resource dynamics as well as the transport of growth/process controlling substrates (nutrients, antibiotics) in the surrounding aqueous phase. I will show that these models can be described by highly-nonlinear density-dependent reaction-diffusion-transport equations comprising a double degeneracy. I will discuss both the hydrostatic and hydrodynamic cases. Well-posedness, long-time dynamics of solutions in terms of global attractors, and asymptotics of their Kolmogorov’s entropy will be treated.


Gerrit Welper, USC

Monday, February 13, 2017, KAP 414, 3:30 PM – 4:30 PM

Interpolation of solutions of hyperbolic PDEs by transformed snapshots

Abstract: In recent years significant research efforts have been devoted to numerical simulations of PDEs with deterministic and random coefficients. Nonetheless, hyperbolic problems remain a challenge. One of the major obstructions is the prevalence of shocks, which require significant computational resources to be efficiently resolved by established methods. We improve their efficiency by introducing transformations of the physical domain that align shock discontinuities. They are computed by optimizing a training error and constructed in a way that avoids unacceptable local minima.


Adam Larios, University of Nebraska

Monday, February 27, 2017, KAP 414, 3:30 PM – 4:30 PM

The Singularity’s Tale

Abstract: One of the famous seven Clay Millennium Prize Problems is to determine whether the 3D Navier-Stokes equations of fluid flow develop a singularity in finite time (i.e., whether the solutions “blow-up”).  A closely related, and potentially more challenging problem, is to decide the blow-up question for the 3D Euler equations of ideal fluid flow.  We will discuss some of the recent history of the search for blow-up of the 3D Euler equations, and build an understanding of the phenomenon by doing live simulations of the simpler 1D Burgers equation.  We will also present recent results on the computational search for blow-up of the 3D Euler equations.


Alexander Kiselev, Rice University

Monday, March 6, 2017, KAP 414, 3:30 PM – 4:30 PM

Regularity and blow up in ideal fluid

Abstract: The incompressible Euler equation of fluid mechanics has been derived in 1755. It is one of the central equations of applied analysis, yet due to its nonlinearity and non-locality many fundamental properties of solutions remain poorly understood. In particular, the global regularity vs finite time blow up question for incompressible three dimensional Euler equation remains open.

In two dimensions, it has been known since 1930s that solutions to Euler equation with smooth initial data are globally regular. The best available upper bound on the growth of derivatives of the solution has been double exponential in time. I will describe a construction showing that such fast generation of small scales can actually happen, so that the double exponential bound is qualitatively sharp.

This work has been motivated by numerical experiments due to Hou and Luo who propose a new scenario for singularity formation in solutions of 3D Euler equation. The scenario is axi-symmetric. The geometry of the scenario is related to the geometry of 2D Euler double exponential growth example and involves hyperbolic points of the flow located at the boundary of the domain. If time permits, I will discuss some recent attempts to gain insight into the three-dimensional fluid behavior in this scenario.


Zaher Hani, Georgia Tech

Monday, March 20, 2017, KAP 414, 3:30 PM – 4:30 PM

Effective dynamics of nonlinear Schroedinger equations on large domains

Abstract: In this talk, we will be mainly concerned with the following question: Suppose we consider a nonlinear dispersive or wave equation on a large domain of characteristic size L; what is the effective dynamics when L is very large? This question is relevant for equations that are naturally posed on large domains (like water waves on an ocean), and in turbulence theories for dispersive equations. It’s not hard to see that the answer is intimately related to the particular time scales at which we study the equation, and one often obtains different effective dynamics on different timescales. After discussing some time scales (and their corresponding effective dynamics) that hold for more-or-less generic dispersive equations, we will try to go further in describing the effective dynamics over much longer time scales. This becomes more equation-dependent, and here we specialize to the nonlinear Schroedinger equation (any power nonlinearity) posed on a large box of size $L$. Our main result is to exhibit a new type of dynamics that appears at a particular large time scale, (that we call the resonant time scale) defined in terms of the size of the domain L and the characteristic size of the initial data. As mentioned, going to such long time scales is partly motivated by turbulence theory for dispersive PDE, aka wave turbulence theory, in which one would like to address the effective dynamics on even longer timescales. We will touch on these topics and time scales as well. This is joint work with Tristan Buckmaster, Pierre Germain, and Jalal Shatah (all at Courant Institute, NYU).


John Hunter, UC Davis

Monday, March 27, 2017, KAP 414, 3:30 PM – 4:30 PM

Contour dynamics and front propagation in the incompressible Euler and SQG equations

Abstract: Vorticity discontinuities in the two-dimensional incompressible Euler equations and temperature discontinuities in the SQG equations support surface waves that decay exponentially away from the discontinuities. These waves have constant linearized frequency on vorticity discontinuities and almost constant wave speed, with a logarithmic correction, on SQG fronts. This talk will describe approximate equations for weakly nonlinear surface waves on planar Euler and SQG fronts, obtained by expansion of suitably regularized contour dynamics equations, and discuss some of their properties.


Michael Aizenman, Princeton University

(CAMS Distinguished Lecturer)

Monday, April 3, 2017, KAP 414, 3:30 PM – 4:30 PM

Stochastic Geometry of Correlations in Stat-Mech and  Quantum Systems

Abstract: Some of the qualitative features of interacting classical and quantum systems can be illuminated through stochastic geometric representations.   In these, the correlations in some of the basic model are presented as mediated through fluctuating clusters and/or random loops. Such representations facilitate insights on a number of phenomena, including: existence of phase transitions related to the onset of long range order, dimension dependence of the critical exponents in Ising type models, the emergence of conformal invariance in critical two dimensional models and relations with the conformally invariant SLE random curves. For one dimensional quantum spin chains a stochastic geometric representation allows us to shed light on the difference between the integer and half integer cases in the spectral (Haldane) gap.In the talk we tread on grounds which were earlier marked by USC Professor Mark Kac, to whose memory this lecture is dedicated.


