## James Bremer : Improved methods for discretizing integral operators

- Presentations ( 217 Views )Integral equation methods are frequently used in the numerical solution of elliptic boundary value problems. After giving a brief overview of the advantages and disadvantages of such methods vis-a-vis more direct techniques like finite element methods, I will discuss two problems which arise in integral equation methods. In both cases, I take a contrarian position. The first is the discretization of integral operators on singular domains (e.g., surfaces with edges and curves with corners). The consensus opinion holds that integral equations given on such domains are exceedingly difficult to discretize and that sophisticated analysis, often specific to a particular boundary value problem, is required. I will explain that, in fact, the efficient solution of a broad class of such problems can be effected using an elementary approach. Exterior scattering problems given on planar domains with tens of thousands of corner points can be solved to 12 digit accuracy on my two year old desktop computer in a matter of hours. The second problem I will discuss is the evaluation of the singular integrals which arise form the discretization of weakly singular integral operators given on surfaces. Exponentially convergent algorithms for evaluating these integrals have been described in the literature and it is widely regarded as a "solved" problem. I will explain why this is not so and describe an approach which yields only algebraic convergence, but nonetheless performs better in practice than standard exponentially convergent methods.

## Ellen Eischen : L-functions, congruences, and applications

- Presentations ( 178 Views )L-functions, certain meromorphic functions that include the Riemann zeta-function, encode important number-theoretic information. The first part of this talk will focus on some striking properties of special values of the Riemann zeta-function and certain other L-functions (namely, congruences modulo powers of a prime number). In the second part of the talk, I will introduce tools that are useful for studying these congruences. These tools have applications not only to number theory, but also to homotopy theory. This will be a colloquium style talk, intended for a broad audience.

## Matthias Heymann : Computing maximum likelihood paths of rare transition events, and applications to synthetic biology

- Presentations ( 176 Views )Dynamical systems with small noise (e.g. SDEs) allow for rare transitions from one stable state into another that would not be possible without the presence of noise. Large deviation theory provides the means to analyze both the frequency of these transitions and the maximum likelihood transition path. The key object for the determination of both is the quasipotential, V(x,y) = inf S_T(phi), where S_T(phi) is the action functional associated to the system, and where the infimum is taken over all T>0 and all paths phi:[0,T]->R^n leading from x to y. The numerical evaluation of V(x,y) however is made difficult by the fact that in most cases of interest no minimizer exists.

In my work I prove an alternative geometric formulation of V(x,y) that resolves this issue by introducing an action on the space of curves ( i.e. this action is independent of the parametrization of phi). In this formulation, a minimizer exists, and we use it to build a flexible algorithm (the geometric minimum action method, gMAM) for finding the maximum likelihood transition curve.

In one application I show how the gMAM can be useful in the newly emerging field of synthetic biology: We propose a method to identify the sources of instabilities in (genetic) networks.

This work was done in collaboration with my adviser Eric Vanden-Eijnden and is the core of my PhD thesis at NYU.

## David Speyer : Matroids and Grassmannians

- Presentations ( 142 Views )Matroids are combinatorial devices designed to encoded the combinatorial structure of hyperplane arrangements. Combinatorialists have developed many invariants of matroids. I will explain that there is reason to believe that most of these invariants are related to computations in the K-theory of the Grassmannian. In particular, I will explain work of mine limiting the complexity of Hacking, Keel and Tevelev's "very stable pairs", which compactify the moduli of hyperplane arrangements. This talk should be understandable both to those who don't know matroids, and to those who don't know K-theory.

## Allan Sly : Mixing in Time and Space

- Presentations ( 130 Views )For Markov random fields temporal mixing, the time it takes for the Glauber dynamics to approach it's stationary distribution, is closely related to phase transitions in the spatial mixing properties of the measure such as uniqueness and the reconstruction problem. Such questions connect ideas from probability, statistical physics and theoretical computer science. I will survey some recent progress in understanding the mixing time of the Glauber dynamics as well as related results on spatial mixing. Partially based on joint work with Elchanan Mossel

## Margaret Beck : Using global invariant manifolds to understand metastability in Burgers equation with small viscosity.

- Presentations ( 117 Views )Finding globally stable states can provide useful information about the behavior of solutions to PDEs: for any initial condition, the solution will eventually approach such a state. However, in some cases, the solution can exhibit long transients in its approach to the state. If the transient is long enough, it may be this behavior, rather than the limiting behavior, that is observed in numerical simulations or experiments. This is referred to as "metastability" and has been found, for example, in the 2D Navier-Stokes equations with small viscosity. A similar phenomenon has been seen in Burgers equation, which can be explained using global invariant manifolds. More precisely, it is shown that in terms of similarity, or scaling, variables there exists a one-dimensional global center manifold of stationary solutions. Through each of these fixed points there exists a one-dimensional, global, attractive, invariant manifold. Metastability corresponds to a fast transient in which solutions approach this 'metastable' manifold, followed by a slow decay along this manifold, and, finally, convergence to the globally stable state.