Colloquia

Richard Askey, University of Wisconsin

Geometry through the School Years
Richard Askey, University of Wisconsin-Madison
November 19, 2009

Abstract:  There has been a lot written about teaching arithmetic to children and the National Mathematics Advisory Panel had a focus on algebra and preparing students to take it.  Geometry has been ignored, yet it is the part of mathematics where our students do most poorly once one gets beyond the stage of names for different figures.  This talk with start with elementary school geometry.

One essential difference between a triangle and a quadrilateral is that a triangle is rigid while a quadrilateral is not.  This can easily be illustrated with fingers.  Other topics will include such things as how the same figures can be used to get the area of a triangle, the angle sum of a triangle and later even the addition formula for sines and cosines, and why there is a factor of 1/2 in the formula for the area of a triangle but 1/3 in the formula for the volume of a pyramid. 

 

 

Alex Freire, University of Tennessee

Networks of curves or surfaces moving by curvature

Alex Freire, University of Tennessee

December 3, 2009

 

Abstract: In the solidification of certain metal alloys, one observes the following pattern: a fast phase separation stage is followed by a much slower coarsening stage, in which the interfaces separating the phases move so as to minimize the total perimeter- that is, move by a curvature law.  This problem can be modeled by a parabolic system with a small parameter. Furthermore, one observes that, at the junctions where three interfaces of the network meet, the angles between them are constant throughout the motion.

 

In the limit when the parameter goes to zero we obtain a differential-geometric evolution problem with non-standard boundary conditions. Even for networks of curves a general global existence theorem is still missing. I will discuss the p.d.e. model, what is known for curve networks and very recent results on local and global existence for triple-junction networks of surfaces moving by mean curvature. There will be movies.

Cristina Caputo, University of Texas at Austin

Title tba
Cristina Caputo, University of Texas at Austin
February 25, 2010

 

Al Baernstein, Washington University in St. Louis

Al Baernstein, Washington University in St. Louis
title tba
March 4, 2010

Maggy Tomova, University of Iowa

Maggy Tomova, University of Iowa
March 18, 2010
title tba

 

Weights in generalizations of Serre's conjecture

Weights in generalizations of Serre's conjecture
Michael Schein, Bar-Ilan University
August 27, 2009

Abstract:  Given a prime p and a modular form, one can associate to it a two-dimensional mod p representation of the absolute Galois group of Q. Given such a mod p representation, J.-P. Serre conjectured in the 1970's when it should arise from a modular form in this way, and, if so, what the weights and levels of the associated modular forms are. We will explain these notions and discuss generalizations of the conjecture to totally real fields and to Galois representations of arbitrary dimension, as well as theorems toward these new conjectures.

Growth Networks

Growth Networks
Raja Kali and Javier Reyes, department of economics, University of Arkansas

Abstract: Is there a way to kindle acceleration in the economic growth rate of a country? What role do trade and comparative advantage play in this process? These are among the most enduring and important questions in economics. This research project aims to use a novel, network-based approach to international trade to make progress in decoding the mystery of growth acceleration.

One of the general results of the literature in complex networks is that high performance networks in many settings (biological, technological, social, economic) have the "small world" property. In other words, in several contexts, the small world seems to be an "optimal" topology. A small world is a network whose topology combines high clustering among nodes with high connectivity (short path length) across nodes. Due to high clustering, such networks are likely to have strong spillovers between nodes, and short path length provides the potential for long range leaps across the network. Both features are advantageous in the context of economic development and growth. Could it be that the key to growth acceleration is whether the pattern of product specialization of a country develops a "small world" topology before the take-off? This could come about because product space and the pattern of product specialization of a country, which are both evolving over time, overlap so as to create the conditions for a small world. If true, then this implies that the country's location in product space and its pattern of product specialization matter for its likelihood of experiencing a growth acceleration. Our research aims to marshal evidence to examine this insight.

On The Critical Group of Finite Projective Planes

On The Critical Group of Finite Projective Planes
Stuart Shirell, University of Arkansas
September 17, 2009

The critical group of a graph on n vertices is defined to be $Z^n/(Image(M))$, where M is the Laplacian matrix associated to the graph. This is an isomorphism invariant and is largely viewed as relatively robust.

Relatively little is known about the critical group in general, however.  For example the critical groups of complete graphs, bipartite complete graphs, and the n-cube are (mostly) known, but little else is known about these groups or how they reflect the symmetry properties of the underlying graph. We determine completely the critical groups of non-degenerate, finite projective planes and show that all projective planes of a given order have isomorphic critical groups.

 

Complex Analysis beyond One Dimension

Mehmet Celik, University of Arkansas at Fort Smith
Complex Analysis beyond One Dimension
October 8, 2009

Abstract: The study of functions defined on multidimensional complex space is called the theory of several complex variables. Classical complex analysis is the study of functions defined on one dimensional complex space. The main object of study in both theories is complex analytic functions. Some parts of the theory of analytic functions, such as the maximum principle, normal limit of nowhere-zero analytic functions, and Cauchy's estimates for derivatives, are essentially the same in all dimensions. The most interesting parts of the multidimensional complex theory are the features that differ from the one dimensional theory. At the beginning of the presentation, we will mention some differences between these two theories from different points of view. Then, we will focus on one particular difference and explore the world of complex analysis beyond one dimension. The talk is accessible to any graduate students in mathematics.

Nonlinear Geometric Optics

Nonlinear Geometric Optics
Jeffrey Rauch, University of Michigan
October 19, 2009

Abstract:  Geometric optics is a family of methods to construct approximate solutions to partial differential equations.  The solutions have a small parameter often representing a wavelength.The approximations are accurate in the short wavelength limit. In this limit, direct numerical simulation is not possible.  The history is long and rich. In this talk we recall some of the  history and basic ideas of the subject and recent advances,  notably for nonlinear hyperbolic equations. For the compressible Euler equations there are solutions with three incoming wave trains and with outgoing waves with velocities dense in the unit sphere.  My work in this area is with J.L. Joly and G. Metivier.

Homogenization of Elliptic Boundary Value Problems

Homogenization of Elliptic Boundary Value Problems
Zhongwei Shen, University of Kentucky
October 22, 2009

Abstract: In this talk I will discuss recent progress on boundary value problems in Lipschitz domains for a family of  second order elliptic equations with rapidly oscillating coefficients, arising the theory of homogenization. This is a joint work with Carlos Kenig.
 

Is Width Additive?

Is Width Additive?
Ryan Blair, University of California, Santa Barbara
November 17, 2009

Abstract: Width is an invariant of knots that depends on the number of minima and maxima of the knot as well as their relative positions. The behavior of width with respect to connected sum has been studied for three decades. Originally, it was conjectured that width is additive under the operation of connect sum. I will outline classic results pertaining to the question and discuss ongoing joint work with Maggy Tomova that suggests width is not additive.