Jonathan Mattingly, Duke University

Monday, April 17, 2017, KAP 414, 3:30 PM – 4:30 PM

Building Natural Lyapunov Functions and stablization by noise

Abstract: I will discuss a number of stochastic systems where question of existence of a stochastic steady state (and invariant measure) or the convergence to equilibrium can be reduced to the proving the  existence of an appropriate Lyapunov function.

This will lead us to consider the questions: How does one build a Natural Lyapunov Function ? Can this be done in a systematic way?

I will consider a number of illustrative examples including the stabilization by noise of an unstable planer vector field and the convergence to equilibrium of a Hamiltonian oscillator with a singular potential, such as a Lennard-Jones potential.

Along the way, I will make connections to stochastic averaging, indeterminacy  and hypercoercivity.


Mark Green, UCLA

(Whiteman Lecturer)

Wednesday, April 19, 2017, Mudd Hall of Philosophy, 101, 4:00 PM – 5:00 PM

The Unreasonable Effectiveness of Bayes’ Theorem

Abstract: Mathematics has a way of turning out to be more useful than one might have expected. A case in point is Bayes’ Theorem, a result about probabilities from the 18th century which has been used for applications as diverse as breaking  the German Enigma codes in World War II, quantifying the link between smoking and lung      cancer, problems in genomics and understanding the human brain.  In this talk, I will discuss the power of probabilistic thinking in general, explain Bayes’ Theorem and give some examples of how it is used in machine learning in general, with some examples from clustering tumors to the automated discovering “topics” in a corpus of scientific papers.


Walter Rusin, Oklahoma State University

Monday, April 24, 2017, KAP 414, 3:30 PM – 4:30 PM

Remarks and observations on active scalar equations

Abstract: We will discuss properties of active scalar equations that can be traced back to the constitutive law relating the drift velocity and the active scalar. In particular, we will focus on the properties of equations where the Fourier symbol of the multiplier generating the velocity is an even or odd function. Some results on existence of solutions will be discussed.

Colloquia for the Fall 2016 Semester

Career Advice Panel

Monday, August 29, 2016, KAP 414, 3:30 PM – 4:30 PM

Panel discussion: “Applying for Grants and Jobs: Information and Advice”

Panelists: Aravind Asok, Jay Bartroff and Aaron Lauda

Moderator: Susan Friedlander

All postdocs and graduate students are strongly encouraged to come and ask questions about applying for grants from organizations such as the NSF, the NSA and the Simons Foundation. We will also discuss applying for academic positions.


Jill Mesirov, UC San Diego

(Joint with Computational Biology)

Monday, September 12, 2016, KAP 414, 3:30 PM – 4:30 PM

Computational Approaches for Genomic Medicine

Abstract: The acceleration of data acquisition is changing the face of biomedical research. Computational approaches can take advantage of these data and bring the promise of improved understanding and treatment of disease.

I will describe our approaches to leverage multiple data types to stratify pediatric brain tumor patients into groups with high and low risk of relapse after treatment; shed light on the functional correlates of genetic variants; and combine data from cell line and mouse models with public drug sensitivity databases to identify a novel candidate therapeutic compound. Finally, I will review the software through which we make our methods available to the research community.


Christian Zillinger, USC

Monday, September 19, 2016, KAP 414, 3:30 PM – 4:30 PM

On linear inviscid damping, boundary effects and blow-up

Abstract: The Euler equations of fluid dynamics are time-reversible equations and possess many conserved quantities, including the kinetic energy and entropy. Furthermore, as shown by Arnold, they even have the structure of an infinite-dimensional Hamiltonian system. Despite these facts, in experiments one observes a damping phenomenon for small velocity perturbations to monotone shearflows, where the perturbations decay with algebraic rates. In this talk, I discuss the underlying phase-mixing mechanism of linear inviscid damping, its mathematical challenges and will sketch how to establish decay with optimal rates for a general class of monotone shear flows and circular flows. Here, a particular focus will be on the setting of a channel with impermeable walls and an annular domain, where boundary effects asymptotically result in the formation of singularities.


Michele Coti-Zelati, University of Maryland

Monday, September 12, 2016, KAP 414, 3:30 PM – 4:30 PM

Deterministic and stochastic aspects of fluid mixing

Abstract: The process of mixing of a scalar quantity into a homogenous fluid is a familiar physical phenomenon that we experience daily. In applied mathematics, it is also relevant to the theory of hydrodynamic stability at high Reynolds numbers – a theory that dates back to the 1830’s and yet only recently developed in a rigorous mathematical setting. In this context, mixing acts to enhance, in certain senses, the dissipative forces. Moreover, there is also a transfer of information from large length-scales to small length-scales vaguely analogous to, but much simpler than, that which occurs in turbulence. In this talk, we focus on the study of the implications of these fundamental processes in linear settings, with particular emphasis on the long-time dynamics of deterministic  systems (in terms of sharp decay estimates) and their stochastic perturbations (in terms of invariant measures).


