Next Talk
 Speaker: 

 Title: 

 Date and time: 
December 31, 1:00 p.m. (light refreshments at 12:45 p.m.) 
 Venue: 

Abstract
This seminar is meant to benefit young mathematicians, particularly graduate students and postdocs.
It aims to accomplish the following:
 provide a venue for talks that young mathematicians will understand
 expose students to areas of research at the Courant Institute
The research talks should be fairly introductory and accessible to students and nonspecialists in the audience.
Schedule Fall 2014
October 10
 Speaker: 
Tristan Buckmaster 
 Title:  Onsager's Conjecture 

 
Abstract
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., Philip Isett and myself related to resolving the second component of Onsager's conjecture. In particular, we will discuss the construction of weak nonconservative solutions to the Euler equations whose Hölder 1/3epsilon norm is Lebesgue integrable in time. 

October 17
 Speaker: 
Paul Bourgade 
 Title:  Universality in random matrix theory 

 
Abstract
Eugene Wigner stated the general hypothesis that the distribution of eigenvalue spacings of large complicated quantum systems is universal, in the sense that it depends only on the symmetry class of the physical system but not on other detailed structures. The simplest case for this hypothesis concerns large but finite dimensional matrices. I will explain some historical aspects random matrix theory, as well as recent techniques developed to prove eigenvalues and eigenvectors universality, for matrices with independent entries from all symmetry classes. The methods are both probabilist (random walks and coupling) and analytic (homogenization for parabolic PDEs with random coefficients). 

October 24
 Speaker: 
Michael O'Neil 
 Title:  Fast algorithms for Gaussian processes 

 
Abstract
Gaussian processes are one class of stochastic process that provides a straightforward generalization of random variables to random functions. These random functions are specified via a pointwise multivariate normal distribution whose variance structure is described using a covariance kernel. Although their use in mathematical statistics, machine learning, and data analysis for regression and classification problems has been ubiquitous, their computational cost has been dominated by dense linear algebra operations (inverses, applications, and determinants) which scale as O(n^2) or O(n^3), where n is the number of pointwise evaluations (or observed evaluations) of the random function. However, by drawing on many ideas central to fast multipole methods, it is possible to design schemes which scale as O(nlogn) or O(n) for many widelyused covariance kernels. In this talk, we will give a brief overview of Gaussian processes, their many applications, and demonstrate the performance of these fast schemes via several numerical examples. 

November 7
 Speaker: 
Georg Stadler 
 Title:  Inverse problems governed by largescale PDE models 

 
Abstract
Inverse problems are a framework to convert measurements into information about a system, which cannot be observed directly. They are at the base of medical imaging, play an important role in weather prediction and in the detection of airplanes and submarines, and are the reason why we know a lot about how our planet looks inside. If the underlying mechanism is mathematically described by a partial differential equation, these problems can become computationally very challenging. As an example, I will show results for a timedomain seismic inverse problem. Due to the size of this problem, a parallel wave simulation code is used, and I will discuss some of the scalability, parallelism, and highperformance computing issues underlying this implementation. 

November 14
 Speaker: 
Alex Mogilner 
 Title:  Cell movement as a free boundary problem 

 
Abstract
Cell migration is one of the fundamental biological phenomena underlying development, wound healing, cancer and immune response. biophysics and biochemistry of cell movement has been actively modeled. I will introduce a model of the cell as a contractile and polymerizing viscoelastic gel with a free boundary. Mechanics of such gel can be in the simplest case described by just two coupled PDEs, but those have to be solved on a free boundary domain. I will show how this model explains both steady motility, and cell turning and symmetry breaking. 

December 5
 Speaker: 
Yuri Bakhtin 
 Title:  Noisy heteroclinic networks and sequential decision making 

 
Abstract
I will talk about small noisy perturbations of systems with multiple saddletype equilibrium points. The goal is to give a precise description of the asymptotic behavior in the limit of vanishing noise and interpret the results in terms of sequential decision making. I will discuss applications to neural systems and psychology. 

If you would like to give a talk or ask a question about the seminar,
please contact one of the seminar organizers:
Previous semesters
Descriptions of earlier talks are
here.
Department of Mathematics
Courant Institute of Mathematical Sciences
New York University
251 Mercer St.
New York, NY 10012