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 Spring 2014
January 31
 Speaker: 
Miles Crosskey, Duke University 
 Title:  Learning and Fast Simulation of Intrinsically LowDimensional Stochastic Dynamical Systems in High Dimensions 

 
Abstract
When simulating multiscale stochastic differential equations (SDEs) in highdimensions, separation of timescales and highdimensionality 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 lowintrinsic 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 highdimensional, 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 
 Title:  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 lefthand side is the 2norm of the matrix \(p(A)\), while the
norm on the righthand 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 
 Title:  The Mathematics and Physics of weak turbulence 

 
Abstract
How does an infinitedimensional 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 nonrigorous 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 
 Title:  Dynamics of ferromagnets: averaging methods, bifurcation diagrams, and thermal noise effects 

 
Abstract
Driving nanomagnets by spinpolarized currents offers exciting prospects in magnetoelectronics, but the response of the magnet to such currents remains poorly understood. For a single domain ferromagnet, I will show that an averaged equation describing the diffusion of energy on a graph captures the lowdamping dynamics of these systems. In particular, I compute the mean times of thermally assisted magnetization reversals in the finite temperature system, giving explicit expressions for the effective energy barriers conjectured to exist. I will then discuss the problem of extending the analysis to spatially nonuniform magnets, leading to a transition state theory for infinite dimensional Hamiltonian systems. 

April 4
 Speaker: 
Mehryar Mohri 
 Title:  MultipleSource Adaptation Problem 

 
Abstract
The problem of adaptation is one of the most important problems in
modern machine learning since, while massive data sets are commonly
accessible, sample points often do not follow the same distribution.
I will discuss some solutions for the problem with remarkable
properties and point out some interesting open algorithmic
problems. Next I will further extend the theory and learning
guarantees using the notion of Renyi divergence.
This is joint work with Y. Mansour and A. Rostami. 

April 18
 Speaker: 
Zahra Sinaei 
 Title:  Convergence of harmonic maps 

 
Abstract
In this talk I will present a compactness theorem for a sequence of harmonic maps which are defined on a converging sequence of Riemannian manifolds. The sequence of manifolds will be considered in the space of compact ndimensional Riemannian manifolds with bounded sectional curvature and bounded diameter, equipped with measured GromovHausdorff topology. 

April 25
 Speaker: 
Daniel L. Stein 
 Title:  Order, Disorder, Symmetry and Complexity 

 
Abstract
One of the deepest scientific questions we can ask is, How might complexity arise? That is, starting from simple, undirected processes subject to physical and chemical laws, how could structures with complex
shapes and patterns arise, and even more perplexing, what processes could give rise to living cells, and how might they then organize themselves into
complex organisms, leading ultimately to such things as brains, consciousness, and societies? We are far from answering these questions at almost any level, but they have attracted increasing attention in the scientific community, and some initial headway has been made. The basic problem can be reframed as one involving the selforganization of microscopic constituents into larger assemblies, in such a way that the process leads to an increase of information, the creation of new patterns, and eventually increasing hierarchical levels of complex structure. The key to understanding these processes cannot be found in any single (natural or social) scientific field but rather in collaborations that cross many disciplinary boundaries.Although we are still at the initial stages of inquiry, new and interesting approaches and points of view have arisen. In this talk I present one that arises from the point of view of physics. We start by describing the (wellunderstood) phenomenon of matter organizing itself into simple ordered structures, like crystals and magnets, and then explore how our ideas are affected when we consider the effects of randomness and disorder, pervasive in the physical world. We will see that randomness and disorder are, paradoxically, essential for more ordered, complex structures to arise. Using these ideas, we provide some hints (but only hints) as to how we can gain a handle on issues related to the increase of complexity. Underlying all of our considerations is the notion of symmetry in physics: where it comes from and how matter "breaks" its inherent symmetry to create new information and everincreasing complexity. 

May 2
 Speaker: 
Leslie Greengard 
 Title:  Quadrature by Expansion: A new method for the evaluation of layer
potentials 

 
Abstract
The practical application of integral equation methods requires the evaluation of boundary integrals with singular, weakly singular or nearly singular kernels in complicated domains. Historically, these issues have been handled either by product integration rules (computed semianalytically), by the construction of corrections to high order nonsingular rules for specific kernels, by singularity subtraction/cancellation, or by kernel regularization and asymptotic analysis. We have developed a systematic, high order approach that works for any singularity (including hypersingular kernels), based only the assumption that the field induced by the integral operator is locally smooth when restricted to either the interior or the exterior. Discontinuities in the field across the boundary are permitted. The scheme, denoted QBX (quadrature by expansion), is easy to implement and compatible with fast hierarchical algorithms such as the fast multipole method. This is joint work with Andreas Kloeckner, Alex Barnett, Michael O'Neil and Charlie Epstein. 

If you would like to give a talk or ask a question about the seminar,
please contact one of the seminar organizers:
Aukosh Jagannath   aukosh [at] cims [dot] nyu [dot] edu 
Irena Vankova   vankova [at] cims [dot] nyu [dot] edu 
Klaus Widmayer   klaus [at] cims [dot] nyu [dot] edu 
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