Konstantin Batygin, California Institute of Technology

Monday, October 10, 2016, KAP 414, 3:30 PM – 4:30 PM

Planet Nine from Outer Space

At the outskirts of the solar system, beyond the orbit of Neptune, lies an expansive field of icy debris known as the Kuiper belt. The orbits of the individual asteroid-like bodies within the Kuiper belt trace out highly elongated elliptical paths, and require hundreds to thousands of years to complete a single revolution around the Sun. Although the majority of the Kuiper belt’s dynamical structure can be understood within the framework of the known eight-planet solar system, bodies with orbital periods longer than about 4,000 years exhibit a peculiar orbital alignment that eludes explanation. What sculpts this alignment and how is it preserved? In this talk, I will argue that the observed clustering of Kuiper belt orbits can be maintained by a distant, eccentric, Neptune-like planet, whose orbit lies in approximately the same plane as those of the distant Kuiper belt objects, but is anti-aligned with respect to those of the small bodies. In addition to accounting for the observed grouping of orbits, the existence of such a planet naturally explains other, seemingly unrelated dynamical features of the solar system.


Marta Lewicka, University of Pittsburg

Monday, October 17, 2016, KAP 414, 3:30 PM – 4:30 PM

Prestrained elasticity and curvature constraints


This lecture is concerned with the analysis of thin elastic films exhibiting residual stress at free equilibria. Examples of such structures and their actuations include: plastically strained sheets; swelling or shrinking gels; growing tissues; atomically thin graphene layers, etc. These and other phenomena can be studied through a variational model, pertaining to the non-Euclidean version of nonlinear elasticity, measuring the deviation of a deformation from isometric immersions of a given non-flat Riemannian metric, rather than the flat Euclidean metric.

In this context, analysis of scaling of the energy minimizers in terms of the film’s thickness leads to the rigorous derivation of a hierarchy of limiting theories, differentiated by the isometry constraints with low regularity. This leads to questions of rigidity and flexibility of solutions to the weak formulations of the related PDEs, including the Monge-Ampere equation.


Michael Wolf, University of Zurich

Monday, October 24, 2016, KAP 414, 3:30 PM – 4:30 PM

Resurrecting Weighted Least Squares

Abstract: We show how asymptotically valid inference in regression models based on the weighted least squares (WLS) estimator can be obtained even when the model for reweighting the data is misspecified. Like the ordinary least squares estimator, the WLS estimator can be accompanied by heteroskedasticity-consistent (HC) standard errors without knowledge of the functional form of conditional heteroskedasticity. First, we provide rigorous proofs under reasonable assumptions; second, we provide numerical support in favor of this approach. Indeed, a Monte Carlo study demonstrates attractive finite-sample properties compared to the status quo, both in terms of estimation and making inference.


Mahir Hadzic, Kings College London

Monday, October 31, 2016, KAP 414, 3:30 PM – 4:30 PM

Generic melting and freezing rates in the classical Stefan problem


Stefan problem is one of the most famous phase transition free-boundary problems modelling the phase transitions between liquids and solids. After an introduction to the Stefan problem we shall provide a  brief discussion of the role of self-similarity in the study of singularity formation in PDEs.

In the second part of the talk we prove the existence of STABLE non self-similar finite time melting and freezing regimes motivated  by the pioneering analysis of Herrero and Velazquez. We introduce a new and canonical functional framework for the study of type II (i.e. non self  similar) blow up for a class of problems including a related construction for the harmonic heat flow studied by Raphael and Schweyer. This is a joint work with P. Raphael.


Roman Shvydkoy, University of Illinois Chicago

Monday, November 7, 2016, KAP 414, 3:30 PM – 4:30 PM

Mechanisms for energy balance restoration in Onsager critical and supercritical flows

Abstract: The celebrated Onsager conjecture of 1949 states that any solution to the incompressible Euler system conserves energy if its regularity is better than 1/3, and that there is a solution with lower regularity that dissipates energy. In recent years the conjecture has attracted a lot of attention due to its intimate connection to regularity problems, problems arising in turbulence, scaling laws, intermittency, etc. In this talk we will discuss basic techniques used to settle the positive part of the conjecture, outline recent developments that helped settle the negative part, and discuss in more detail the mechanisms behind survival of the energy balance relation in the Onsager critical and supercritical flows.


Michael Holst, UC San Diego

Monday, November 14, 2016, KAP 414, 3:30 PM – 4:30 PM

A Look at Some Mathematics Research Problems in General Relativity

Abstract: In 1915, Einstein (and Hilbert) formulated the equations of general relativity,
the Einstein Equations, as a geometric description of what we experience as gravity, displacing Newton’s theory of gravity.  Already by 1916, it was realized that the equations predict that gravitationally interacting bodies will emit radiation at the speed of light that will carry energy away from the interaction.  Unlike other types of radiation, this radiation is actually a “ripple” in space (and time).  Massive interactions that occur at relatively close ranges, such as in-spiraling and colliding black holes, emit enough radiation that they could potentially be detected by a device that we could build using late 20th/early 21st century technology.  Such a device was built, called LIGO (Laser Interferometer Gravitational-Wave Observatory).  At more than $600M, it is the most ambitious and expensive NSF project ever undertaken. On 11 February 2016, it was announed that LIGO detected a clear, unambiguous, loud and violent inspiral, collision, merger, and ringdown of a binary black hole pair, each of which had a solar mass in the range 10-50, with roughly the equivalent of three solar masses in energy released as gravitational radiation. This radiation traveled outward from the collision at the speed of light, reaching the LIGO detectors on earth roughly 1.3 billion years


