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Colloquia for the Fall 2015 Semester

Juhi Jang
Monday, August 31 KAP 414 3:30 PM - 4:30 PM
On the kinetic Fokker-Planck equation with absorbing barrier

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
Monday, September 14 KAP 414 3:30 PM - 4:30 PM
The Euler equations with a free interface

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 KAP 414 3:30 PM - 4:30 PM
Career Panel Discussion: "Planning your career: questions and advice"

Panelists: Andrea Appel, Eric Friedlander, Susan Montgomery, 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
Wednesday, September 30 KAP 414 3:30 PM - 4:30 PM
Special Time
Multilayer Networks and Applications

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 KAP 414 3:30 PM - 4:30 PM
Ergodicity Results for Stochastic Boussinesq Equations

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
Wednesday, October 7 KAP 414 3:30 PM - 4:30 PM
Special Time
Long term behaviour of maximal entropy solutions for 2D Euler equation

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
Monday, October 12 KAP 414 3:30 PM - 4:30 PM
CAMS Distinguished Lecturer
Algorithms for Overcoming the Curse of Dimensionality for Certain Hamilton-Jacobi Equations Arising in Control Theory and Elsewhere

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 KAP 414 3:30 PM - 4:30 PM
Optimal geometries on surfaces

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
Monday, November 2 KAP 414 3:30 PM - 4:30 PM
The regularity of the 2D Muskat equations with finite slope

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 RTH 217 3:00 PM - 4:00 PM
Special Time & Location
Moving Fast and Slow: Feedforward and feedback control in insect locomotion

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 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
Monday, January 26 KAP 414 3:30 PM - 4:30 PM
CAMS Distinguished Lecturer
How Attempting To Answer A Physics Question Led Me to Graph Theory, Error-Correcting Codes, Coxeter Algebras, and Algebraic Geometry

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 KAP 414 3:30 PM - 4:30 PM
Existence of a solution to an equation arising from Mean Field Games

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 KAP 414 3:30 PM - 4:30 PM
Energy asymptotics for dissipative waves

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
Monday, March 9 KAP 414 3:30 PM - 4:30 PM
Non-Markovian Reduced Equations for Stochastic PDEs

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
Monday, March 23 KAP 414 3:30 PM - 4:30 PM
Wake Signature Detection

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

Monday, April 13 Gerontology Courtyard 3:15 PM - 4:00 PM
CAMS Distinguished Lecturer
You are cordially invited to attend

Emmanuel Candes
Stanford University, Joint with the Marshall School of Business
Monday, April 13 Gerontology Auditorium 4:00 PM - 5:00 PM
CAMS Distinguished Lecturer
Around the Reproducibility of Scientific Research: A Knockoff Filter for Controlling the False Discovery Rate

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 KAP 414 4:45 PM - 5:30 PM
Special Time
Simons Foundation Discussion

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.

Yuri Tschinkel
Director of the MPS Division of the Simons Foundation and Professor at the Courant Institute
Wednesday, April 15 KAP 414 3:30 PM - 4:30 PM
Special Time
Geometry of Numbers

I will discuss Minkowski's geometric ideas and their modern incarnations.

Anthony Suen
Hong Kong Institute of Education
Monday, April 27 KAP 414 3:30 PM - 4:30 PM
Existence of intermediate weak solution to the equations of multi-dimensional chemotaxis systems

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
Monday, May 4 KAP 414 3:30 PM - 4:30 PM
CAMS Distinguished Lecturer
Learning Genetic Risk Models Using Distance Covariance

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.