Past Colloquia

28 February 2014
16:30
Professor Camillo De Lellis
Abstract

The Plateau's problem, named after the Belgian physicist J. Plateau, is a classic in the calculus of variations and regards minimizing the area among all surfaces spanning a given contour. Although Plateau's original concern were $2$-dimensional surfaces in the $3$-dimensional space, generations of mathematicians have considered such problem in its generality. A successful existence theory, that of integral currents, was developed by De Giorgi in the case of hypersurfaces in the fifties and by Federer and Fleming in the general case in the sixties. When dealing with hypersurfaces, the minimizers found in this way are rather regular: the corresponding regularity theory has been the achievement of several mathematicians in the 60es, 70es and 80es (De Giorgi, Fleming, Almgren, Simons, Bombieri, Giusti, Simon among others).

In codimension higher than one, a phenomenon which is absent for hypersurfaces, namely that of branching, causes very serious problems: a famous theorem of Wirtinger and Federer shows that any holomorphic subvariety in $\mathbb C^n$ is indeed an area-minimizing current. A celebrated monograph of Almgren solved the issue at the beginning of the 80es, proving that the singular set of a general area-minimizing (integral) current has (real) codimension at least 2. However, his original (typewritten) manuscript was more than 1700 pages long. In a recent series of works with Emanuele Spadaro we have given a substantially shorter and simpler version of Almgren's theory, building upon large portions of his program but also bringing some new ideas from partial differential equations, metric analysis and metric geometry. In this talk I will try to give a feeling for the difficulties in the proof and how they can be overcome.

31 January 2014
16:30
Professor Vladimir Markovic
Abstract

The surface subgroup problem asks whether a given group contains a subgroup that is isomorphic to the fundamental group of a closed surface. In this talk I will survey the role that the surface subgroup problem plays in some important solved and unsolved problems in the theory of 3-manifolds, the geometric group theory, and the theory of arithmetic manifolds.

15 November 2013
16:30
Professor Kazuya Kato
Abstract

The height of a rational number a/b (a,b integers which are coprime) is defined as max(|a|, |b|). A rational number with small (resp. big) height is a simple (resp. complicated) number. Though the notion height is so naive, height has played a fundamental role in number theory. There are important variants of this notion. In 1983, when Faltings proved the Mordell conjecture (a conjecture formulated in 1921), he first proved the Tate conjecture for abelian varieties (it was also a great conjecture) by defining heights of abelian varieties, and then deducing Mordell conjecture from this. The height of an abelian variety tells how complicated are the numbers we need to define the abelian variety. In this talk, after these initial explanations, I will explain that this height is generalized to heights of motives. (A motive is a kind of generalisation of abelian variety.) This generalisation of height is related to open problems in number theory. If we can prove finiteness of the number of motives of bounded height, we can prove important conjectures in number theory such as general Tate conjecture and Mordell-Weil type conjectures in many cases.

29 April 2013
16:30
George Papanicolaou
Abstract
<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
16:30
Professor Anand Pillay
Abstract

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
16:30
Abstract

 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
16:30
Bruce Kleiner
Abstract
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
16:30
Professor Steven Strogatz
Abstract
<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
16:30
Stephan Luckhaus
Abstract
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.

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