# ACGT Seminar

The Algebra, Combinatorics, Geometry, and Topology (ACGT) Seminar meets on Tuesdays at 12:45-2:00pm in Room 164 of the Adel Mathematics Building. If you are interested/willing to give a talk, please contact Dana C. Ernst or Michael Falk, ACGT Co-coordinators.

# Schedule Spring 2018

Note that talks are listed in reverse chronological order.

## Weak maps of matroids and higher resonance IV

Date: 4/10/18

Speaker: Mike Falk (NAU)

## Weak maps of matroids and higher resonance III

Date: 4/3/18

Speaker: Mike Falk (NAU)

Abstract: I’ll define weak maps of matroids and Orlik-Solomon algebras, I’ll construct a weak map of the braid matroid based on a representation via bit strings, and examine the consequences for higher resonance in the braid arrangement. This talk is independent of the previous parts.

## Weak maps of matroids and higher resonance II

Date: 3/27/18

Speaker: Mike Falk (NAU)

Abstract: I’ll recap part I, in which the various aspects of degree-one resonance were surveyed. Then I’ll define weak maps of matroids, and construct a weak map of the braid matroid based on a representation via bit strings, and examine the consequences for higher resonance in the braid arrangement. This talk is independent of part I.

## Error-correcting codes

Date: 3/13/18

Speaker: Noah Aydin (Kenyon College)

Abstract: One of the most important and challenging problems in coding theory is to construct codes with good parameters. A promising and fruitful approach has been to focus on the class of quasi-twisted (QT) codes which includes cyclic and constacyclic codes as a special case. In this talk, we will discuss some recent results, both computational and theoretical, related to QT codes. Although constacyclic codes and QT codes have been the subject of numerous studies and computer searches over the last few decades, we have been able to discover a new fundamental result about the structure of constacyclic codes which was instrumental in our comprehensive search method for new QT codes. We also introduce a generalization of QT codes that we call multi-twisted (MT) codes and prove a number of facts about them. Finally, we will briefly mention some recent work on codes over various rings.

### Modules, fields of definition, and the Culler-Shalen norm

Date: 3/6/18

Speaker: Charlie Katerba (Montana State University)

### Weak maps of matroids and higher resonance I

Date: 2/27/18

Speaker: Mike Falk (NAU)

### Structure of braid graphs for reduced words in Coxeter groups

Dates: 2/13/18, 2/21/18

Speaker: Dana Ernst (NAU)

Abstract: Every element $w$ of a Coxeter group $W$ can be written as an expression in the generators, and if the number of generators in an expression (including multiplicity) is minimal, we say that the expression is reduced. Every pair of reduced expressions for the same group element are related by a sequence of commutations and so-called braid moves. We say that two reduced expressions are braid equivalent if they are related by a sequence of braid moves. Given a reduced expression $\overline{w}$ for a group element $w$, we can form the corresponding braid graph whose vertices are the reduced expressions that are braid equivalent to $\overline{w}$ and two reduced expressions are connected by an edge if there is a single braid move that converts one reduced expression into the other. In this talk, we will describe the overall structure of braid graphs for reduced expressions in Coxeter groups of type $A$ and $B$ in terms of products of graphs.

### A taxonomy of quadrilaterals

Date: 1/30/18

Speaker: Jim Swift (NAU)

### An introduction to the theory of square-free monomial ideals and Stanley-Reisner rings

Speaker: Mike Falk (NAU)

Date: 1/23/18

Abstract: A square-free monomial ideal is an ideal in a multi-variable polynomial ring that is generated by monomials with no repeated factors. These naturally correspond to abstract simplicial complexes, and their algebraic structure is intimately connected to the combinatorial structure of the associated complex. For instance, they were originally used to characterize the face vectors of simplicial polytopes with rational vertices. I will define the relevant concepts, display some examples, and sketch the associated homological algebra and combinatorics.