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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.

Colloquia for the Fall 2014 Semester

Monday, September 8 KAP 414 3:30 PM - 4:30 PM
Career Advice Panel

Panelists: Francis Bonahon, Eric Friedlander, Jason Fulman, Cymra Haskell, Paul Sobaje
Moderator: Kenneth Alexander

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

Anna Mazzucato
Penn State University
Monday, September 15 KAP 414 3:30 PM - 4:30 PM
Optimal mixing by incompressible flows

I will discuss mixing of passive scalars by incompressible flows and measures of optimal mixing. In particular, I will present recent results concerning examples of flows that achieve the optimal theoretical rate in the case of flows with prescribed energy or enstrophy budget. These examples are related to loss of regularity for solutions of transport equations.

Marco Sammartino
University of Palermo, visiting USC
Monday, October 6 KAP 414 3:30 PM - 4:30 PM
Navier-Stokes Equations in the Zero Viscosity Limit: Boundary Layers, Separation and Blow Ups

The appearance of a boundary layer (BL) is a ubiquitous phenomenon in applied mathematics: a BL occurs when the presence of a small parameter causes a sharp transition between the perturbed and the unperturbed regime. The concept of BL was introduced by Ludwig Prandtl to give an explanation to D'Alembert's paradox; Prandtl's 1904 paper would prove to be one of the most important fluid dynamics paper ever written.
However, despite more than a century of investigations, many problems raised by Prandtl's BL theory still remain unsolved. Among them we mention the lack of a fully satisfactory mathematical theory of Prandtl's equations and the problem of the convergence, in the zero viscosity limit, of the Navier-Stokes solutions to the Euler solutions.
In this talk after giving a review of some of the results that have been recently obtained in this area we shall consider an incompressible flow interacting with a boundary without assuming that the initial datum satisfies the no-slip condition at the boundary. A typical case when this situation occurs is the impulsively started disk. Other instances widely studied in the literature are when a vortical configuration, which is a steady solution of the Euler equations (like the thick core vortex or the vortex array), is assumed to interact instantaneously with a solid boundary.
Focusing our analysis on the Navier-Stokes equations on a half-space, we shall construct the initialboundary layer corrector in the form of a Prandtl solution with incompatible data. This corrector is the first term of an asymptotic series that we shall prove to approximate, in the zero viscosity limit and for a short time, the Navier-Stokes solutions. Assuming analytic regularity in the tangential direction, we shall prove that this time does not depend on the viscosity.

Nets Katz
Monday, October 13 KAP 414 3:30 PM - 4:30 PM
On the three dimensional Kakeya problem

We discuss new ideas for obtaining lower bounds on the Hausdorff dimension of Kakeya sets. We discuss joint work in progress with Josh Zahl.
Sometimes less is more.

Charles Doering
University of Michigan
Monday, October 20 KAP 414 3:30 PM - 4:30 PM
Wall to wall optimal transport

How much stuff can be transported by an incompressible flow containing a specified amount of kinetic energy or enstrophy? We study this problem for steady 2D flows focusing on passive tracer transport between two parallel impermeablewalls, employing the calculus of variations to find divergence-free velocity field with a given intensity budget that maximize transport between the walls. The maximizing velocity fields, i.e. the optimal flows, consist of arrays of (convection-like) cells. Results are reported in terms of the Nusselt number Nu, the convective enhancement of transport normalized by the flow-free diffusive transport, and the Peclet number Pe, the dimensionless gauge of the strength of the flow. For both energy and enstrophy constraints we find that as Pe increases, the maximum transport is achieved by cells of decreasing aspect ratio. For each of the two flow intensity constraints, we also consider buoyancy-driven flows the same constraint to see how the scalings for transport reported in the literature compare with the absolute upper bounds. This work provides new insight into both steady optimal transport and turbulent transport, an increasingly lively area of research in geophysical, astrophysical, and engineering fluid dynamics. This is joint work with Pedram Hassanzadeh
(Berkeley/Harvard) and Gregory P. Chini (University of New Hampshire) published in *Journal of Fluid Mechanics **751*, 627-662 (2014).

