Department of Mathematics, WUSTL -
Math Club, Fall 2012 -- Spring 2013

Washington University in St. Louis, Cupples I Hall, One Brookings Drive, St. Louis MO. 63130

All meetings will be in Room 199, Cupples I.

Talks will run from 5:40 to 6:30 pm, and will be followed by free pizza.

Date: September 10, 2012

Movie; "The Geometry of Eero Saarinen's Gateway Arch"

Date: September 24, 2012

Professor Matt Kerr; "Finite Fourier transforms and Bernoulli polynomials"

Abstract: I've often wondered why undergraduate courses in linear algebra don't cover finite Fourier transforms, considering future mathematicians might use them in number theory and engineers in MATLAB. What I'll try to explain in this talk is what the two things in my title are, and how together they give you a beautiful way to compute the sums of a nice big set of infinite series -- elementary number theory at its best. I'll say something about more applied uses of finite FT's too.

Talk notes

Date: October 8, 2012

Professor Ari Stern "Simulating dynamical systems: classic methods and modern challenges"

Abstract: Ordinary differential equations (ODEs) are central to many areas of mathematics, and have a vast range of applications in science and engineering. However, most nonlinear ODEs cannot be solved in closed form. Fortunately, all hope is not lost: "numerical integrators" allow us to simulate these dynamical systems, obtaining approximate solutions to an arbitrary degree of accuracy. This talk will introduce a few classic methods for numerical integration, along with the theory used to analyze their stability and convergence. I will also discuss some recent research developments in the area of "geometric numerical integration," explaining why certain methods perform much better than others for simulating physical systems.

Date: October 22, 2012

Professor Anton Weisstein; "The Beauty of Untidiness: an Overview of Mathematical Biology"

Abstract: The long-standing collaboration between mathematics and physics has yielded enormous benefits to both fields. By contrast, the complexity of most biological systems has made them far harder to mathematize, leading to the life sciences being viewed as essentially non-mathematical. However, the development of technologies such as massively parallel genomic sequencing and ultrafast molecular modeling have generated new biological questions that require more specialized mathematical analysis. Just as the study of planetary movements stimulated the development of trigonometry and calculus, these new biological questions offer opportunities for advances in graph theory, statistical inference, and multiscale modeling. In this talk, I will give an overview of mathematical biology, focusing on five specific areas of collaboration. No specialized biology background is assumed.

Date: November 5, 2012

Professor Victor Wickerhauser; "What Haar, Walsh, Hadamard and Rademacher did with 0, 1, and -1"

Abstract: The four mentioned mathematicians found efficient ways to express arbitrary functions as linear combinations of simple functions taking just the values 0, 1, and -1. We will look at their ingenious constructions and discover some of the beautiful connections among their ideas.

Date: November 26, 2012

Professor John Shareshian; "Some divergent series studied by Euler, and permutation statistics"

Abstract: For a positive integer n, a permutation of n is a list of the integers 1 through n in any of the n! possible orders. A permutation statistic is a function that assigns a nonnegative integer to each permutation.

          Certain permutation statistics are called ``Eulerian", due to their connection with work of L. Euler on divergent series. One example of an Eulerian statistic is the excedance statistic, which assigns to each permutation w of n the number of elements i in {1,...,n} such that the number found in the i^th position of w is larger than i. For example, the permutation 13542 has two excedances, found in the second and third positions.

          Other statistics are called ``Mahonian", as the first difficult results on such statistics were found by P.A. MacMahon. One example of a Mahonian statistic is the inversion number, which assigns to each permutation w of n the number of pairs (i,j) of elements of {1,...,n} such that i is less than j but i appears after j in w. For example, 13542 has 4 inversions, namely, the pairs (2,3), (2,4), (2,5) and (4,5).

          After describing Euler's work and its connection to Eulerian statistics, I will (time permitting) discuss modern work on joint distributions involving one Eulerian statistic and one Mahonian statistic. In such work, given a Mahonian statistic f and an Eulerian statistic g, one tries to understand, for each n, the two-variable polynomial obtained by summing, over all permutations w of n, the monomial q^f(w) t^g(w).

Date: January 28, 2013

Professor Mark Alford; "Field theory, the Higgs particle and superconducting metals"

Abstract: The recently discovered Higgs particle and the long-known superconductivity of a cold metal are two aspects of the same basic phenomenon, which is spontaneous symmetry breaking. I will discuss how physicists understand this phenomenon in terms of field theory.

Date: 11-Feb

Professor David Levine; "Nash, Hirsch, and all that"

Abstract: What is a Nash equilibrium, why do economists care about it, and what do entropy and retracts have to do with it? All these questions and more will be answered.

Date: 25-Feb

Professor Jeff Gill; "The Variable Effect of War on Longterm Childhood Mental Health Outcomes"

Abstract: While children are routinely exposed to armed conflicts ranging from minor skirmishes to full-scale national wars, there is relatively little scholarship on the psychological and emotional consequences they face in either the short or long term.  Through continued partnership with the AMPATH network of clinics (a collaborative effort between Indiana University and Moi University Schools of Medicine), we have access to the health clinic patient population in Eldoret, western Kenya.  How does the severity and variety of exposure to political violence affect children differently?  Can this causal process be modeled with a monotonic biological gradient?  Are there clustering and transmission patterns that can be identified?  Do there exist interaction effects between subjects, outcomes, or geographic regions that are supported by empirical data?  We propose using a case-control study to test the effects of violent conflict, understanding sampling effects and potential bias, estimating expected precision and validity, as well as obtaining demographic, political, and geographic background data.  The core of the statistical analysis is the specification of a Bayesian hierarchical model to directly incorporate grouping that results from clustering of effects, geography, and affliction, as well as demographics.

Date: 25-Mar

Professor John P. Cunningham; "R¹⁰⁰ is a big place"

Abstract: Remarkable things happen in high-dimensional Euclidean space. I will discuss some examples of this phenomenon, known to some as the curse of dimensionality, to others as the blessing of dimensionality, and to others as just weird. I will talk about intuition-breakdown in high-dimensional statistics, an issue which is becoming increasingly important in the modern world of huge data sets and important statistical challenges. We will discuss hypercubes, hyperspheres, high-dimensional Gaussian vectors, Fisher's LDA, correlation, concentration of measure, etc. I presume a basic knowledge of statistics, linear algebra, and calculus, and I will try to keep measure theory out of it.

Date: 8-Apr

Professor Heinz Schaettler; CANCELLED

Date: 22-Apr

Professor Kilian Weinberger; "What is Machine Learning"

In this talk I give a brief introduction of the discipline of Machine Learning, its history and the problems it attempts to solve. In addition, I provide a brief overview over a few simple approaches (nearest neighbors, logistic regression and artificial neural networks) and demonstrate them live on several examples.