Past Colloquia

29 April 2013
George Papanicolaou
<p><span>The quantification and management of risk in financial markets</span><br /><span>is at the center of modern financial mathematics. But until recently, risk</span><br /><span>assessment models did not consider the effects of inter-connectedness of</span><br /><span>financial agents and the way risk diversification impacts the stability of</span><br /><span>markets. I will give an introduction to these problems and discuss the</span><br /><span>implications of some mathematical models for dealing with them.</span><span>&nbsp;</span></p>
22 February 2013
Professor Anand Pillay

There are many recent points of contact of model theory and other 
parts of mathematics: o-minimality and Diophantine geometry, geometric group 
theory, additive combinatorics, rigid geometry,...  I will probably 
emphasize  long-standing themes around stability, Diophantine geometry, and 
analogies between ODE's and bimeromorphic geometry.

9 November 2012

 Many geophysical flows over topography can be modeled by two-dimensional
depth-averaged fluid dynamics equations.  The shallow water equations
are the simplest example of this type, and are often sufficiently
accurate for simulating tsunamis and other large-scale flows such
as storm surge.  These hyperbolic partial differential equations
can be modeled using high-resolution finite volume methods.  However,
several features of these flows lead to new algorithmic challenges,
e.g. the need for well-balanced methods to capture small perturbations
to the ocean at rest, the desire to model inundation and flooding,
and that vastly differing spatial scales that must often be modeled,
making adaptive mesh refinement essential. I will discuss some of
the algorithms implemented in the open source software GeoClaw that
is aimed at solving real-world geophysical flow problems over
topography.  I'll also show results of some recent studies of the
11 March 2011 Tohoku Tsunami and discuss the use of tsunami modeling
in probabilistic hazard assessment.

8 June 2012
Bruce Kleiner
A map betweem metric spaces is a bilipschitz homeomorphism if it is Lipschitz and has a Lipschitz inverse; a map is a bilipschitz embedding if it is a bilipschitz homeomorphism onto its image. Given metric spaces X and Y, one may ask if there is a bilipschitz embedding X--->Y, and if so, one may try to find an embedding with minimal distortion, or at least estimate the best bilipschitz constant. Such bilipschitz embedding problems arise in various areas of mathematics, including geometric group theory, Banach space geometry, and geometric analysis; in the last 10 years they have also attracted a lot of attention in theoretical computer science. The lecture will be a survey bilipschitz embedding in Banach spaces from the viewpoint of geometric analysis.
4 May 2012
Professor Steven Strogatz
<p><span>&nbsp;</span><span>Consider a fully-connected&nbsp;social&nbsp;network of people, companies,</span><br /><span>or&nbsp;countries, modeled as an undirected complete graph with real numbers on</span><br /><span>its&nbsp;edges. Positive edges link friends; negative edges link enemies.</span><br /><span>I'll&nbsp;discuss two simple models of how the edge weights of such&nbsp;networks</span><br /><span>might&nbsp;evolve over time, as they seek a balanced state in which "the enemy of</span><br /><span>my&nbsp;enemy is my friend." The mathematical techniques involve elementary</span><br /><span>ideas&nbsp;from linear algebra, random graphs, statistical physics, and</span><br /><span>differential&nbsp;equations. Some motivating examples from international</span><br /><span>relations and&nbsp;social&nbsp;psychology will also be discussed.&nbsp;This is joint work</span><br /><span>with Seth Marvel, Jon&nbsp;Kleinberg, and Bobby Kleinberg.</span><span>&nbsp;</span></p>
2 March 2012
Stephan Luckhaus
What is a phase transition? The first thing that comes to mind is boiling and freezing of water. The material clearly changes its behaviour without any chemical reaction. One way to arrive at a mathematical model is to associate different material behavior, ie., constitutive laws, to different phases. This is a continuum physics viewpoint, and when a law for the switching between phases is specified, we arrive at pde problems. The oldest paper on such a problem by Clapeyron and Lame is nearly 200 years old; it is basically on what has later been called the Stefan problem for the heat equation. The law for switching is given e.g. by the melting temperature. This can be taken to be a phenomenological law or thermodynamically justified as an equilibrium condition. The theory does not explain delayed switching (undercooling) and it does not give insight in structural differences between the phases. To some extent the first can be explained with the help of a free energy associated with the interface between different phases. This was proposed by Gibbs, is relevant on small space scales, and leads to mean curvature equations for the interface – the so-called Gibbs Thompson condition. The equations do not by themselves lead to a unique evolution. Indeed to close the resulting pde’s with a reasonable switching or nucleation law is an open problem. Based on atomistic concepts, making use of surface energy in a purely phenomenological way, Becker and Döring developed a model for nucleation as a kinetic theory for size distributions of nuclei. The internal structure of each phase is still not considered in this ansatz. An easier problem concerns solid-solid phase transitions. The theory is furthest developped in the context of equilibrium statistical mechanics on lattices, starting with the Ising model for ferromagnets. In this context phases correspond to (extremal) equilibrium Gibbs measures in infinite volume. Interfacial free energy appears as a finite volume correction to free energy. The drawback is that the theory is still basically equilibrium and isothermal. There is no satisfactory theory of metastable states and of local kinetic energy in this framework.
10 February 2012
Professor Karen Vogtmann
Free groups, free abelian groups and fundamental groups of closed orientable surfaces are the most basic and well-understood examples of infinite discrete groups. The automorphism groups of these groups, in contrast, are some of the most complex and intriguing groups in all of mathematics. I will give some general comments about geometric group theory and then describe the basic geometric object, called Outer space, associated to automorphism groups of free groups. This Colloquium talk is the first of a series of three lectures given by Professor Vogtmann, who is the European Mathematical Society Lecturer. In this series of three lectures, she will discuss groups of automorphisms of free groups, while drawing analogies with the general linear group over the integers and surface mapping class groups. She will explain modern techniques for studying automorphism groups of free groups, which include a mixture of topological, algebraic and geometric methods.
4 November 2011
Professor John W.M Bush

Yves Couder and co-workers have recently reported the results of a startling series of experiments in which droplets bouncing on a fluid surface exhibit several dynamical features previously thought to be peculiar to the microscopic realm. In an attempt to 

develop a connection between the fluid and quantum systems, we explore the Madelung transformation, whereby Schrodinger's equation is recast in a hydrodynamic form. New experiments are presented, and indicate the potential value of this hydrodynamic approach to both visualizing and understanding quantum mechanics.


24 June 2011
Professor Sir Vaughan Jones

Voiculescu showed how the large N limit of the expected value of the trace of a word on n independent hermitian NxN matrices gives a well known von Neumann algebra. In joint work with Guionnet and Shlyakhtenko it was shown that this idea makes sense in the context of very general planar algebras where one works directly in the large N limit. This allowed us to define matrix models with a non-integral  number of random matrices. I will present this work and some of the subsequent work, together with future hopes for the theory.