Past Colloquia

6 June 2014
Prof. Michael Brenner

Evolution by natural selection has resulted in a remarkable diversity of organism morphologies. But is it possible for developmental processes to create “any possible shape?” Or are there intrinsic constraints? I will discuss our recent exploration into the shapes of bird beaks. Initially, inspired by the discovery of genes controlling the shapes of beaks of Darwin's finches, we showed that the morphological diversity in the beaks of Darwin’s Finches is quantitatively accounted for by the mathematical group of affine transformations. We have extended this to show that the space of shapes of bird beaks is not large, and that a large phylogeny (including finches, cardinals, sparrows, etc.) are accurately spanned by only three independent parameters -- the shapes of these bird beaks are all pieces of conic sections. After summarizing the evidence for these conclusions, I will delve into our efforts to create mathematical models that connect these patterns to the developmental mechanism leading to a beak. It turns out that there are simple (but precise) constraints on any mathematical model that leads to the observed phenomenology, leading to explicit predictions for the time dynamics of beak development in song birds. Experiments testing these predictions for the development of zebra finch beaks will be presented.

Based on the following papers:

28 February 2014
Professor Camillo De Lellis

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
Professor Vladimir Markovic

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
Professor Kazuya Kato

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
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>