CAMS Distinguished Colloquia for the Fall 2023 Semester
Eitan Tadmor (University of Maryland, College Park), Monday, November 13th, KAP 414, 3:30 PM – 4:30 PM |
We discuss a novel swarm-based gradient descent (SBGD) method for non-convex optimization. The swarm consists of agents identified with positions, x, and masses, m. There are three key aspects to the SBGD dynamics. (i) communication: persistent transition of mass from high to lower ground; (ii) marching: mass-dependent marching in directions randomly aligned with gradient descent; and (iii) time stepping protocol which decreases with m.The interplay between positions and masses leads to dynamic distinction between `leaders’ and `explorers’: heavier agents lead the swarm near local minima with small time steps; lighter agents use larger time steps to explore the landscape in search of improved global minimum, by reducing the overall ‘loss’ of the swarm.Convergence analysis and numerical simulations demonstrate the effectiveness of SBGD method as a global optimizer. |
CAMS Distinguished Colloquia for the Spring 2023 Semester
Tomasz Mrowka (Massachusetts Institute of Technology), Tuesday, January 17th, KAP 414, 2:00 PM-3:00 PM |
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 (Massachusetts Institute of Technology), Tuesday, January 17th, KAP 414, 3:30 PM-4:30 PM |
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. |
Svetlana Jitomirskaya (UCI and Georgia Tech), Monday, April 24th, KAP 414, 3:30 PM-4:30 PM |
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. |
CAMS Distinguished Colloquia for the Fall 2019 Semester
Tadashi Tokieda (Stanford University), Monday, September 9th, Irani Hall 101, 3:30 PM – 4:30 PM |
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. |
Terence Tao (The James and Carol Collins Chair in Mathematics at UCLA), Monday, September 16th, Irani Hall 101, 3:30 PM – 4:30 PM |
We survey some recent developments towards the infamous global regularity problem for the Navier-Stokes equations for incompressible viscous fluids. |
CAMS Distinguished Colloquia for the Spring 2019 Semester
Susan Holmes (Professor of Statistics, Stanford University) Monday, February 25th, Irani Hall 101 3:30 – 4:30 pm. |
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. |
CAMS Distinguished Colloquia for the Spring 2018 Semester
Marcelo Viana (Director, Institute Nacional de Mathematica Pure e Aplicada), Wednesday, January 17th, KAP 410, 3:30 PM – 4:30 PM |
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. |
CAMS Distinguished Colloquia for the Fall 2017 Semester
H. Thomas Banks (N.C. State University), Monday, October 23rd, KAP 414, 3:30 PM -4:30 PM |
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. |
CAMS Distinguished Colloquia for the Spring 2017 Semester
Michael Aizenman (Princeton University), Monday, April 3rd, KAP 414, 3:30 PM -4:30 PM |
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. |
CAMS Distinguished Colloquia for the Fall 2016 Semester
Linda Petzold (UC, Santa Barbara), Monday, November 28th, KAP 414, 3:30 PM -4:30 PM |
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. |
CAMS Distinguished Colloquia for the Spring 2016 Semester
Daniel Spielman (Yale University), Monday, May 9th, KAP 414, 3:30 PM -4:30 PM |
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. |
CAMS Distinguished Colloquia for the Fall 2015 Semester
Stanley Osher (UCLA), Monday, October 12th, KAP 414, 3:30 PM -4:30 PM |
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 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”. |
CAMS Distinguished Colloquia for the Spring 2015 Semester
Sylvester Gates (University of Maryland), Monday, January 26th, KAP 414, 3:30 PM -4:30 PM |
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. |
Emmanuel Candes (Stanford University, Joint with the Marshall School of Business), Monday,April 13th, Gerontology Auditorium 4:00 PM – 5:00 PM |
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. |
Grace Wahba (University of Wisconsin), Monday, May 4th, KAP 414, 3:30 PM -4:30 PM |
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. |
CAMS Distinguished Colloquia for the Spring 2014 Semester
Luis Caffarelli (UT Austin), Monday, March 3rd, KAP 414, 3:30 PM -4:30 PM |
In this lecture I will review work that concerns the behavior of surfaces and fronts in a periodic media that is highly oscillatory: minimal surfaces, whose area is weighted by a periodic factor, capillary drops sitting in a composite surface, the effective speed of flame propagation in periodic media. |
CAMS Distinguished Colloquia for the Fall 2013 Semester
Ruth Williams (UCSD), Monday, October 21st, KAP 414, 3:30 PM -4:30 PM |
