Previous WIMAW events

The talks listed as part of our "Trends in Mathematics Women's Speaker Series" have been primarily funded by the WISELI "Celebrating Women in Science & Engineering Grant Program" with additional support from departmental sources, secured by rotating groups of our female graduate students. Many thanks to the generous support from WISELI and VIGRE, without which many of our programs would not have been possible.

Monica Vazirani*
University of California, Davis
May 1, 2009
"The combinatorial and representation-theoretic properties of p-cores and their generalization"
Abstract: The irreducible representations of the symmetric group are indexed by partitions. Those representations which remain both irreducible and projective upon reduction modulo p are indexed by p-cores: those partition from which no rim hooks (or ribbons) of size p can be removed. They also correspond to cosets of the finite symmetric group in the affine symmetric group, equivalently to the root lattice, or to the extremal vectors in the basic representation of the affine Lie algebra sl_p. One can also ask which representations remain irreducible (but not necessarily projective) when reduced mod p, or more generally, which modules for the related Hecke algebra stay irreducible on specialization to a p-th root of unity. This was answered by Fayers in general, and James-Mathas for those modules not equal to their own radical (the p-regular partitions). We give here two new combinatorial descriptions of those partitions, which naturally generalize the description of p-cores. The first is related to the spin of corresponding ribbon tableaux. The second is given in the context of crystal graphs. We also enumerate such partitions, a key ingredient for which is a bijection between p-cores and (p-1)-cores that has several lovely combinatorial descriptions.

 
 
Carolyn Gordon*
Dartmouth University
March 23, 2009
"When you can't hear the shape of a manifold"
Abstract: Mark Kac's question "Can you hear the shape of a drum?" asks the extent to which the geometry of a plane domain, viewed as a vibrating membrane, is encoded in the Dirichlet eigenvalue spectrum of the associated Laplacian, equivalently, in the characteristic frequencies of vibration. More generally, one asks the extent to which spectral data on a Riemannian manifold encodes the geometry. We will survey techniques for constructing manifolds with the same spectral data and look at examples in order to identify geometric invariants that are not spectrally determined.
 
 
Jeanne Clelland
University of Colorado, Boulder
February 16, 2009
"Constructing topologically distinct energy-critical curves in the path space of the Euclidean line"

Abstract: We begin by considering a natural energy functional on the space of maps from the real line into the Euclidean line. The Euler–Lagrange equation for this functional is a second–order PDE which changes type from hyperbolic to parabolic, and initial value problems with certain types of singularites are ill-posed where the initial data intersects the parabolic locus. However, this PDE has a large set of first-order intermediate equations, which in some cases can be used to “repair” the ill-posed initial value problem and construct global solutions with various types of singularities. (This work is joint with George Wilkens and Marek Kossowski.)
Thanks to Professor Gloria Mari-Beffa for organizing Professor Wade's visit.  
 
Chuu-Lian Terng
University of California, Irvine
October 10, 2008
Thanks to Professor Gloria Mari-Beffa for organizing Professor Terng's visit and lunch.  
 
Ruth Williams*
University of California, San Diego
May 9, 2008
"Stochastic delay differential equations with state constraints"
Abstract: Deterministic dynamic models with delayed feedback and state constraints arise in modeling Internet rate control and biochemical reactions involving transcription and translation. Much of the analysis of such models has focused on local stability analysis of equilibrium points or on proving global asymptotic stability of such points. There is interest in understanding when such systems can have sustained oscillations, and what effect noise has on system behavior. Here we consider a one dimensional stochastic delay differential equation as a simple prototype model for a noisy deterministic system with delayed (negative) feedback. We obtain sufficient conditions for the noiseless model to have slowly oscillating periodic solutions and for the noisy model to have stationary solutions. (This talk is based on joint work with Michael Kinnally)
 
 
Aissa Wade
Penn State University
March 28, 2008
"Stability of singular points of Poisson manifold"

Abstract: Let M be a smooth finite-dimensional manifold. Every Poisson structure on M gives rise to a natural foliation of M whose leaves are symplectic manifolds (possibly of different dimensions). A leaf L of a Poisson structure on M is stable if every nearby Poisson structure admits a nearby leaf L', which is diffeomorphic to L. In this talk, we discuss stability of symplectic leaves of a Poisson structure. Of particular interest is the case of zero-dimensional leaves, called singular points. We will give sufficient conditions for stability of singular points. We will provide some examples for Poisson manifolds of dimension less than 5.
Thanks to Professor Gloria Mari-Beffa for organizing Professor Wade's visit.  
 
Maria Westdickenberg*
Georgia Institute of Technology
March 7, 2008
"Noise-Driven Rare Events and Action Minimization"
Abstract: A small noise can have a dramatic effect, overpowering deterministic dynamics and leading to so-called rare events. We will recall some results from the Freidlin-Wentzell theory of large deviations, and describe some recent results, drawing connections with applied mathematics and the calculus of variations.
 
 
Rachel Kuske*
University of British Columbia
April 20, 2007
"Transients+ instabilities+ noise = structure?"
Abstract: Transient or unstable behavior is often ignored in considering long time dynamics in the deterministic world. However, stochastic effects can change the picture dramatically, so that the transients can dominate the long range behavior. Coherence resonance is one relatively simple example of this transformation, and we consider others such as noise-driven synchronization in networks, asset pricing in periodically modulated environments, sustained recurrence of disease, and amplitude-driven phase dynamics. The questions that arise in these contexts illustrate the influence of multiple time scales, cooperation of both discrete and continuous aspects in the dynamics, and the remnants of underlying bifurcation structure visible through the noise.
 
 
Karen Uhlenbeck*
University of Texas, Austin
April 27, 2007
"On the Space-Time Monopole Equation."
Abstract: The Euclidean self-dual Yang Mills equations and their reduction to the Euclidean monopole equations are important equations in differential geometry. The self-dual Yang Mills equations can also be formulated in R^{2,2} equations with such signatures are not usually interesting in geometry. However, the reduction of the equation to a three-dimensional monopole equation with signature (2,1) yields a fascinating wave equation, which inherits the algebraic properties of an integrable system the Yang-Mills equations. We give a brief outline of geometric aspects of these equations and in particular discuss the scattering and inverse scattering theory.
 
 

 
*Part of our Trends in Mathematics series.