Discrete Geometry & Combinatorics Seminar: Spring 2015

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The Combinatorics of Symmetric Group Representations

Michael Dougherty | April 13

• The representation theory of the symmetric group can be described combinatorially using diagrams called Young tableaux. This description is very pleasing, but it can be hard to find explicit examples of decompositions into irreducible representations in the literature. This talk will focus on a surprisingly nice basis for the symmetric group algebra using these techniques, as well as some of the current research efforts on this topic. No background is necessary, although general awareness of representations may be useful.

Partitions, Polytopes, and Catalan Combinatorics

Jon McCammond | April 20

• As every budding combinatorialist knows, there are many, many structures with Catalan many elements and some of my favorites are the non-crossing partition lattice and the vertices/faces of the simple/simplicial associahedron. In this talk I will review these objects and their generalizations to arbitrary finite Coxeter groups and beyond. I will also describe a new and somewhat surprising topological connection between them. In the fall I gave a talk on this research in the seminar with the title "Noncrossing hypertrees" but I failed to defined them before the talk ended. So maybe a more appropriate title would be "Noncrossing hypertrees 2: now with more noncrossing hypertrees.

A Generalization of the BNS Invariant

Daryl Cooper | April 27

• Given a group G the BNS invariant is a certain subset of Hom(G,R). It is connected to a simple geometry/topology picture as well as algebra.
  
  Tillmann and I generalized it to a function f : Hom(G,R) ---> (non-negative integers) such that f^{-1}(0) is the BNS invariant.
  
  The talk will not assume any prior knowledge.

Constructing Lie Algebras from Dynkin Diagrams

Gordon Kirby | May 4

• The classification of complex semisimple Lie algebras is generally reduced to classifying irreducible root systems via connected Dynkin diagrams. In my talk I am interested in going the reverse direction and explicitly recovering a Lie algebra from a Dynkin diagram. This can be done in multiple ways, a few of which I will discuss, including playing the "Mutation and Numbers Games" on a graph corresponding to Dynkin diagram.

Daily Life in Different Geometries

Steve Trettel | May 18

• This talk grew out of thinking through one of the funniest exercises I've ever seen in a math textbook (Ex 2.7.9 in Thurston's book, for those interested): "Describe the sensation of a person in S^3 being left-multiplied by a 1-parameter subgroup". A lot of the time we think about the three sphere or hyperbolic space by using a model but these are badly distorted and make it difficult to visualize what life would be like "inside" one of these spaces. So, this week I'll give an informal introduction to their intrinsic geometry. We'll start by reviewing what it feels like to live inside R^3, and then see what changes when we introduce either positive or negative curvature.
  
  For motivation, here's some of the questions we will try to answer in this talk:
  • Why do onions feel hollow in hyperbolic space?
  • Why is it hard to package a ream of paper in the 3-sphere?
  • Why are bikes are better than cars in curved spaces?
  • Can you tell what geometry you live in by watching your friend back away from you?
  • Why can't you feel how fast you're going in Euclidean space?
  • Is it always dark in hyperbolic space?