Mathematics Colloquium Spring 2013
McHenry Building - Room 4130
Refreshments served at 3:30 - McHenry Building 4161
For further information please contact the Mathematics Department at 459-2969
April 9, 2013
Shintani domains and group cohomology
Samit Dasgupta, University of California, Santa Cruz
In 1976, Takuro Shintani introduced a geometric method of writing down explicit fundamental domains for certain group actions. These fundamental domains are disjoint unions of simplicial cones. While Shintani showed the existence of these fundamental domains, he did not write down explicit formulae for the cones involved. This was done by Pierre Colmez under certain conditions in 1988.In this talk I will focus on a cocycle property satisfied by the Shintani construction that I am studying in joint work with Pierre Charollois and Matthew Greenberg. This cocycle property allows one to prove a generalization of Colmez's theorem to the construction of explicit "signed" fundamental domains without any conditions; this generalization was recently proved independently by Francisco Diaz y Diaz and Eduardo Friedman using topological degree theory.
If time permits, I will mention an application of the Shintani cocycle to number theory, and in particular to the study of p-adic L-functions.
April 16, 2013
Regularity and decay estimates of dissipative equations
Hantaek Bae, University of California, Davis
We establish Gevrey regularity of solutions to dissipative equations. The main tools are Gevrey estimates in Sobolev spaces (using Kato-Ponce estimates in Gevrey space) and in Besov spaces (using the Littlewood-Paley decomposition). As an application, we obtain temporal decay rate of solutions for a large class of equations, especially the Navier-Stokes equations. This is join work with Professor Eitan Tadmor and Animikh Biswas.April 23, 2013
Integrable Models in Statistical Physics & Their Universality
Craig Tracy, University of California, Davis
The 2D Ising model, random matrix models and the asymmetric simple exclusion process (ASEP) are three examples of "integrable'' stochastic models. We explain how these three examples have led to more general theories; and in the case of ASEP, experimental verification.
April 30, 2013
A soothing invisible hand: optimal control with moderation potentials
Debra Lewis, University of California, Santa Cruz
Bounded control regions complicate the analysis of optimal control systems: solutions often lose differentiability when they move onto or off the boundary. Penalty functions can be used to enforce the control bounds, but the function choice heavily influences the solutions. Reinterpreting the penalty as an incentive for submaximal control investment guides the selection of a control cost term moderating the response in generalized time minimization problems. We describe a class of smooth, bounded 'moderation incentives' for which Pontryagin's Maximum Principle determines an unparameterized Hamiltonian system. We focus on a two-parameter family with dogleg control response curves - one extreme is the kinetic energy-style control cost frequently used in geometric optimal control, with piecewise linear response, while the other yields the controls determined by a traditional logarithmic penalty function.May 7, 2013
Helicoidal minimal surfaces of every genus
David Hoffmann, Stanford
The helicoid was shown to be a minimal surface -one whose mean curvature vanishes - by J-P Meusnier around 1776. Soon after the notion of minimality was made precise. Until recently the helicoid was the only known properly embedded minimal surface with finite topology and infinite total curvature. Eleven years ago another such surface was found.
It has genus one and is asymptotic to the helicoid. This lecture is about the recent proof (joint with Brian White (Stanford) and Martin Traizet (Tours)) of the existence of properly embedded minimal surfaces of any genus, asymptotic to the helicoid. (Shown (*)). The proof involves establishing the existence of a family of helicoid-like minimal surfaces with handles within a family of Riemannian three-manifolds that limit to Euclidean space. Degree-theoretic methods establish existence of the family. The limit of the family is a minimal surface in Euclidean space asymptotic to a helicoid. The delicate problem is making sure that the handles of the surfaces do not drift away to infinity in the limit, leaving a surface of genus zero or one. The solution to this problem involves viewing the escaping handles as point masses that exert an attractive force on one another, then analyzing the possible stable con
figurations of such points. The speaker will try as best he can to avoid technicalities and to explain the underlying ideas.
May 14, 2013
The search for the exotic.
Terry Gannon, University of Alberta
There is a general belief that the collection of (sufficiently nice) vertex operator algebras (VOAs) should be more or less equivalent to the collection of (sufficiently nice) subfactors. This is an exciting metaphor, because some things that are clear for VOAs are much more obscurefor subfactors, and vice versa. In particular, it turns out to be much easier to construct and classify subfactors. In my talk I'll discuss what the recent explosion of progress in subfactor construction and classification suggests for VOAs.
May 21, 2013
TBA
May 28, 2013
How to chase the Jacobian conjecture.
Martin Weissman, University of California, Santa Cruz
The Jacobian conjecture states that a polynomial map from
C^n to C^n with nowhere-vanishing Jacobian determinant is a bijection.
After giving some background on polynomial maps on affine space, and
reductions of the Jacobian conjecture to simpler cases, I will discuss
tools for chasing the conjecture. Specifically, I will introduce
techniques from representation theory, model theory, and
combinatorics. Audience members are warned that chasing the Jacobian
conjecture can become a bad habit that crowds out more promising
research, and the whole conjecture might be false.
June 4, 2013
Singularly Beautiful Algebraic Curves
Joel Langer, Case Western Reserve University
The theorems of Gauss on constructible n-gons and Abel on uniform subdivision of the Bernoulli lemniscate place the circle and lemniscate among only a handful of algebraic curves known to possess such nice subdivision properties.
For these curves, unit speed parameterization (or its norm) extends to meromorphic (elementary or elliptic) functions on the complex plane. Such parameterizations are already rare, as may be seen from the ‘polyhedral geometry’ on a (complex) curve C; this is defined via the meromorphic quadratic differential on C induced by dx2 + dy2.
The required behavior of this quadratic differential forces rather special singularities of C and it follows, e.g., that Bernoulli lemniscates are the only curves of degree at most four with compact polyhedral geometry. In this talk, such results and related examples will be illustrated via a graphical technique for visualizing the (real) foci and polyhedral geometry of an algebraic curve.