Why do LIGO scientists know that this is what LIGO detected?  The answer is that the signal LIGO detected was shown, after extensive data analysis and computer simulations of the Einstein evolution and constraint equations, to be a very close match to simulations of wave emission from a very particular type of binary collision.  In this lecture, we will examine some fundamental mathematics research questions involving the Einstein constraint equations.  We begin with an overview of the most useful mathematical formulation of the constraint equations, and then summarize the known existence, uniqueness, and multiplicity results through 2008.  We then present a number of new existence and multiplicity results developed since 2008 that substantially change the solution theory for the constraint equations. We then shift gears and consider Petrov-Galerkin type approximation methods for developing “provably good” numerical methods for solving this type of system.  We examine how one proves rigorous error estimates for particular classes of numerical methods, including both classical finite element methods and newer methods from the finite element exterior calculus.

About the speaker:

Michael Holst is a Professor of Mathematics and Physics at UCSD and holds a Chancellor’s Associates Endowed Chair.  He is a core faculty member in both the Center for Computational Mathematics (CCoM) and the Center for  Astrophysics and Space Sciences (CASS).  He received a Ph.D. from the University of Illinois in 1993.


Linda Petzold, UC, Santa Barbara

Monday, November 28, 2016, KAP 414, 3:30 PM – 4:30 PM

The Emerging Roles and Computational Challenges of Stochasticity in Biological Systems

Abstract: In recent years it has become increasingly clear that stochasticity plays an important role in many biological processes.  Examples include bistable genetic switches, noise enhanced robustness of oscillations, and fluctuation enhanced sensitivity or “stochastic focusing”..  Numerous cellular systems rely on spatial stochastic noise for robust performance.   We examine the need for stochastic models, report on the state of the art of algorithms and software for modeling and simulation of stochastic biochemical systems, and identify some computational challenges.


Stefan Steinerberger, Yale University

Monday, December 5, 2016, KAP 414, 3:30 PM – 4:30 PM

Mysterious Interactions between Analysis and Number Theory

Abstract: I will discuss three different topics that connect classical analysis with number theory in an unexpected way.  (1) A new type of Poincare inequality on the Torus that is optimal in all sort of ways, scales, exponents,…  (2a) If the Hardy-Littlewood maximal function of a function f(x) is easy to compute, the function is f(x) = sin(x) or, equivalently, (2b) if f(x) is periodic and the trapezoidal rule is sharp on all intervals of length 1, then the function is trigonometric. This statement is clearly very elementary but the only proof I could find has to use some nontrivial number theory and I am not sure why!  (3) Strange, unexplained (and pretty!) patterns that appear in an old integer sequence of Stanislaw Ulam from the 1960s [Prize Money: $200 dollars for an explanation/proof].

Colloquia for the Spring 2016 Semester

Mohammed Ziane, USC

Monday, February 1, 2016, KAP 414, 3:30 PM – 4:30 PM

Some regularity results for the 3D-Navier-Stokes equations


In this talk, we will present three types of regularity results on the 3D Navier-Stokes equations. First we discuss, some conditions on one component of the velocity that guarantee the regularity of Leray’s weak solutions. These conditions are in the spirit of Prodi-Serrin conditions.

The second type of results is based on anisotropy and fast oscillations, and gives global regular solutions with large BMO^-1 norms. Finally we give a partial regularity result in the sense of Caffarelli, Kohn, Nirenberg, which is based on only one component of the velocity.


CANCELLED – Michael Shelley, Courant Institute

(Joint with AME)

Wednesday, February 10, 2016, KAP 414, 3:30 PM – 4:30 PM


Benedict Gross, Harvard University

(Whiteman Lecture)

Monday, February 22, 2016, Gerontology: Leonard Davis Auditorium located in 124, 3:30 PM – 4:30 PM

How large is n! = n(n-1)(n-2)…3.2.1 ?

Abstract: The number n! (pronounced “n factorial”) occurs in many counting problems. For example, that 52! is the number of ways to shuffle a deck of cards. This number grows very rapidly with n, and mathematicians of the 17th century used the new methods of calculus to estimate it. After reviewing some of this work, I’ll discuss Euler’s Gamma function, which interpolates the function F(n) = (n-1)! to the real numbers, as well as a more recent analog.


Gunnar Carlsson, Stanford University

Monday, February 29, 2016, KAP 414, 3:30 PM – 4:30 PM

Topology and the Big Data Problem

Abstract: In recent years there has been a lot of attention given to “Big Data”.  In fact, many of the problems that need to be addressed relate not to the “Big”, but rather in the inherent complexity of much of the important data that is being produced.  What this means is that there is a need for an organizing principle for data analysis.  One such organizing principle uses methods from Topology, the mathematical discipline which concerns itself with the study of shape, or rather the higher dimensional generalization of shape.  The methods give rise to new methods of modeling data, as well as feature creation and invariants of the shape of the data which are readily interpretable.  We will discuss these ideas, with numerous examples from various areas within the sciences and industry.


Shanghua Teng, Computer Science, USC

Monday, Monday, March 7, 2016, KAP 414, 3:30 PM – 4:30 PM

Through the Lens of the Laplacian Paradigm: Big Data and Scalable Algorithms — a Pragmatic Match Made On Earth


In the age of Big Data, efficient algorithms are in higher demand now more than ever before. While Big Data takes us into the asymptotic world envisioned by our pioneers, the explosive growth of problem size has also significantly challenged the classical notion of efficient algorithms:

Algorithms that used to be considered efficient, according to polynomial-time characterization, may no longer be adequate for solving today’s problems. It is not just desirable, but essential, that efficient algorithms should be scalable. In other words, their complexity should be nearly linear or sub-linear with respect to the problem size. Thus, scalability, not just polynomial-time computability, should be elevated as the central complexity notion for characterizing efficient computation.

