Zoom link for all seminar meetings: https://usc.zoom.us/j/98591406199
Monday, January 9th,
3:30 - 4:30 pm, KAP 414
Title: 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.
Tuesday, January 17th,
2:00 - 3:00 pm, KAP 414
Title: 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.
Tuesday, January 17th,
2:00 - 3:00 pm, KAP 414
Title: 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.
Monday, January 23rd,
3:30 - 4:30 pm, KAP 414
Title: 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.
Monday, February 6th,
3:30 - 4:30 pm, KAP 414
Title: 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.
Monday, February 13th,
3:30 - 4:30 pm, KAP 414
Title: 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.
Monday, February 27th,
3:30 - 4:30 pm, KAP 414
Title: 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.
Monday, March 6th,
3:30 - 4:30 pm, KAP 414
Title: 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.
Monday, March 20th,
3:30 - 4:30 pm, KAP 414
Title: 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.
Monday, March 27th,
3:30 - 4:30 pm, KAP 414
Title: 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.
Monday, April 24th,
3:30 - 4:30 pm, KAP 414
Title: 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.