Mathematics Colloquium Winter 2013
For further information please contact the Mathematics Department at 459-2969
January 8, 2013
January 15, 2013
Modules for elementary abelian p-groups
University of Aberdeen, Scotland
Many aspects of the representation theory and cohomology of finite groups are controlled by the elementary abelian subgroups. So a great deal of attention has been spent over recent years on understanding the representation theory of elementary abelian p-groups. My intention in this lecture is to introduce the subject, beginning with Chouinard's theorem and Dade's lemma, and then concentrate on developments over the last five or ten years.
January 22, 2013
Local Symmetry in Wallpaper Functions and Geodesics on Orbifolds
Frank A. Farris
Santa Clara University
The symmetry group of the image shown (*) is called p6 by crystallographers. It is generated by one translation and one rotation through 60 degrees. Characterizing the pattern by its group invariance ignores a salient feature: the apparent reflection symmetry of the turtle-shaped figures. The explanation of this additional, local symmetry, involves such far-flung ideas as closed geodesics of orbifolds, eigenvalues of the Laplacian, and quadratic number fields.
January 29, 2013
Physics Department, Stanford University
February 12, 2013
Theoretical issues in single molecule studies
Applied Math & Statistics, UCSC
We describe the mathematical framework of Langevin and Fokker-Planck equations for modeling molecular motors. We introduce several peculiar behaviors of molecular motors in comparison with macroscopic motors. Those include time scale of inertia, thermal excitation, and fluctuations in velocity. We then discuss the Stokes efficiency for molecular motors and study its connection to thermodynamics efficiency.
February 19, 2013
Counting characters in finite groups
Mathematics Department, University of Wisconsin
Associated with an arbitrary finite group G, there is a set of objects denoted Irr(G), the irreducible characters of G. Each irreducible character has a degree, which is a positive integer dividing the order of G, and it is a classical result that the number of irreducible characters of G is equal to the number of conjugacy classes of G. Now fix a prime number p. Two still unproved conjectures about an arbitrary finite group G are the McKay conjecture and the Alperin weight conjecture. The McKay conjecture describes the number of irreducible characters of G having degree not divisible by p, and the Alperin weight conjecture concerns characters at the opposite extreme, with degree divisible by the full p-part of the order of G. We will discuss these two conjectures and also a comparatively easy counting theorem for permutation groups that bears a striking formal resemblance to the Alperin weight conjecture. This permutation group result yields a character counting theorem related to groups of automorphisms of an arbitrary finite groups.
February 26, 2013
Hypersurfaces in hyperbolic space with support function
Mathematics Department, UCSC
In this talk I will introduce a new analytic tool for the study of hypersurfaces in hyperbolic space. This tool was developed in the spirit of the AdS/CFT correspondence of of quantum gravity and mathematical physics. I will illustrate how an immersed hypersurface can be unfolded into an embedded one along the normal flow when it has a global support function. As a corollary we obtain a strong Bernstein theorem for complete, immersed, constant mean curvature hypersurfaces in hyperbolic space.
March 5, 2013
Helicoidal minimal surfaces of every genus
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 in (*)). 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 configurations of such points. The speaker will try as best he can to avoid technicalities and to explain the underlying ideas.
March 12, 2013
Regularization of the Big Bang
A new result is described which shows how the big bang singularity can be regularized. This is accomplished by modeling the big bang using a Friedmann differential equation for the scale factor, r, as a function of time, t, where r=0 corresponds to the big bang. It is shown that the Friedmann equation can be reduced to a differential equation describing motion in a central force field. A McGehee regularization transformation can be applied which shows how to smoothly extend r=r(t) through r=0 as a function of t, yielding a unique branch extension. This is possible provided that a key parameter, called the equation of state, w, satisfies relative prime number conditions. From a mathematical perspective, this implies our universe can be viewed as an extension of a previous universe that collapsed. This approach holds a lot of promise as a new tool in cosmology. The implications of this result is discussed for other cosmological models. This methodology was previously applied to the motion of a test particle about a black hole (with F. Pretorius).