In this talk, I will discuss the emerging Laplacian Paradigm, which has led to breakthroughs in scalable algorithms for several fundamental problems in network analysis, machine learning, and scientific computing. I will focus on three recent applications: (1) PageRank Approximation (and identification of network nodes with significant PageRanks). (2) Random-Walk Sparsification. (3) Scalable Newton’s Method for Gaussian Sampling.


Nathan Glatt-Holtz, Virginia Tech

Monday, March 21, 2016, KAP 414, 3:30 PM – 4:30 PM

Asymptotic Coupling and Applications for Nonlinear Stochastic Partial Differential Equations

Abstract: We introduce the notion of asymptotic coupling and explain how this formalism provides a conceptually simple means of proving unique ergodicity in certain stochastic systems whose deterministic counterpart possesses a finite number of determining modes.


Suncica Canic, University of Houston

Monday, March 28, 2016, KAP 414, 3:30 PM – 4:30 PM

Fluid-composite structure interaction and blood flow


Fluid-structure interaction (FSI) problems arise in many applications. The widely known examples are aeroelasticity and biofluids. In biofluidic applications, including the interaction between blood flow and cardiovascular tissue, the coupling between the fluid and structure is highly nonlinear because the density of the structure (tissue) and the density of the fluid (blood) are roughly the same. In such problems, geometric nonlinearities of the fluid-structure interface and significant exchange in the energy between the moving fluid and structure play important roles in the physical and mathematical description of the underlying biological problem. The problems are further exacerbated by the fact that the walls of major arteries are composed of several layers, each with different mechanical characteristics. No mathematical results exist so far that analyze existence of solutions to fluid-structure interaction problems in which the structure is composed of several different layers. In this talk we summarize the main difficulties in studying the underlying problem, and present a computational scheme based on which the existence of a weak solution to this class of FSI problems was obtained. Our results reveal a new physical regularizing mechanism in FSI problems: inertia of the fluid-structure interface with mass regularizes evolution of the FSI solution. This means that in our large (muscular) arteries, the inner-most layer of arterial walls, which consists of an elastic lamiae covered with endothelial cells, smooths out the propagation of the pressure wave in the cardiovasuclar system. All theoretical results will be illustrated with numerical examples.

This is a joint work with Boris Muha (University of Zagreb, Croatia), and with Martina Bukac (Notre Dame University).


Ioan Bejenaru, UC San Diego

Monday, April 18, 2016, KAP 414, 3:30 PM – 4:30 PM

Multilinear Restriction Theory

Abstract: I will introduce the linear and multilinear restriction theory and their relations with various fields in Mathematics: Harmonic Analysis, PDE, Number Theory, Incidence Geometry and Geometric Analysis. I will talk in more detail about the effect of the underlying geometry in the context of multilinear theory.


Mihaela Ifrim, UC Berkeley

Monday, April 25, 2016, KAP 414, 3:30 PM – 4:30 PM

Two dimensional water waves in holomorphic coordinates


This is joint work with Daniel Tataru, and in parts with Benjamin Harrop-Griffits and John Hunter. My talk is concerned with the irrotational infinite/finite depth water wave equations in two space dimensions, with either gravity or surface tension. I will also make some remarks on constant vorticity (infinite depth) case when only gravity is considered.

We consider this problem expressed in position-velocity potential holomorphic coordinates. Viewing this problem(s) as a quasilinear dispersive equation, we develop new methods which will be used to prove enhanced lifespan of solutions and also global solutions for small and localized data. For the gravity water waves there are several results available; they have been recently obtained by Wu, Alazard-Burq-Zuily and Ionescu-Pusateri using different coordinates and methods. In the capillary water waves and constant vorticity cases, we were the first to establish a global result, respectively a cubic lifespan existence of smooth and localized solutions. Our goal is to improve the understanding of these problems by providing a single setting for all the above cases, and presenting simpler proofs. The talk will try to be self contained.


Daniel Spielman, Yale University

Monday, May 9, 2016, KAP 414, 3:30 PM – 4:30 PM

Laplacian Matrices of Graphs: Algorithms and Applications


The Laplacian matrices of graphs arise in many fields including Machine Learning, Computer Vision, Optimization, Computational Science, and of course Network Analysis.  We will explain what these matrices are and why they arise in so many applications.  We then will survey recent progress on the design algorithms that allow us to solve such systems of linear equations in nearly linear time.

In particular, we will show how fast algorithms for graph sparsification directly lead to fast Laplacian system solvers.  As an application, we will explain how Laplacian system solvers can be used to quickly solve linear programming problems arising from natural graph problems.

Colloquia for the Fall 2015 Semester

Juhi Jang, USC

Monday, August 31, 2015, KAP 414, 3:30 PM – 4:30 PM

On the kinetic Fokker-Planck equation with absorbing barrier

Abstract: We discuss the well-posedness theory of classical solutions to the Kolmogorov equation, a simplest kinetic Fokker-Planck equation in bounded domains with absorbing boundary conditions. We show that the solutions are smooth up to the boundary away from the singular set and they are Holder continuous up to the singular set. This is joint work with H.J. Hwang, J. Jung and J.L. Velazquez.


