Tuesday, April 24, 2012 - 16:00 , Location: Skiles 005 , Prof. Andrea Bertozzi , UCLA Math , Organizer: Sung Ha Kang
There is an extensive applied mathematics literature developed for problems in the biological and physical sciences. Our understanding of social science problems from a mathematical standpoint is less developed, but also presents some very interesting problems. This lecture uses crime as a case study for using applied mathematical techniques in a social science application and covers a variety of mathematical methods that are applicable to such problems. We will review recent work on agent based models, methods in linear and nonlinear partial differential equations, variational methods for inverse problems and statistical point process models. From an application standpoint we will look at problems in residential burglaries and gang crimes. Examples will consider both "bottom up" and "top down" approaches to understanding the mathematics of crime, and how the two approaches could converge to a unifying theory.
Thursday, April 19, 2012 - 11:05 , Location: Skiles 006 , Boris Khesin , IAS/University of Toronto , Organizer:
We show that the LIA approximation of the incompressible Euler equation describes the skew-mean-curvature flow on vortex membranes in any dimension. This generalizes the classical binormal, or vortex filament, equation in 3D. We present a Hamiltonian framework for higher-dimensional vortex filaments and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively. This framework, in particular, allows one to define the symplectic structures on the spaces of vortex sheets.
Thursday, April 5, 2012 - 11:05 , Location: Skiles 006 , Frank Sottile , Texas A&M , Organizer: Anton Leykin
Building on work of Jordan from 1870, in 1979 Harris showed that a geometric monodromy group associated to a problem in enumerative geometry is equal to the Galois group of an associated field extension. Vakil gave a geometric-combinatorial criterion that implies a Galois group contains the alternating group. With Brooks and Martin del Campo, we used Vakil's criterion to show that all Schubert problems involving lines have at least alternating Galois group. My talk will describe this background and sketch a current project to systematically determine Galois groups of all Schubert problems of moderate size on all small classical flag manifolds, investigating at least several million problems. This will use supercomputers employing several overlapping methods, including combinatorial criteria, symbolic computation, and numerical homotopy continuation, and require the development of new algorithms and software.
Tuesday, March 13, 2012 - 11:05 , Location: Skiles 006 , Jeff Kahn , Mathematics, Rutgers University , email@example.com , Organizer: Prasad Tetali
Thresholds for increasing properties are a central concern in probabilistic combinatorics and elsewhere. (An increasing property, say F, is a superset-closed family of subsets of some (here finite) set X; the threshold question for such an F asks, roughly, about how many random elements of X should one choose to make it likely that the resulting set lies in F? For example: about how many random edges from the complete graph K_n are typically required to produce a Hamiltonian cycle?) We'll discuss recent progress and lack thereof on a few threshold-type questions, and try to say something about a ludicrously general conjecture of G. Kalai and the speaker to the effect that there is always a pretty good naive explanation for a threshold being what it is.
Thursday, March 8, 2012 - 11:05 , Location: Skiles 006 , F. Alberto Grünbaum , University of California, Berkeley , Organizer: Plamen Iliev
I will review the well known method (pushed mainly by Karlin and McGregor) to study birth-and-death processes with the help of orthogonal polynomials. I will then look at several extensions of this idea, including ¨poker dice¨ (polynomials in several variables) and quantum walks (polynomials in the unit circle).
Thursday, February 23, 2012 - 11:05 , Location: Skiles 006 , Alexander Shapiro , ISyE, Georgia Tech , Organizer: Anton Leykin
In many practical situations one has to make decisions sequentially based on data available at the time of the decision and facing uncertainty of the future. This leads to optimization problems which can be formulated in a framework of multistage stochastic programming. In this talk we consider risk neutral and risk averse approaches to multistage stochastic programming. We discuss conceptual and computational issues involved in formulation and solving such problems. As an example we give numerical results based on the Stochastic Dual Dynamic Programming method applied to planning of the Brazilian interconnected power system.
Tuesday, February 14, 2012 - 11:00 , Location: Skiles 005 , Ron Douglas , Texas A&M University , Organizer: Brett Wick
An intesting class of bounded operators or algebras of bounded operators on Hilbert spaces, particularly on Hilbert spaces of holomorphic functions, have a natural interpretation in terms of concepts from complex geometry. In particular, there is an intrinsic hermitian holomorphic vector bundle and many questions can be answered in terms of the Chern connection and the associated curvature. In this talk we describe this setup and some of the results obtained in recent years using this approach. The emphasis will be on concrete examples, particularly in the case of Hilbert spaces of holomorphic functions such as the Hardy and Bergman spaces on the unit sphere in C^n.
Wednesday, February 8, 2012 - 11:05 , Location: Skiles 005 , Jeff Kahn , Mathematics, Rutgers University , Organizer: Prasad Tetali
Pardon the inconvenience. We plan to reschedule later...
Friday, December 9, 2011 - 16:00 , Location: Skiles 006 , Benson Farb , University of Chicago , Organizer: John Etnyre
There will be a tea 30 minutes before the colloquium.
Tom Church, Jordan Ellenberg and I recently discovered that the i-th Betti number of the space of configurations of n points on any manifold is given by a polynomial in n. Similarly for the moduli space of n-pointed genus g curves. Similarly for the dimensions of various spaces of homogeneous polynomials arising in algebraic combinatorics. Why? What do these disparate examples have in common? The goal of this talk will be to answer this question by explaining a simple underlying structure shared by these (and many other) examples in algebra and topology.