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 Colloquia for the Summer 2015 Semester Grace Wahba University of Wisconsin Monday, May 4 KAP 414 3:30 PM - 4:30 PM CAMS Distinguished Lecturer To be Announced 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 UCLA Monday, March 9 KAP 414 3:30 PM - 4:30 PM To be Announced 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, 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. Emmanuel Candes Stanford University, Joint with the Marshall School of Business Monday, April 13 KAP 414 3:30 PM - 4:30 PM CAMS Distinguished Lecturer To be Announced Igor Kukavica USC Monday, April 27 KAP 414 3:30 PM - 4:30 PM To be Announced 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 SobajeModerator: Kenneth AlexanderAll 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 Caltech 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 UCLA 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. Colloquia for the Summer 2014 Semester Thanasis Fokas Cambridge University Thursday, May 8 KAP 414 3:30 PM - 4:30 PM Boundary Value Problems and Medical Imaging In the late 60s a new area emerged in mathematical physics known as "Integrable Systems". Ideas and techniques of "Integrability" have had a significant impact in several areas of mathematics, science and engineering, from the proof of the Schottky problem in algebraic geometry, to optical communications. In this lecture, two such implications will be reviewed: (a) A novel method for analysing boundary value problems, which unifies the fundamental contributions to the analytical solution of PDEs of Fourier, Cauchy and Green, and also constructs a non-linearization of some of these results. This method has led to the emergence of new numerical techniques for solving linear elliptic PDEs in polygonal domains. (b) A new approach for solving the inverse problems arising in certain important medical imaging techniques, including Single Photon Emission Computerised Tomography (SPECT).
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