Stochastic models of processing networks arise in a wide variety of applications in science and engineering, e.g., in high-tech manufacturing, transportation, telecommunications, computer systems, customer service systems, and biochemical reaction networks. These “stochastic processing networks” typically have entities, such as jobs, vehicles, packets, customers or molecules, that move along paths or routes, receive processing from various resources, and that are subject to the effects of stochastic variability through such variables as arrival times, processing times and routing protocols. Networks arising in modern applications are often heterogeneous in that different entities share (i.e., compete for) common network resources. Frequently the processing capacity of resources is limited and there are bottlenecks, resulting in congestion and delay due to entities waiting for processing. The control and analysis of such networks present challenging mathematical problems. This talk will explore the effects of resource sharing in stochastic networks and describe associated mathematical analysis based on elegant fluid and diffusion approximations. Illustrative examples will be drawn from biology and telecommunications. |
CAMS Distinguished Colloquia for the Spring 2013 Semester
Ronald Graham, Monday, March 4th, KAP 414, 2:00 PM -3:00 PM, Special Time |
The mystery of magic and the art of juggling have surprising links to interesting ideas from mathematics. In this talk, I will illustrate some of these connections. |
Fan Chung Graham (UC San Diego), Monday, March 4th, KAP 414, 3:30 PM -4:30 PM |
We will discuss some recent developments in several new directions of spectral graph theory, including random walks for directed graphs, ranking algorithms, graph gauge theory, network games, graph limits and graphlets, for example. |
CAMS Distinguished Colloquia for the Fall 2012 Semester
Bradley Efron (Standford University), Monday, October 15th, GFS 106, 3:45 PM -4:45 PM, Special Time & Location |
Bayesian methods require a catalog of prior experience for the interpretation of statistical evidence. In the absence of prior information, empirical Bayes methods rely instead on a catalog of cases similar to the problem of interest. The crime rate in one small city, for example, may be estimated by modifying its observed rate with evidence from other cities. I will give some examples that show how powerful the empirical Bayes approach can be in practice, both for estimation and testing. The use of “other” cases then raises the question of just which others are relevant, and how their information bears on the case of interest. |
CAMS Distinguished Colloquia for the Spring 2012 Semester
Edward Witten (Institute for Advanced Study ), Wednesday, March 28th, SAL 101, 3:45 PM -4:45 PM, Special Time & Location |
In this talk, I will sketch a new approach to Khovanov homology of knots and links based on counting the solutions of certain elliptic partial differential equations in four and five dimensions. The equations are formulated on four and five-dimensional manifolds with boundary, with a rather subtle boundary condition that encodes the knots and links. The construction is formally analogous to Floer and Donaldson theory in three and four dimensions. It was discovered using quantum field theory arguments but can be described and understood purely in terms of classical gauge theory. |
CAMS Distinguished Colloquia for the Fall 2011 Semester
Nancy Kopell (Boston University), Monday, Dec 5th, GER Auditorium, 3:30 PM -4:30 PM, Special Location |
Rhythms of the nervous system are produced in all cognitive states, and have been shown to be highly associated with a myriad of cognitive tasks. Thus, changes in these rhythms, however they come about, are likely to change the ability to do such tasks. This talk focuses on the beta (12-30 Hz) and alpha (9-11) rhythms, and pathological states due to anesthesia and PD; it is about three related studies, the latter two emerging from the first one. The first concerns an early stage of anesthesia, in which, paradoxically, the subject gets more excited and disoriented. With low propofol, the brain rhythms show an increase in beta oscillations, which in normal awake state is associated with brain functions including motor preparation and higher-order processing. The second concerns the beta oscillations associated with abnormal motor control in Parkinson’s disease. The relationship between the two phenomena can be seen from the underlying physiology using modeling as well as experiments. Finally, the first story led to looking at higher doses of propofol at which consciousness is lost, and uses experimental data to get new ideas about the physiological basis for the loss of consciousness. Again, the focus on relevant physiology of the rhythms is what led, though modeling, to the new insights. Applications to other states of consciousness, such as coma, may be discussed. |
CAMS Distinguished Colloquia for the Spring 2011 Semester
Dennis Sullivan (SUNY and CUNY Graduate Center ), Monday, March 7th, KAP 414, 3:30 PM -4:30 PM |
