Colloquia for the Fall 2015 Semester


Juhi Jang
USC
Monday, August 31
KAP 414
3:30 PM  4:30 PM

On the kinetic FokkerPlanck equation with absorbing barrier
We discuss the wellposedness theory of classical solutions to the Kolmogorov equation, a simplest kinetic FokkerPlanck 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
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
Oxford
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 motortask 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 RayleighBenard convection perturbed by an additive noise). First we will discuss ergodicity and mixing results in the twodimensional periodic domain with a spatially degenerate stochastic forcing. These results generalize recent progress of Hairer and Mattingly on hypoellipticity for infinitedimensional 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 GlattHoltz (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
UCLA
Monday, October 12
KAP 414
3:30 PM  4:30 PM
CAMS Distinguished Lecturer

Algorithms for Overcoming the Curse of Dimensionality for Certain HamiltonJacobi Equations Arising in Control Theory and Elsewhere
It is well known that time dependent HamiltonJacobiIsaacs 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
Princeton
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 energyconserving biped, through a muscleactuated hexapod driven by a neural central pattern generator, to reduced phaseoscillator 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 GrossPitaevskii (GP) hierarchy, which is an infinite system of coupled linear nonhomogeneous 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\"osSchleinYau. 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 KlainermanMachedon and the approach that we developed with ChenHainzlSeiringer 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, ChenTzirakis and ChenHainzlSeiringer.


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, ErrorCorrecting Codes, Coxeter Algebras, and Algebraic Geometry
We discuss how a still unsolved problem in the representation theory of Superstring/MTheory 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 socalled 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. ReyesSouto.



Mickael Chekroun
UCLA
Monday, March 9
KAP 414
3:30 PM  4:30 PM

NonMarkovian 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. Backwardforward systems will be introduced to give access to such PMs as pullback limits depending — through the nonlinear terms — on (approximations of) the timehistory of the dynamics on the low modes. These auxiliary systems will be used for the effective derivation of nonMarkovian reduced stochastic differential equations from Markovian SPDEs. The nonMarkovian 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 nonMarkovian terms allow in certain circumstances to restore in a striking way the missing information due to the lowmode projection, namely to parameterize what is not observed. Noiseinduced large excursions or noiseinduced transitions will serve as illustrations.



Geoffrey Spedding
USC A&ME
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, initiallyturbulent 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 discoveriesis 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, applicationbased 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 multidimensional chemotaxis systems
We prove the globalintime 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 $(n_0n_\infty,c_0)$ 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.

