# Next Talk

Speaker: Rodrigo Trevino TBA December 11, 1:00 p.m. (light refreshments at 12:45 p.m.) WWH 1302

#### Abstract

TBA

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 non-specialists in the audience.

# Schedule Fall 2015

## October 9

Speaker: Charles S. Peskin Fluid-Structure Interaction by the Immersed Boundary Method Abstract An example of an immersed boundary is a heart valve leaflet. These thin membranes are immersed in blood and move at the local fluid velocity while they simultaneously apply forces to the surrounding fluid that prevent backflow when the valve is closed, and shear the forward flow when the valve is open to create vortices that subsequently promote efficient valve closure. The immersed boundary (IB) method was created to deal with this problem, and has since grown into a generally useful tool for the computer simulation of fluid-structure interaction, especially in biology. In this talk I'll introduce the IB method in the context of the heart, and then show movies that illustrate various ways in which the IB method has been generalized and applied.

## October 23

Speaker: Ofer Zeitouni Extremal processes for some Gaussian random fields Abstract Consider a (locally finite) random configuration ${\bf X}=\{X_i\}$ with values in $R$ (such a collection is called a \textit{point process}). Let $Z_i$ be i.i.d. random variables, independent of ${\bf X}$. Assume that the distribution of ${\bf X}$ is invariant under the transformation $X_i\mapsto X_i+Z_i$. Liggett has identified all possible such distributions, as mixtures of Poisson processes with (constant or exponential) intensities. Recently, this identification has played an important role in identifying the point process of extremes of certain Gaussian random fields. I will describe Liggett's results and explain how it is applied to the study of such extremes, as well as applications to the study of the Gaussian free field and to certain spin-glass systems. All terms will be defined in the talk.

## October 30

Speaker: Jonathan Goodman Monte Carlo sampling for Bayesian statistics with complex models: theory, algorithms, applications. Abstract A basic problem in statistics is to make inferences about physical parameters from experimental data. Because of measurement and modeling errors, data do not determine parameter values exactly. Bayesian statistics represents the remaining parameter uncertainty as a “posterior” probability distribution of the parameters conditional on the measurements and prior information. For high dimensional problems, it is hard to represent this posterior except as a collection of random samples from it. This talk discusses the problem of producing such samples. We describe MCMC ( Markov chain Monte Carlo) samplers, which are guaranteed to “work” in principle but may be too slow in practical applications. There are several theoretical approaches to understanding the convergence of MCMC samplers, including Sobolev and log Sovolev inequalities, the beautiful methods of the Lovasz school (which are based on “Cheeger’s inequality”), and work of Hairer and Stuart. Each of these has led to better samplers for specific problems. Some of my work is motivated by an analogy between MCMC samplers and optimization algorithms. One idea from optimization is that a good method for generic problems should be invariant under affine transformations. This allows the method to work well for poorly conditioned problems. I present two affine invariant sampling algorithms, one of which is the basis of a popular software package that will be described later this afternoon by David Hogg. Line search is another method from optimization that we have imported to MCMC samplers. I will list several open problems and opportunities.

## November 6

Speaker: Benjamin Harrop-Griffiths Modified scattering for the modified Korteweg-de Vries equation (mKdV) Abstract The Korteweg-de Vries (KdV) equation arises as an asymptotic limit of numerous dispersive systems and together with its generalizations has a wide range of physical applications including fluid mechanics, plasma physics and nonlinear optics. In this talk we will provide a brief introduction to the generalized KdV family of equations with particular emphasis on their asymptotic behavior. In particular we will discuss why solutions to the modified KdV (mKdV) do not scatter to linear waves, even for small initial data and will sketch a proof of modified scattering using the method of testing by wave packets. This robust approach does not rely on the inverse scattering transform and hence may be applied to short range perturbations of the mKdV as well as numerous other non-integrable equations.

## November 13

Speaker: Tom Trogdon Corner singularities, Gibbs phenomenon and the Unified Transform Method Abstract Abstract: Consider solving a linear, constant-coefficient evolution PDE in one spatial dimension where the initial data vanishes on the negative half line (x < 0). One can interpret this solution, restricted to x > 0, t > 0, as the solution of an initial-boundary value problem where the boundary data is not compatible with the initial data. This solution exhibits a corner singularity. Furthermore, in a dispersive and non-dissipative setting such a solution typically exhibits Gibbs-like high-oscillation and non-vanishing overshoot as t tends to zero. In this talk, I will discuss the explicit solution of general (1+1)-dimensional initial-boundary value problems using the so-called Unified Transform Method, the behavior of corner singularities and their relation to the classical Gibbs phenomenon. I will also discuss the computation of these singular solutions.

## November 20

Speaker: Alena Pirutka Rationality: reasonable or not Abstract An algebraic variety is rational if it has an open subset isomorphic to an open of an affine space. However, these varieties are still quite difficult to understand. For instance, it could be very tricky to determine whether a given variety is rational or not. In this talk we will discuss various properties related to the rationality, provide examples where these properties hold and give some ideas on methods used for these questions.

## December 11

Speaker: Rodrigo Trevino TBA Abstract TBA

# Contact Info

Monty Essidessid [at] cims [dot] nyu [dot] edu
Alex Kaiserkaiser [at] cims [dot] nyu [dot] edu
Reza Gheissarireza [at] cims [dot] nyu [dot] edu

## Previous semesters

### Spring 2011 schedule

Descriptions of earlier talks are here.

Department of Mathematics
Courant Institute of Mathematical Sciences
New York University
251 Mercer St.
New York, NY 10012