Discrete Geometry & Combinatorics Seminar: Spring 2015
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Angles in Polytopes
| April 15
- Everyone learns in school that the measures of the angles in a euclidean triangle add up to pi. Are there corresponding restrictions on the dihedral angles and/or solid angles in a tetrahedron? In this talk I will describe some basic results on angles in high-dimensional polytopes and the various equations they satisfy. To be able to state these classical results (due to Sommerville and MacMullen) I will also introduce the notion of the incidence algebra of a poset as well as moebius inversion and zeta functions.
Groups, Trees and R-trees
| April 22
- This is an introductory talk. A tree is a connected graph with no loops.
An R-tree is a generalization that allows every point to be a vertex. If T is a tree
or R-tree there is a group Aut(T) of automorphisms of T. Automorphisms are of
two kinds depending on whether or not there is a fixed point. This gives a geometry
(T,Aut(T)). One may study a homomorphism from a given group G into Aut(T).
I will describe some applications to topology and group theory.
Noncrossing Partitions and the Milnor Fiber of the Braid Arrangement
| April 29
- In the 1960s, Milnor proved that certain intersections of balls with complex hypersurfaces can be expressed as fiber bundles. In particular, the complement of a central (non-affine) arrangement of complex hyperplanes is a fiber bundle over the circle. However, in even the "nicest" such setting, the braid arrangement, the homology of the fiber is unknown in general. In this talk we will discuss a way of leveraging the combinatorial structure of noncrossing partitions to create a geometrically appealing simplicial complex for the Milnor fiber. Knowledge of fundamental groups and covering spaces will be useful, although the content is otherwise introductory.
Tiling with Polyominoes
Gordon Kirby | May 6
- A polyomino is a shape made by connecting a certain number of equal sized squares, each joined together with at least on other square along an edge. The question of whether a region in the plane can be perfectly tiled using polyominoes drawn from a finite set of polyominoes is a well-established tiling problem. This talk will give an overview of necessary conditions for such a tiling to exist using combinatorial group-theoretic invariants from Conway and Lagarias' paper.
Braid Groups and Euclidean Simplices
Elizabeth Leyton Chisholm
| May 13
- Room Change: South Hall 6635
- In the early 2000s, Daan Krammer and Stephen Bigelow independently proved that braid groups are linear. They used the Lawrence-Krammer-Bigelow (LKB) representation for generic values of its variables q and t. The t variable is related to the Garside structure of the braid group used in Krammer's algebraic proof. The q variable, associated with the dual Garside structure of the braid group, has received less attention.
In this talk we give a geometric interpretation of the q portion of the LKB representation in terms of an action of the braid group on the space of non-degenerate euclidean simplices. In our interpretation, braid group elements act by systematically reshaping (and relabeling) euclidean simplices. The reshapings associated to the simple elements in the dual Garside structure of the braid group are of an especially elementary type that we call relabeling and rescaling.
Graphs and Quadratic Forms
Larry Gerstein | May 20
- The structure of an undirected graph is completely determined by a symmetric matrix: its adjacency matrix with respect to an ordering of its vertices; and that matrix can be used to define a quadratic form. The main purpose of the talk is to raise this question: "What can quadratic forms tell us about graphs?" As an initial answer, the theory of quadratic forms will be applied to the graph isomorphism problem. No knowledge of quadratic forms will be assumed, but the essential definitions and facts will be sketched.
Classifying Local Representations of the Braid Group
Joe Ricci | May 27
- Let s_1, ..., s_n be the standard generators of B_n and x_1, ..., x_n those of F_n. This talk will investigate local representations of the braid group into Aut(F_n) that is, ones where the generator s_i maps <x_i, x_i+1> into itself and acts as the identity on all other x_j’s. A great example comes from the action of B_n on a disc with n punctures. A classification of such representations will presented as well as generalizations and connections to other types of representations.
Conway's Proof of the Classification of Surfaces
| June 3
- It is a well-loved fact that each closed connected surface is homeomorphic to either the sphere, a connect sum of tori, or a connect sum of real projective planes. This has been known since the 1860's. In the early 1990's, Conway developed what he called the "Zero Irrelevancy Proof" (ZIP). This talk will cover Conway's proof.