# ACGT Seminar

The Algebra, Combinatorics, Geometry, and Topology (ACGT) Seminar meets on Tuesdays at 12:45-2:00pm in Room 146 of the Adel Mathematics Building. If you are interested/willing to give a talk, please contact Dana C. Ernst, ACGT Coordinator.

# Schedule Spring 2017

Note that talks are listed in reverse chronological order.

### Memories from Toronto and Elementary Topoi

Speaker: Philip Doi (graduate student at NAU)

Dates: 4/11/2017

Abstract: The conception of a topos results in a mathematical object that is complex enough to codify nontrivial structure; oft described as a category “where one can do mathematics”. For instance, the category of sets, given as set, pervades most of modern mathematics in some way, with its existence presumed indirectly by the Zermelo-Fraenkel-Choice axioms. There are other examples, some of which are experienced subliminally in maths where category theory is less explicit. This presentation will include some historical motivation as well as the story of how I came across the notion of a topos when discussing a certain Toronto problem with a category theorist. Since I am approaching the subject from the perspective of set theoretic topology and as a layperson, I’ll sketch out the formalization of elementary topoi, showing their use in describing problems in axiomatic set theory. Additionally, we can demonstrated the desirability of having a topos by contrasting topoi to a few interesting non-examples such as the category of NF sets (induced by W. V. Quine’s New Foundations set theory).

### Permutations with triple inversions and volume under 3D lattice paths

Speaker: Julie Baum and Krista Young (graduate students at NAU)

Dates: 3/28/2017, 4/4/2017

Abstract: A permutation $w=w(1)w(2)\cdots w(n)\in S_n$ is said to have a descent in position $k$ if $w(k)>w(k+1)$ and an inversion of $w$ is a pair $i<j$ such that $w(i)>w(j)$. It is well-known that there is a bijection between the set of permutations in $S_n$ that have at most a descent in position $k$ and the set of 2D lattice paths from $(0,0)$ and $(k,n-k)$. This bijection has the property that a permutation with $m$ inversions corresponds to a lattice path having area $m$ under the path. In this talk, we will explore a bijection between permutations with at most two descents in fixed positions and certain 3D lattice paths. In this case, volume “under” the path will correspond to a type of generalized inversion, which we refer to as a “triple inversion”.

### A Cubic Residue Version of Chapman’s Evil Determinant Problem

Speaker: Ryan Wood (undergraduate at NAU)

Dates: 2/28/2017, 3/6/2017

Abstract: In 2001, R. Chapman conjectured that a special infinite class of matrices, constructed using quadratic residue symbols, had constant determinant values. This conjecture, known as Chapman’s Evil Determinant Problem, was resolved in 2014. In this series of talks, we will present a generalization–involving cubic residues–of Chapman’s problem. Basic knowledge of number theory and linear algebra is helpful but not required.

### Factorizations of the longest element: wiring diagrams and higher Bruhat orders

Speaker: Michael Falk (NAU)

Dates: 1/31/2017, 2/7/2017, 2/14/2017, 2/21/2017

Abstract: Having established last time the representation of rank-three oriented matroids matroids by wiring diagrams, we’ll finish that discussion by showing that wiring diagrams in the uniform case correspond to factorizations of the longest permutation. We will then describe the higher Bruhat orders, and at least state the result relating the number of elements to such factorizations. An interpretation of this result is that the number of factorizations is the number of chambers of a certain hyperplane arrangement.

### Oriented matroids, wiring diagrams, factorizations of the long permutation

Speaker: Michael Falk (NAU)

Date: 1/24/2017

Abstract: We will explain the relationship between oriented matroids, arrangements of pseudo-spheres, and, in rank three, wiring diagrams, and use this to establish the bijective correspondence between rank-three oriented matroids and admissible sequences of permutations. Specializing to the case of uniform oriented matroids, one deduces that the number of isomorphism classes, up to re-orientation, of uniform rank-three oriented matroids on $n+1$ points is equal to the number of factorizations of the longest element in the Coxeter system of type $A$.