# 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 Dynamics of ferromagnets: averaging methods, bifurcation diagrams, and thermal noise effects Abstract Driving nanomagnets by spin-polarized 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 low-damping 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 non-uniform magnets, leading to a transition state theory for infinite dimensional Hamiltonian systems.

## April 4

Speaker: Mehryar Mohri Multiple-Source 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 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 n-dimensional Riemannian manifolds with bounded sectional curvature and bounded diameter, equipped with measured Gromov-Hausdorff topology.

## April 25

Speaker: Daniel L. Stein 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 self-organization 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 (well-understood) 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 ever-increasing complexity.

## May 2

Speaker: Leslie Greengard 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 semi-analytically), by the construction of corrections to high order non-singular 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.

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