Igor Kukavica, USC

Monday, September 14, 2015, KAP 414, 3:30 PM – 4:30 PM

The Euler equations with a free interface

Abstract: We address the local existence of solutions for the water wave problem. For the space dimensions three, we show that the local in time existence holds for initial velocities belonging to $H^{2.5+\delta}$, where $\delta>0$ is arbitrary, with the initial vorticity in $H^{2+\delta}$. The result is joint with A. Tuffaha and V. Vicol.


Career Panel Discussion

Monday, September 21, 2016, KAP 414, 3:30 PM – 4:30 PM

Panel discussion: “Planning your career: questions and advice”

Panelists: Andrea Appel, Eric Friedlander, Susan Montgomery and Stanislav Minsker

Moderator: Susan Friedlander

All graduate students and postdocs are encouraged to come and ask questions about positioning themselves for their future careers.


Mason Porter, Oxford

Monday, September 30, 2015, KAP 414, 3:30 PM – 4:30 PM

Multilayer Networks and Applications

Abstract: Networks provided a powerful representation of complex systems of interacting entities. One of the most active areas of network science, with an explosion of publications during the last few years, is the study of “multilayer networks,” in which heterogeneous types of entities can be connected via multiple social ties that change in time. Multilayer networks include multiple subsystems and “layers” of connectivity, and it is important to take such multilayer features into account to try to improve our understanding of complex systems. In this talk, I’ll give an overview of multilayer networks. I will introduce some ideas for how to find dense sets of nodes known as “communities” in multilayer networks and how this can lead to insights in applications such as political party realignment in voting networks and motor-task learning in functional brain networks. I will also discuss how to measure important nodes in multilayer networks, with an example describing the measurement of the quality of mathematics programs over time, and will end by presenting a few of the current challenges in the study of multilayer networks.


Geordie Richards, University of Rochester

Monday, October 5, 2015, KAP 414, 3:30 PM – 4:30 PM

Ergodicity Results for Stochastic Boussinesq Equations

Abstract: We will review some recent results on invariant measures for stochastic Boussinesq equations (model equations for Rayleigh-Benard convection perturbed by an additive noise). First we will discuss ergodicity and mixing results in the two-dimensional periodic domain with a spatially degenerate stochastic forcing. These results generalize recent progress of Hairer and Mattingly on hypoellipticity for infinite-dimensional systems. Then, with a less degenerate forcing but more physical boundary conditions, we present a simplified proof of ergodicity, and discuss some singular parameter limits.

This talk is based on joint works with Nathan Glatt-Holtz (Virginia Tech), Juraj Foldes (Universite Libre de Bruxelles) and Enrique Thomann (Oregon State University).


Juraj Földes, Université Libre de Bruxelles

Monday, October 7, 2015, KAP 414, 3:30 PM – 4:30 PM

Long term behaviour of maximal entropy solutions for 2D Euler equation

Abstract: Two dimensional turbulent flows for large Reynold’s numbers can be approximated by solutions of incompressible Euler’s equation. As time increases, the solutions of Euler’s equation are increasing their disorder; however, at the same time, they are limited by the existence of infinitely many invariants. Hence, it is natural to assume that the limit profiles are functions which maximize an entropy given the values of conserved quantities. Such solutions are described by methods of Statistical Mechanics and are called maximal entropy solutions. Nevertheless, there is no general agreement in the literature on what is the right notion of the entropy. We will show that on symmetric domains, independently of the choice of entropy, the maximal entropy solutions with small energy respect the geometry of the domain.
This is a joint work with Vladimír Šverák (University of Minnesota).


Stanley Osher, UCLA

(CAMS Distinguished Lecturer)

Monday, October 12, 2015, KAP 414, 3:30 PM – 4:30 PM

Algorithms for Overcoming the Curse of Dimensionality for Certain Hamilton-Jacobi Equations Arising in Control Theory and Elsewhere

Abstract: It is well known that time dependent Hamilton-Jacobi-Isaacs partial differential equations (HJ PDE) play an important role in analyzing continuous dynamic games and control theory problems. An important tool for such problems when they involve geometric motion is the level set method. The cost of these algorithms, and, in fact, all PDE numerical approximations is exponential in the space dimensions and time.

In this work we propose and test methods for solving a large class of HJ PDE without the use of grids or numerical approximations. For this wide class, which includes many linear control problems, we can obtain methods which are rapidly convergent, low memory, easily parallelizable and apparently very low complexity in dimension. We can evaluate the solution in many dimensions at between 10(-4) to 10(-8) seconds per evaluation on a laptop.

In addition, as a step needed in our procedure, we have developed a new and equally fast and efficient method to find the closest point xopt lying in the union of compact convex sets in Rn, (n large) to any point x exterior to this set.

The term “curse of dimensionality” was coined by Richard Bellman in 1957 when he considered problems in dynamic optimization.

Osher’s research interests include scientific computing, applied PDE, shock capturing methods, and image processing techniques.

Osher’s many honors and awards include membership of the National Academy of Sciences, Fellow of the American Academy of Arts and Sciences, Fellow of SIAM, Fellow of the AMS, honorary degrees from Hong Kong and ENS in Paris, the ICIAM Pioneer Prize and the SIAM Kleinman Prize. Most recently Osher received the Carl Friedrich Gauss Prize whose citation credited “his far ranging inventions that have changed our conception of physical, perceptual and mathematical concepts, giving us new tools to apprehend the world”.