If one has an AT (Algebraic Topology) model of a system of fields and operations in Riemannian geometry, there is a natural way to construct derived models at each scale of resolution. In addition there are transition mappings between these derived models at different scales.The process of constructing derived models is based on the key idea of AT: chain homotopy equivalences between chain complexes. If a nonlinear PDE among the original system of fields and operations can be reformulated in the derived models, one can obtain a system of finite energy or finite scale models which are correlated by structure mappings. Incompressible Navier Stokes evolution in 3D can be described by the differential algebra of differential forms, the Hodge star operator and the projections of the Hodge decomposition. These objects are naturally interpreted in AT. The lecture will discuss this AT approach to deriving computational fluid models. |
CAMS Distinguished Colloquia for the Fall 2010 Semester
Charles Fefferman (Princeton University), Friday, December 3rd, KAP 414 3:30 PM – 4:30 PM |
The problem concerns the evolution of the interfaces between two or more fluids in a porous medium. The talk presents new phenomena arising when at least three fluids are present. (Joint work with several coauthors) |
CAMS Distinguished Colloquia for the Spring 2010 Semester
James Glimm, Monday, March 22nd, Andrus Gerontology Center, 4:00 PM – 5:00 PM |
Turbulent mixing is an important aspect of a number of practical problems, often combined with combustion or some other reaction. Due to the importance of this problem, considerable effort has been invested in verification (mathematical correctness of numerical solutions) and validation (correctness and applicability of the equations to be solved). A standard test problem of this class is Rayleigh-Taylor mixing, the problem of a heavy fluid over a light one, mixing under the acceleration force of gravity. |
CAMS Distinguished Colloquia for the Fall 2009 Semester
Margaret Wright, (Courant Institute, NYU), Friday, October 9th, Andrus Gerontology Center, 3:30 PM – 4:30 PM |
Non-derivative methods for optimization have had a sometimes rocky relationship for more than 50 years with applied mathematicians who specialize in optimization. Although practitioners have never wavered in their fondness for non-derivative methods, their mathematical foundations were mostly lacking until the late 1980s. Since then, significant progress has been made concerning theoretical underpinnings, but several perplexing mysteries remain. In addition, there has been continuing and lively controversy about which methods are “most effective” on real-world applications, with disagreements about both the selection of test problems and the choice of criteria for assessing computational results. This talk will briefly survey the current state of the art, trying along the way to highlight a few of the interesting open questions. |
CAMS Distinguished Colloquia for the Spring 2009 Semester
Terence Tao (UCLA), Joint with the Whiteman Lectures, Thursday, February 19th, Gerentology Auditorium, 3:30 PM – 4:30 PM |
Suppose one wants to recover an unknown signal x in Rn from a given vector Ax=b in Rm of linear measurements of the signal x. If the number of measurements m is less than the degrees of freedom n of the signal, then the problem is underdetermined and the solution x is not unique. However, if we also know that x is sparse or compressible with respect to some basis, then it is a remarkable fact that (given some assumptions on the measurement matrix A) we can reconstruct x from the measurements b with high accuracy, and in some cases with perfect accuracy. Furthermore, the algorithm for performing the reconstruction is computationally feasible. This observation underlies the newly developing field of compressed sensing. In this talk we will discuss some of the mathematical foundations of this. |
George Papanocolauo (Stanford University), Friday, April 17, Gerentology Auditorium, 3:30 PM – 4:30 PM |
It is somewhat surprising at first that it is possible to locate a network of sensors from cross correlations of noise signals that they record. This is assuming that the speed of propagation in the ambient environment is known and that the noise sources are sufficiently diverse. If the sensor locations are known and the propagation speed is not known then it can be estimated from cross correlation information. Although a basic understanding of these possibilities had been available for some time, it is the success of recent applications in seismology that have revealed the great potential of correlation methods, passive sensors and the constructive use of ambient noise in imaging. I will introduce these ideas in an interdisciplinary, mathematical way and show that a great deal can be done with them. Things become more complicated, and a mathematically more interesting, when the ambient medium is also strongly scattering. I will end with a review of what is known so far in this case, and what might be expected. |
CAMS Distinguished Colloquia for the Fall 2008 Semester
Perci Diaconis (Stanford University), Friday, October 24th, KAP 249, 3:30 PM – 4:30 PM |
The usual process of “carries” when adding numbers turns out to have interesting mathematics hidden in it. It begins with an “amazing” matrix discovered by Holte, which has close connections to the usual way of mixing cards by riffle shuffling. The connections give new results for addition and for shuffling. This is joint work with Jason Fulman. |