# Next Talk

Speaker: Pierre Germain The Mathematics and Physics of weak turbulence March 14, 1:00 p.m. (light refreshments at 12:45 p.m.) WWH 1302

#### Abstract

How does an infinite-dimensional Hamiltonian system evolve as time goes to infinity - think for instance of inviscid surface waves in a water tank? Mathematically, this fascinating question is very far from being understood - we are even lacking a good conjecture! In non-rigorous terms, an answer is provided by the theory of weak turbulence, which was partly confirmed by experiments. I will present some aspects of these questions.

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 Spring 2014

## January 31

Speaker: Miles Crosskey, Duke University Learning and Fast Simulation of Intrinsically Low-Dimensional Stochastic Dynamical Systems in High Dimensions Abstract When simulating multiscale stochastic differential equations (SDEs) in high-dimensions, separation of timescales and high-dimensionality can make simulations computationally very expensive. The size of time steps are dictated by the micro scale properties, while interesting behavior often occurs on the macro scale. This forces us to take many time steps in order to learn about the macro scale behavior. In this talk I will present a general framework for using micro scale simulations to automatically learn accurate macro scale models of certain SDEs. This method is particularly efficient when the SDE and the macro scale has low-intrinsic dimension, i.e. a small number of effective degrees of freedom. The learned macro scale model can then be used for fast computation and storage of long simulations. I will discuss various examples, both low- and high-dimensional, as well as results about the accuracy of the fast simulators we construct, and its dependency on the number of short paths of the original simulator available to the learning algorithm.

## February 14

Speaker: Michael L. Overton Investigation of Crouzeix's Conjecture via Optimization Abstract Crouzeix's conjecture is a fascinating open problem in matrix theory. We present a new approach to its investigation using optimization. Let $$p$$ be a polynomial of any degree and let $$A$$ be a square matrix of any order. Crouzeix's conjecture is the inequality $$\|p(A)| \leq 2 \|p\|_{W(A)}.$$ Here the left-hand side is the 2-norm of the matrix $$p(A)$$, while the norm on the right-hand side is the maximum of $$|p(z)|$$ over $$z\in W(A)$$, the field of values (or numerical range) of $$A$$. It is known that the conjecture holds if 2 is replaced by 11.08 (Crouzeix 2007). Joint work with Anne Greenbaum, Adrian S. Lewis and Lloyd N. Trefethen

## March 14

Speaker: Pierre Germain The Mathematics and Physics of weak turbulence Abstract How does an infinite-dimensional Hamiltonian system evolve as time goes to infinity - think for instance of inviscid surface waves in a water tank? Mathematically, this fascinating question is very far from being understood - we are even lacking a good conjecture! In non-rigorous terms, an answer is provided by the theory of weak turbulence, which was partly confirmed by experiments. I will present some aspects of these questions.

## March 28

Speaker: Katherine Newhall TBA Abstract TBA

## April 4

Speaker: Mehryar Mohri TBA Abstract TBA

## April 18

Speaker: Gérard Ben Arous TBA Abstract TBA

## April 25

Speaker: Daniel Stein TBA Abstract TBA

## May 2

Speaker: Leslie Greengard TBA Abstract TBA

# Contact Info

Aukosh Jagannathaukosh [at] cims [dot] nyu [dot] edu
Irena Vankovavankova [at] cims [dot] nyu [dot] edu
Klaus Widmayerklaus [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