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, ACGT Coordinator.

Schedule Fall 2016

Note that talks are listed in reverse chronological order.

Discriminantal arrangements, higher Bruhat order, and uniform oriented matroids

Speaker: Michael Falk (NAU)

Dates: 12/6/2016

Abstract: This final ACGT of the semester will be a continuation of Falk’s talks from earlier in the semester and will tie together with ideas from Ernst’s talks.

A two-sided Coxeter complex

Speaker: T. Kyle Petersen (DePaul University)

Date: 11/29/2016

Abstract: The Coxeter complex of a finite reflection group is an abstract simplicial complex whose faces are left cosets of standard parabolic subgroups. It can also be realized as a triangulation of a sphere. I will present a triangulation of a sphere whose cells are naturally related to the two-sided cosets of parabolic subgroups. Several nice properties of the Coxeter complex carry over to the two-sided case, including combinatorial models for faces and numerical invariants like the h-polynomial. I will keep the discussion focused on the case of the symmetric group, so the talk should be accessible even to those unfamiliar with Coxeter groups in general.

On the size of a braid class of a permutation

Speaker: Dana C. Ernst (NAU)

Dates: 11/8/2016, 11/15/2016, 11/23/2016

Abstract: Recall that the symmetric group $S_n$ is generated by the adjacent 2-cycles $s_1:=(1,2)$, $s_2:=(2,3)$, $\dots$, $s_{n-1}:=(n-1,n)$. That is, every element in $S_n$ can be written in a word using the alphabet $s_1,s_2,\ldots,s_{n-1}$. If we express a permutation as a product of adjacent 2-cycles in the most efficient way possible, then we call the expression a reduced expression. There may be many different reduced expressions for a given permutation, but all of them can be written in terms of the same number of adjacent 2-cycles occurring in the product (called the length). In addition, recall that disjoint cycles commute, which implies that $s_is_j=s_js_i$ iff $|i-j|>1$. It turns out that $S_n$ also satisfies $s_is_js_i=s_js_is_j$ iff $|i-j|=1$. Replacing $s_is_j$ with $s_js_i$ when $|i-j|>1$ is called a commutation move. On the other hand, replacing $s_is_js_i$ with $s_js_is_j$ when $|i-j|=1$ is called a braid move. According to Matsumoto’s Theorem, any two reduced expressions for the same permutation are related via a sequence of commutation and braid moves. We say that two reduced expressions are commutation (respectively, braid) equivalent if we can obtain one from the other via a sequence of commutation (respectively, braid) moves. Both relations determine an equivalence relation on the set of reduced expressions for a group element. The corresponding equivalence classes are called commutation classes and braid classes, respectively. Counting, or even bounding, the number of commutation classes for an arbitrary element remains an open problem. In addition, the maximum size of a commutation class is known. However, in her MS thesis, D. Zollinger provides reasonable bounds on the size of a braid class for an arbitrary permutation. But I don’t understand her proof. In this talk, I will provide all of the necessary background and sketch a couple of her proofs in the hopes that I will better understand them and/or that you will help me fill in the details that I am missing.

Scattered Spaces and Classifying General Toronto Spaces

Speaker: Phillip Doi (NAU graduate student)

Dates: 10/25/2016, 11/1/2016

Abstract: In the realm of point-set topology, a Toronto space is a topological space, homeomorphic to each of its full cardinality subspaces. Under certain conditions, Toronto spaces are easy to classify, particularly if they are not Hausdorff (not T2). The challenge we consider lies in classifying Hausdorff Toronto spaces of uncountable cardinalities. We will explore results and open questions about consistency that concern aleph one, T2, Toronto spaces, as well as their relationship to scatter spaces. In this matter, we can come to natural generalizations of the original definition. Furthermore, some new (albeit straightforward) results will be presented, which may have gone unnoticed in the cataloging of folklore about these spaces.

When is $PX(n,k)$ Cayley?

Speaker: Steve Wilson (NAU emeritus)

Dates: 9/27/2016, 10/4/2016, 10/11/2016, 10/18/2016

Abstract: We will start by just talking about Cayley graphs and digraphs. If $G$ is a group, a Cayley graph of $G$ is a way to make pictorial the inner workings of the group, to understand its actions in a visual way. We will show some examples and see how the graph can illuminate the group. Then we will ask the question: given a graph, how can you tell if it is a Cayley graph for some group? And what group? And how? With that expertise established, we will turn our attention to the hated-and-feared Praeger-Xu graphs $PX(n,k)$. We will talk about the symmetries of the graph, and why its large group of symmetries makes it, counter-intuitively, difficult to handle. Then, after we have done all of that, we will seek some partial answers to the title question.

Grassmannians, families of arrangements, and Bruhat order

Speaker: Michael Falk (NAU)

Dates: 9/6/2016, 9/13/2016, 9/20/2016

Abstract: In this series of three lectures, we will study the space of generic hyperplane arrangements as a subset of the Grassmannian, motivating the (dual) definition(s) of Manin-Schechtman discriminantal arrangements, which generalize the Coxeter arrangements of type $A$. The so-called higher Bruhat order is defined using these arrangements, and is related to the infamous Heroin Hero problem for Coxeter groups; we will try to elucidate the connection. In the third lecture we’ll investigate possible generalization of the construction to a certain good compactification of the space of generic arrangements.