Colloquium SeriesSpring 2013Unless otherwise noted, talks will be at 3:50 PM in O'Connor 103. Also, there will be refreshments before each talk in O'Connor 31 at 3:30 PM. Tuesday April 16th Speaker: Martin Weissman, UC Santa Cruz Title: The arithmetic of arithmetic hyperbolic Coxeter's groups. Abstract: In the 1990s, John Conway used a ternary regular tree to visualize the classical theory of binary quadratic forms. After covering some highlights of Conway's "topograph," I will introduce a more general framework of arithmetic Coxeter's groups. These yield more general "topographs" which have been used most recently by Savin and Bestvina to study binary Hermitian forms. This talk will require no familiarity with group theory or quadratic forms  only addition, subtraction, and multiplication of integers will be used for the majority of the time.
Tuesday, April 30th Speaker: David Uminsky, University of San Francisco Title: On Why Nature Rarely Assembles into Spheres Abstract: Soap bubbles on their own naturally select a sphere as their preferred shape as it is the solution which best balances the desire to minimize surface area while capturing a fixed amount of volume. Despite their natural beauty Nature rarely selects empty shells, or spheres, as the preferred shape to assemble into especially when particles to talk to one another do so over different length scales. Two notable exceptions are virus self assembly and the spontaneous assembly of macroions into super molecular spherical structures call "Blackberries." In this talk we will show how mathematics is just the right tool to explain this phenomena and help us predict when spheres will be the favored structure. The same tools will allow us to design nano particles to assemble into a variety of spherical patterns.
Tuesday, May 14th Speaker: Corey Irving, Santa Clara University Title: Generalized Barycentric Coordinates: An Introduction and an Application to Algebraic Geometry Abstract: Barycentric coordinates allow one to express points of a polygon as convex combinations of the vertices. For triangles and simplices these coordinates are uniquely defined and well known. However, for ngons with n>3, they are not unique and less well known. For the first part of the talk we discuss various ways to define barycentric coordinates for general ngons. The second part will focus on one type of barycentric coordinates, Wachspress coordinates, which are rational functions on the polygon, and we examine the algebraic variety they define. The first part of the talk will only assume basic freshman mathematics knowledge. The second part will assume some abstract algebra and very basic algebraic geometry. This is a summary of dissertation work done at Texas A&M where my advisor was Frank Sottile.
Tuesday, May 28th Speaker: Tom Church, Stanford University Title: Combinatorial stability and representation stability Abstract: If you choose a squarefree polynomial f(T) in F_q[T] uniformly at random, it will have slightly less than one linear factor on average. The exact value of this expectation depends on the degree of f(T), but as deg f(T) goes to infinity, the expectation stabilizes and converges to 1  1/q + 1/q^2  1/q^3 + ... = q / (q+1).
In joint work with J. Ellenberg and B. Farb, we proved that the stabilization of this combinatorial formula, and other statistics like it, is equivalent to a representationtheoretic stability in the cohomology of braid groups. I will describe how combinatorial stability for many different geometric counting problems can be converted to questions of representation stability in topology, and vice versa. These problems include not just statistics of squarefree polynomials, but also statistics of eigenvectors in finite matrix groups like GL_n(F_q). This talk will assume no background, and is intended for a general mathematical audience.
Tuesday, June 4th Speaker: Stephen Michael Miller, Santa Clara University Title: The distance from a point to its opposite along the surface of a box Abstract: Given a point (the “spider”) on a rectangular box, we would like to ﬁnd a formula for the minimal distance along the surface to its opposite point (the “ﬂy”  the reﬂection of the spider across the center of the box). Without loss of generality, we can assume that the box has dimensions 1 × a × b with the spider on one of the 1 × a faces (with a ≤ 1). The shortest path between the points is always a line segment for some planar ﬂattening of the box by cutting along edges. We then partition the 1 × a face into regions, depending on which faces this path traverses. This choice of faces determines an algebraic distance formula in terms of a, b, and suitable coordinates imposed on the face. We then partition the set of points (a, b) by isotopy of the borders of the 1 × a face’s regions and a labelling of these regions.
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Abstracts of previous talks are available here.
