# CMU Algebra and Combinatorics Seminar

## Spring 2017

### Meeting Times

Fridays, 11:15am–12:15pm, in Pearce 226.

### Schedule

 Date Speaker Affiliation Title 03/17 Jonathan Judge Trinity College A Class of Simple Modules for KLR Algebras 03/24 Cam McLeman University of Michigan, Flint On Missing Class Groups 04/07 Akalu Tefera Grand Valley State University On Proofs of Certain Combinatorial Identities 04/14 Gene Freudenburg Western Michigan University Embeddings and cancellation in affine algebraic geometry 04/28 Heather Jordon Albion College Cycle Decompositions of Complete Graphs and Circulants

### Abstracts

Speaker: Jonathan Judge
Title: A Class of Simple Modules for KLR Algebras
Abstract: The modules of families of Khovanov-Lauda-Rouquier (KLR) algebras yield a categorification of quantum groups. Many features of quantum group theory carry over to the KLR setting, including dual canonical bases and crystal bases. We provide a construction for a family of simple modules of KLR algebras. From this family of simple modules, one may obtain the building blocks of families of modules corresponding to the dual canonical and crystal bases.

Speaker: Cam McLeman
Title: On Missing Class Groups
Abstract: The ideal class group of a number field quantifies the extent to which its ring of integers fails to have the well-beloved unique factorization property. As such, this group is of tremendous importance in nearly all aspects of modern number theory, and yet so great is our ignorance that we do not yet know, for example, if there are infinitely many number fields with trivial class group. This talk will introduce the "arithmetic statistics" approach to the subject, blending analytic number theory, probability, partition theory, and some supercomputing to elucidate the distribution of class groups over the family of quadratic imaginary number fields. In particular, we offer a concise litmus test to decide whether a given $p$-group is likely to occur even once as the class group of any such field.

Speaker: Akalu Tefera
Title: On Proofs of Certain Combinatorial Identities
Abstract: In this talk we present combinatorial proofs of Ruehr's Identities. Furthermore, computer assisted proofs of some of Ruehr's identities will be discussed. We also present some open problems.

Speaker: Gene Freudenburg
Title: Embeddings and cancellation in affine algebraic geometry
Abstract: Affine algebraic geometry over the field $\mathbb{C}$ is the study of subvarieties of affine $\mathbb{C}^n$, and of polynomial algebras over $\mathbb{C}$. The Embedding Problem asks whether a given affine variety $V$ admits distinct embeddings in $\mathbb{C}^n$. That is, if $f\colon V \to \mathbb{C}^n$ and $g\colon V \to \mathbb{C}^n$ are algebraic embeddings, does there exist an algebraic automorphism $\alpha$ of $\mathbb{C}^n$ such that $g = \alpha\circ f$ ? The Cancellation Problem, which descends from Zariski's cancellation problem for fields, asks whether the condition that $V\times\mathbb{C}^n$ and $W\times\mathbb{C}^n$ are isomorphic implies $V$ and $W$ are isomorphic. An important case of each problem is the case where $V$ itself is an affine space. This talk will discuss the status of each of these problems, in addition to the affine Cremona groups, which are groups of algebraic automorphisms of $\mathbb{C}^n$, $n\ge 0$.

Speaker: Heather Jordon
Title: Cycle Decompositions of Complete Graphs and Circulants
Abstract: In this talk, we will discuss several different methods for decomposing complete graphs or almost complete graphs into cycles of a fixed length. We will also discuss how these techniques apply to circulant graphs.