Richard Schoen, Stanford University and UCI

Monday, October 19, 2015, KAP 414, 3:30 PM – 4:30 PM

Optimal geometries on surfaces

Abstract: The problem of finding surface geometries (metrics) of a given area which maximize their lowest eigenvalue has been studied for over 50 years. Despite some spectacular successes the problem is still not well understood for most surfaces. In this Colloquium, we will describe this question and the results which have been obtained including very recent progress.


Vlad Vicol, Princeton

Monday, November 2, 2015, KAP 414, 3:30 PM – 4:30 PM

The regularity of the 2D Muskat equations with finite slope

Abstract: We consider the 2D Muskat equation for the interface between two constant density fluids in an incompressible porous medium, with velocity given by Darcy’s law. We establish that as long as the slope of the interface between the two fluids remains bounded and uniformly continuous, the solution remains regular. We provide furthermore a global regularity result for small initial data: if the initial slope of the interface is sufficiently small, there exists a unique solution for all time. This is joint work with P. Constantin, R. Shvydkoy, and F. Gancedo.


Phil Holmes, Princeton University

Monday, November 9, 2015, RTH 217 , 3:00 PM – 4:00 PM

Moving Fast and Slow: Feedforward and feedback control in insect locomotion

Abstract: I will describe mathematical models for running insects, from an energy-conserving biped, through a muscle-actuated hexapod driven by a neural central pattern generator, to reduced phase-oscillator models that capture the dynamics of noisy gaits and external perturbations, and provide estimates of coupling strengths between legs. I will argue that both simple models and large simulations are necessary to understand biological systems, and end by describing some current experiments on fruit flies that cry out for new and improved models.


Natasa Pavlovic, University of Texas

Monday, November 16, 2015, KAP 414, 3:30 PM – 4:30 PM

From quantum many body systems to nonlinear dispersive PDE, and back


The derivation of nonlinear dispersive PDE, such as the nonlinear Schr\”{o}dinger (NLS) from many body quantum dynamics is a central topic in mathematical physics, which has been approached by many authors in a variety of ways. In particular, one way to derive NLS is via the Gross-Pitaevskii (GP) hierarchy, which is an infinite system of coupled linear non-homogeneous PDE that describes the dynamics of a gas of infinitely many interacting bosons, while at the same time retains some of the features of a dispersive PDE.

In the talk we will discuss the process of going from a quantum many body system of bosons to the NLS via the GP. The most involved part in such a derivation of NLS consists in establishing uniqueness of solutions to the GP, which was originally obtained by Erd\”os-Schlein-Yau. A key ingredient in their proof is a powerful combinatorial method that resolves the problem of the factorial growth of number of terms in iterated Duhamel expansions. In the talk we will focus on approaches to the uniqueness step that are motivated by the perspective coming from nonlinear dispersive PDE, including the approach of Klainerman-Machedon and the approach that we developed with Chen-Hainzl-Seiringer based on the quantum de Finetti’s theorem. Also we will look into what else the nonlinear PDE such as the NLS can tell us about the GP hierarchy and quantum many body systems, following results that we obtained with Chen, Chen-Tzirakis and Chen-Hainzl-Seiringer.

Colloquia for the Spring 2015 Semester

Sylvester Gates, University of Maryland

(CAMS Distinguished Lecturer)

Monday, January 26, 2015, KAP 414, 3:30 PM – 4:30 PM

How Attempting To Answer A Physics Question Led Me to Graph Theory, Error-Correcting Codes, Coxeter Algebras, and Algebraic Geometry

Abstract: We discuss how a still unsolved problem in the representation theory of Superstring/M-Theory has led to the discovery of previously unsuspected connections between diverse topics in mathematics.


Wilfrid Gangbo, Georgia Tech

Monday, February 2, 2015, KAP 414, 3:30 PM – 4:30 PM

Existence of a solution to an equation arising from Mean Field Games

Abstract: We construct a small time strong solution to a nonlocal Hamilton–Jacobi equation introduced by Lions, the so-called master equation, originating from the theory of Mean Field Games. We discover a link between metric viscosity solutions to local Hamilton–Jacobi equations studied independently by Ambrosio–Feng and G–Swiech, and the master equation. As a consequence we recover the existence of solutions to the First Order Mean Field Games equations, first proved by Lions. We make a more rigorous connection between the master equation and the Mean Field Games equations. (This talk is based on a joint work with A. Swiech).


Jerome Goldstein, University of Memphis

Monday, February 9, 2015, KAP 414, 3:30 PM – 4:30 PM

Energy asymptotics for dissipative waves

Abstract: Topics include sharp results on equipartition of energy, overdamping, and asymptotic parabolicity. These are for linear waves, and these problems have a long history, the newest being asymptotic parabolicity, which was born in G I Taylor’s 1922 paper. This is joint work with G. Reyes-Souto.


Mickael Chekroun, UCLA

Monday, March 9, 2015, KAP 414, 3:30 PM – 4:30 PM

Non-Markovian Reduced Equations for Stochastic PDEs

Abstract: In this talk, a novel approach to deal with the parameterization problem of the “small” spatial scales by the “large” ones for stochastic partial differential equations (SPDEs) will be discussed. This approach relies on stochastic parameterizing manifolds (PMs) which are random manifolds aiming to provide — in a mean square sense — approximate parameterizations of the small scales by the large ones. Backward-forward systems will be introduced to give access to such PMs as pullback limits depending — through the nonlinear terms — on (approximations of) the time-history of the dynamics on the low modes. These auxiliary systems will be used for the effective derivation of non-Markovian reduced stochastic differential equations from Markovian SPDEs. The non-Markovian effects are here exogenous in the sense that they result from the interactions between the external driving noise and the nonlinear terms, given a projection of the dynamics onto the modes with low wavenumbers. It will be shown that these non-Markovian terms allow in certain circumstances to restore in a striking way the missing information due to the low-mode projection, namely to parameterize what is not observed. Noise-induced large excursions or noise-induced transitions will serve as illustrations.


