Colloquium

Research seminars

VIGRE Working Groups

Clifford lectures

Michael Wolf, Rice University

"Minimal Desingularizations of Planes in Space"

Abstract:

We prove that there is only one way, due to Scherk in 1835, to 'desingularize' the intersection of two planes in space and obtain a periodic minimal surface as a result.  The proof is mostly an exercise in, and an introduction to, the basic theory of moduli spaces:  we translate the geometry of minimal surface in space into a statement about a moduli space of flat structures on Riemann surfaces, and then discuss the deformation theory of and degenerations in this moduli space to prove the result. Naturally, we'll explain all of the terms.

POSTPONED due to Hurricane Ike...reschedule date April 2, 2009

Patrick Rabier, Univ of Pittsburgh

"Bifurcation of Periodic Solutions in Nonlinear Evolution Problems with Periodic Forcing"

Abstract:

click to read abstract

Mohan Putcha, North Carolina State University, Raleigh

"Reductive Monoids"

Abstract:

A reductive monoid M is the Zariski closure of a reductive group G. We will discuss various combinatorial structures associated with their study. First of all the ‘diagonal’ idempotents form the face lattice of a polytope. The second invariant is the cross-section lattice Λ obtained from an idempotent cross-section of the G x G-orbits of M. We illustrate these structures when M is the closure of the images of various natural representations of Mn(k), the multiplicative monoid of all n x n matrices over k. Next the Bruhat-Renner decomposition of M gives rise to a ?nite monoid R, which for Mn(k), is the rook monoid (=symmetric inverse semigroup), rich in combinatorial and algebraic structure. Finally there is a decomposition of M related to conjugacy classes that gives rise to a ?nite conjugacy poset C. This combinatorial structure is of yet, very little understood. For the matrix monoid Mn(k), C consists of partitions of m, m ≤ n, ordered by a generalization of the dominance order on partitions of n.

Workshop on Informatic Phenomena

Christopher Fuchs , Perimeter Institute for Theoretical Physics, Waterloo, Canada

"Charting the Shape of Quantum State Space"

Physicists have become accustomed to the idea that a theory's content is always most transparent when written in coordinate-free language.  But sometimes the choice of a good coordinate system can be quite useful for settling deep conceptual issues.  This is particularly so for an information-oriented or Bayesian approach to quantum foundations:  One good coordinate system may (eventually!) be worth more than a hundred blue-in-the-face arguments.  This talk will motivate and chronicle the search for one such class of coordinate systems for finite dimensional operator spaces, the so-called Symmetric Informationally Complete measurements.  The desired class will take little more than five minutes to define, but the quest to construct these objects will carry us down a 35 year journey, with the most frenzied activity only recently.  If time permits, I will turn the tables and discuss how one might hope to get the formal content of quantum mechanics out of the very existence of such a coordinate system.

Duong Phong , Columbia University

"Non-linear parabolic flows in Kaehler geometry"

Abstract:

A major theme in geometry is to characterize a geometric structure by a ``canonical metric", that is, a metric with best curvature properties. A well-known example is the uniformization theorem, which characterizes a complex structure on a surface by a metric with constant scalar curvature. Canonical metrics can be viewed as the fixed points of a suitable flow of metrics. Their existence reduces then to the problem of long-time existence and convergence of such flows. In this talk, we provide a self-contained survey of some of these flows, including the Donaldson heat flow, the Kaehler-Ricci flow, and the Calabi flow. The emphasis will be on the many open problems, and in particular on the conjectures relating the existence of canonical metrics to the algebraic- geometric notion of stability in geometric invariant theory.

Bojan Popov, Texas A&M University 

"Terrain Data Reconstruction and approximating PDEs via L1 -minimization"

Abstract:

In this talk we will consider a class of L1-based minimization methods for solving the following problems:

(1)Digital elevation maps (DEM) for natural and urbanterrain;

(2)First order Partial Differential Equations (PDEs);

(3)Super-resolution (SR) problems. We will describe each of the three situations. In the case of DEM, various numerical examples will be given. For 1rst order PDEs, a convergence theory will be presented. In the last case, SR problems, we will give a new way to enhance the resolution of digital images. The talk will be accessible to all graduate and advanced undergraduate students.

Bruce Sagan, MSU/NSF

"Rational generating functions and compositions"

(joint work with Anders Björner)

Abstract:

A composition of the nonnegative integer n is a way of writing n as an ordered sum.  So the compositions of 3 are 1+1+1, 1+2, 2+1, and 3 itself.  It is well-known (and easy to prove) that if c_n is the number of compositions of n then c_n = 2^{n-1} for n at least 1 and c_0 = 1.  Equivalently, we have the generating function

sum_{n >= 0} c_n x^n = (1-x)/(1-2x)

which is a rational function.  We show that this is a special case of a more general family of rational functions associated with compositions.  Our techniques include the use of formal languages. Surprisingly, identities from the theory of hypergeometric series are needed to do some of the computations.

Lex Renner, University of Western Ontario

"The H-polynomial of a Group Embedding"

Abstract:

The Poincare polynomial of a Weyl group calculates the Betti numbers of the projective homogeneous space G/B, while the h-vector of a simple polytope calculates the Betti numbers of the corresponding rationally smooth toric variety. There is a common generalization of these two extremes called the H-polynomial. It applies to projective, homogeneous spaces, toric varieties and, much more generally, to anyalgebraic variety X where there is a connected, solvable, algebraic group acting with a finite number of orbits. We illustrate this situation by describing the H-polynomials of certain projective G x G -varieties X, where G is a semisimple group and B is a Borel subgroup of G. This description is made possible  by finding an appropriate cellular decomposition for X and then describing the cells combinatorially in terms of the underlying monoid of B x B - orbits. The most familiar example here is the wonderful compactification of a semisimple group of adjoint type.

Thanksgiving Holiday

Ben Schmidt, University of Chicago

"Blocking light in closed manifolds"

Abstract: To what extent does the collision of light in a closed Riemannian manifold M determine the Riemannian metric on M?  I'll discuss conjectures and related results that aim to characterize locally symmetric Riemannian manifolds of nonnegative curvature in terms of collisions of light rays

Gregg Musiker, MIT

"Elliptic curves and chip-firing games on wheel graphs"

In this talk, I will begin by discussing chip-firing games on graphs, and how for a given graph G, this gives rise to a group stucture whose order equals the number of spanning trees on G.  In the second part, I will describe elliptic curves over finite fields, and how such objects also have group structures.  For a family of graphs obtained by deforming the sequence of wheel graphs, the cardinalities of these groups satisfy a nice reciprocal relationship with the orders of elliptic curves as we consider field extensions.  I will finish by discussing other surprising ways that these group structures are analogous.  Some of this research was completed as part of my dissertation work at  the University of California, San Diego under Adriano Garsia's guidance