Monday, Jan 22 3:30 pm - 4:30 pm
Groupes Reductifs sur un Corps Local II
A surefire way to find new results about old thingsI will tell the story of how my PhD thesis advisor, Norberto Kerzman, and his mentor, Eli Stein, discovered a new property of the centuries old Cauchy integral and how it has influenced the way I think about complex analysis. I have tried to use the Kerzman-Stein modus operandi in my own research, and once in a while, it has led me to find shiny new things in moldy corners of the basement of complex analysis.
Wednesday, Jan 24 1:30 pm - 2:20 pm
Some Cases of the Lex-Plus-Powers Conjecture.
Wednesday, Jan 24 1:30 pm - 2:30 pm
Scattering symplectic geometryAbstract: Scattering-symplectic manifolds are manifolds equipped with a type of minimally degenerate Poisson structure that is not too restrictive so as to have a large class of examples, yet restrictive enough for standard Poisson invariants to be computable. We will define scattering-symplectic geometry, providing plenty of examples, and discuss connections with contact geometry.
Wednesday, Jan 24 1:30 pm - 2:45 pm
Long Time Behavior of Brownian Motion in Tilted Periodic Potentials
We will investigate various limits concerning the long time average velocity V of a Brownian particle diffusing on a periodic potential. The prototype model is Langevin dynamics which incorporates inertia (mass) and friction. The key feature of the current work is the consideration of an additional macroscopic tilt, F. The goal is to understand how the average velocity V depends on F. Interesting thresholds for the value of F can be obtained, in particular under the limit of vanishing friction and noise. Using the averaging theory of Freidlin-Wentzel, the current work provides rigorous mathematical justification of some formulas obtained by Risken. An earlier, fundamental result by Tanaka for Brownian particle diffusing on a random (Brownian) potential will also be discussed. This is a joint work with Liang Cheng.
Wednesday, Jan 24 3:30 pm - 4:30 pm
GKZ-systems and mixed Hodge modulesI will define GKZ-systems, and talk a little about their properties from the algebraic, analytic, and combinatorial point of view. Then I will discuss a theorem of Gelfand et al, and a sharpening by Mathias Schulze and myself, on the question which GKZ-systems arise as (D-module-)direct image of a natural D-module on a torus. In such cases, the GKZ-system inherits a mixed Hodge module structure. I will then explain work with Thomas Reichelt that computes this MHM structure on a class of GKZ-systems that comes up naturally in mirror symmetry. Very few of such explicitly computed structures seem to be known.
An Introduction to Topological Data Analysis with a Focus on Persistent HomologyAbstract: Topological data analysis applies techniques from topology to the study of large data sets. One of the most useful tools of TDA is persistent homology, which provides a method of determining the global structure of a discrete data set. From searching for cancer biomarkers to studying swarming behavior of animals, persistent homology has been used by researchers to study a wide variety of problems outside mathematics.
Instabilities in shallow water wave modelsAbstract: Slow modulations in wave characteristics of a nonlinear, periodic traveling wave in a dispersive medium may develop non-trivial structures which evolve as it propagates. This phenomenon is called modulational instability. In the context of water waves, this phenomenon was observed by Benjamin and Feir and, independently, by Whitham in Stokes' waves. I will discuss a general mechanism to study modulational instability of periodic traveling waves which can be applied to several classes of nonlinear dispersive equations including KdV, BBM and regularized Boussinesq type equations.
Friday, Jan 26 3:30 pm - 4:30 pm
Constructive Invariant Theory and Noncommutative Rank
If G is a group acting on a vector space V by linear transformations, then the invariant polynomial functions on V form a ring. In this talk we will discuss upper bounds for the degrees of generators of this invariant ring. An example of particular interest is the action of the group SL_n x SL_n on the space of m-tuples of n x n matrices by simultaneous left-right multiplication. In this case, Visu Makam and the speaker recently proved that invariants of degree at most mn^4 generate the invariant ring. We will explore an interesting connection between this result and the notion of noncommutative rank.
Research Areas: Commutative rings and algebras
Graduate Student Invited Colloquium Speaker, Prof. Chelsea Walton, University of Illinois at Urbana Champaign, MATH 175
Tuesday, Jan 30 3:30 pm - 4:20 pm
Wednesday, Jan 31 3:30 pm - 4:30 pm
GKZ-systems and mixed Hodge modulesThis will be a continuation of the last talk.
Thursday, Feb 1 1:30 pm - 2:20 pm
Tuesday, Feb 20 3:30 pm - 4:20 pm
Automorphic Forms and Representation Theory Seminar, Prof. Manish Patnaik, University of Alberta, UNIV 317
Thursday, Feb 22 1:30 pm - 2:20 pm
Tuesday, Mar 6 3:30 pm - 4:20 pm
Tuesday, Mar 27 3:30 pm - 4:20 pm
Tuesday, Apr 3 3:30 pm - 4:20 pm
Tuesday, Apr 10 3:30 pm - 4:20 pm
Tuesday, Apr 17 3:30 pm - 4:20 pm
Thursday, Apr 19 1:30 pm - 2:20 pm
Tuesday, Apr 24 3:30 pm - 4:20 pm