- Speaker: Eric Hanson
- Title: Picture Groups and Polygonal Lattices
- Abstract: A finitely presented group, called the picture group, can be associated to any finite polygonal lattice. The motivating example is the torsion lattice of any finite-dimensional algebra (over a field) satisfying certain finiteness properties. In this talk, we will define what it means for a lattice to be polygonal and how to build the corresponding picture group. We will then give an overview of a recent result showing that for the torsion lattice of a Nakayama algebra, the corresponding picture group is CAT(0). No familiarity with either the theory of lattices or the representation theory of algebras will be assumed. As much as possible, representation-theoretic content will be kept light in this talk. This is joint work with Kiyoshi Igusa.
- Speaker: Duncan Levear
- Title: A Bijection for Shi arrangement faces
- Abstract: A collection of hyperplanes induces a cell structure on n-dimensional space. We consider the combinatorial question of counting the number of cells of each dimension. There is a rich history behind counting the highest-dimensional cells (called the regions), but the lower-dimensional faces have received less attention. In this talk, we will survey the combinatorics of hyperplane arrangements, and describe our recent work, which places the faces of the Shi arrangement in explicit correspondence with a set of decorated binary trees. Our bijection is a generalization of a bijection for regions defined by Bernardi in 2015.
- Speaker: Ian Montague
- Title: Decomposing Knot Floer Homology
- Abstract: In this talk, I will describe an approach to define explicitly computable invariants of tangles in B^3 using the framework of Douglas-Lipshitz-Manolescu's cornered invariants, which extends Heegaard Floer Homology to an (almost) twice-extended TQFT. No prior knowledge of Heegaard Floer Homology will be assumed in this talk, and technical statements will be passed over in favor of drawing pictures whenever possible.
- Speaker: Zach Larsen
- Title: Basics of Gröbner Bases and Elimination Theory
- Abstract: In contrast to the scheme-theoretic approach to algebraic geometry, we will go back to basics and look at some fundamental techniques for dealing with systems of polynomial equations. Choosing particularly nice generating sets, called Gröbner bases, not only makes computations much easier but also illuminates the general theory. We will look at a few examples, and then prove some important theoretical results, including the fact that projective varieties are complete.
- Speaker: Rose Morris-Wright
- Title: A curve complex for Artin groups
- Abstract: One important tool in the study of mapping class groups is the curve complex. When considering braid groups as mapping class groups this complex has an algebraic interpretation involving the parabolic subgroups of the braid group. Recently I have been working to develop a complex which would generalize the curve complex for braid groups to Artin groups of FC type. In this talk, I will first briefly review some important facts about the curve complex and then discuss how the construction of this complex can be extended to Artin groups.
- Speaker: Duncan Levear
- Title: Chaos of Card Shuffling: The Invasion of Total Randomness
- Abstract: We give a streamlined proof of a classical theorem: every shuffling scheme will eventually randomize a deck of cards. The main ingredient is the group algebra. We will also explain some seminal results from the last century in the theory of card shuffling.