Geoffrey Spedding, USC A&ME

Monday, March 23, 2015, KAP 414, 3:30 PM – 4:30 PM

Wake Signature Detection

Abstract: The various regimes of strongly stratified flows have been studied extensively in theory, laboratory and numerical experiment. In the case of stratified, initially-turbulent wakes, the particular applications have drawn the research into high Froude and Reynolds number regimes (an internal Froude number is a ratio between timescales of turbulent motions vs. the restoring buoyancy forces, and a Reynolds number can be viewed as a ratio of timescales of advection vs. diffusion), that quite surprisingly have turned out to have rather general application. If, as seems likely, the conditions for making persistent flows with robust pattern are widespread, then we may consider the generation of, and search for, geometric pattern as being a phenomenon that is almost ubiquitous. Here we consider cases that range from island wakes that persist for more than 10,000 km to copepod tracks that have initial scales on the order of mm. Similarities and analogies will be noted in a somewhat qualitative fashion, in the hopes of inspiring future work.


Reception: Emmanuel Candes, CAMS Distinguished Lecturer

Monday, April 13, 2015, Gerontology Courtyard, 3:15 PM – 4:00


Emmanuel Candes, Stanford University

(CAMS Distinguished Lecturer, Joint with the Marshall School of Business)

Monday, April 13, 2015, Gerontology Auditorium, 4:00 PM – 5:00 PM

Around the Reproducibility of Scientific Research: A Knockoff Filter for Controlling the False Discovery Rate

Abstract: The big data era has created a new scientific paradigm: collect data first, ask questions later. When the universe of scientific hypotheses that are being examined simultaneously is not taken account, inferences are likely to be false. The consequence is that follow up studies are likely not to be able to reproduce earlier reported findings or discoveries. This reproducibility failure bears a substantial cost and this talk is about new statistical tools to address this issue. Imagine that we observe a response variable together with a large number of potential explanatory variables, and would like to be able to discover which variables are truly associated with the response. At the same time, we need to know that the false discovery rate (FDR)—the expected fraction of false discoveries among all discoveries—is not too high, in order to assure the scientist that most of the discoveries are indeed true and replicable. We introduce the knockoff filter, a new variable selection procedure controlling the FDR in the statistical linear model whenever there are at least as many observations as variables. This method achieves exact FDR control in finite sample settings no matter the design or covariates, the number of variables in the model, and the amplitudes of the unknown regression coefficients, and does not require any knowledge of the noise level. This work is joint with Rina Foygel Barber.


Yuri Tschinkel, Director of the MPS Division of the Simons Foundation and
Professor at the Courant Institute

Wednesday, April 15, 2015, KAP 414, 3:30 PM – 4:30 PM

Geometry of Numbers

Abstract: I will discuss Minkowski’s geometric ideas and their modern incarnations.


Yuri Tschinkel, Director of the MPS Division of the Simons Foundation and Professor at the Courant Institute

Wednesday, April 15, 2015, KAP 414, 4:45 PM – 5:30 PM

Simons Foundation Discussion

Abstract: The Simons Foundation Division for Mathematics and the Physical Sciences (MPS) seeks to extend the frontiers of basic research. The Division’s primary focus is on mathematics, theoretical physics and theoretical computer science. The division awards grants primarily through competitive, open, application-based procedures.


Anthony Suen, Hong Kong Institute of Education

Monday, April 27, 2015, KAP 414, 3:30 PM – 4:30 PM

Existence of intermediate weak solution to the equations of multi-dimensional chemotaxis systems

Abstract: We prove the global-in-time existence of intermediate weak solutions of the equations of chemotaxis system in a bounded domain of $\mathbb{R}^2$ or $\mathbb{R}^3$ with initial chemical concentration small in $H^1$. No smallness assumption is imposed on the initial cell density which is in $L^2$. We first show that when the initial chemical concentration $c_0$ is small only in $H^1$ and $(n_0-n_\infty,c_0)$ is smooth, the classical solution exists for all time. Then we construct weak solutions as limits of smooth solutions corresponding to mollified initial data. Finally we determine the asymptotic behavior of the global solutions.


Grace Wahba, University of Wisconsin

(CAMS Distinguished Lecturer)

Monday, May 4, 2015, KAP 414 3:30 PM – 4:30 PM

Learning Genetic Risk Models Using Distance Covariance

Abstract: We extend an approach suggested by Li, Zhong and Zhu (2012) to use distance covariance (DCOV) as a variable selection method by providing the DCOV Variable Selection Theorem, which gives a principled stopping rule for a greedy variable selection algorithm. We apply the resulting DCOV Variable Selection Method in two genetic based classification problems with small sample size and large vectors of gene expression data.

The first problem involves the well known SBRCT (Small Blue Round Cell Tumor) childhood Leukemia data, which involves gene expression data from four different types of Leukemia, and it is well known that these data are easy to classify.

The second involves Ovarian Cancer data from The Cancer Genome Atlas, and involves Ovarian Cancer patients that are either sensitive or resistant to a platinum based cancer chemotherapy. The Ovarian Cancer data presents a difficult classification problem.