Tristan Buckmaster
Courant Institute
Monday, October 27 KAP 414 3:30 PM - 4:30 PM
Onsager's Conjecture

In 1949, Lars Onsager in his famous note on statistical hydrodynamics conjectured that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy; conversely, he conjectured the existence of solutions belonging to any Hölder space with exponent less than 1/3 which dissipate energy. The first part of this conjecture has since been confirmed (cf. Eyink 1994, Constantin, E and Titi 1994). During this talk we will discuss recent work by Camillo De Lellis, László Székelyhidi Jr., Phil Isett and myself related to resolving the second component of Onsager's conjecture. In particular, we will discuss the construction of weak non-conservative solutions to the Euler equations whose Hölder $1/3-\epsilon$ norm is Lebesgue integrable in time.

Michael Wolf
University of Zurich
Monday, November 3 KAP 414 3:30 PM - 4:30 PM
Spectrum Estimation: A Unified Framework for Covariance Matrix Estimation and PCA in Large Dimensions

Covariance matrix estimation and principal component analysis (PCA) are two cornerstones of multivariate analysis. Classic textbook solutions perform poorly when the dimension of the data is of a magnitude similar to the sample size, or even larger. In such settings, there is a common remedy for both statistical problems: nonlinear shrinkage of the eigenvalues of the sample covariance matrix. The optimal nonlinear shrinkage formula depends on unknown population quantities and is thus not available. It is, however, possible to consistently estimate an oracle nonlinear shrinkage, which is motivated on asymptotic grounds. A key tool to this end is consistent estimation of the set of eigenvalues of the population covariance matrix (also known as the spectrum), an interesting and challenging problem in its own right. Extensive Monte Carlo simulations demonstrate that our methods have desirable finite-sample properties and outperform previous proposals.

David Levermore
University of Maryland
Monday, November 10 KAP 414 3:30 PM - 4:30 PM
Scattering Theory for the Boltzmann Equation and the Arrow of Time (joint work with Claude Bardos, Irene Gamba, and Francois Golse)

We develop a scattering theory for a class of eternal solutions of the Boltzmann equation posed over all space. In three spatial dimensions each of these solutions has thirteen conserved quantities. The Boltzmann entropy has a unique minimizer with the same thirteen conserved values. This minimizer is a local Maxwellian that is also a global solution of the Boltzmann equation --- a so-called global Maxwellian. We show that each of our eternal solutions has a streaming asymptotic state as time goes to minus or plus infinity. However it does not converge to the associated global Maxwellian as time goes to infinity unless it is that global Maxwellian. The Boltzmann entropy decreases as time increases, but does not decrease to its minimum as time goes to infinity. Said another way, the final step in the traditional argument for the heat death of the universe is not valid.

Inwon Kim
Monday, November 17 KAP 414 3:30 PM - 4:30 PM
Congested crowd motion and Quasi-static evolution

In this talk we investigate the relationship between a quasi-static evolution and a transport equation with a drift potential, where the density is transported with a constraint on its maximum. The latter model, in a simplified setting, describes the congested crowd motion with a density constraint. When the drift potential is convex, the crowd density is likely to aggregate, and thus if the initial density starts as a patch (i.e. if it is a characteristic function of some set) then it is expected that the density evolves as a patch. We show that the evolving patch satisfies a Hele-Shaw type equation. We will also discuss preliminary results on general initial data.

Ngoc Mai Tran
University of Texas
Friday, November 21 KAP 414 3:00 PM - 4:00 PM
Special Time
Special Colloquium: Random permutations and random partitions

I will talk about various problems related to random permutations and random partitions. In particular, I discuss size-biased permutations, which have applications to statistical sampling. Then I will talk about random partitions obtained from projections of polytopes. These are related to random polytopes and zeros of random tropical polynomials.

Joseph Neeman
University of Texas
Friday, November 21 KAP 414 4:30 PM - 5:30 PM
Special Time
Special Colloquium: Gaussian noise stability

Given two correlated Gaussian vectors, X and Y, the noise stability of a set A is the probability that both X and Y fall in A. In 1985, C. Borell proved that half-spaces maximize the noise stability among all sets of a given Gaussian measure. We will give a new, and simpler, proof of this fact, along with some extensions and applications. Specifically, we will discuss hitting times for the Ornstein-Uhlenbeck process, and a noisy Gaussian analogue of the "double bubble" problem.