# Algebra and Combinatorics Seminar

## Fall 2019

 Date: August 30, 2019 Time: 3:00pm Location: BLOC 628 Title: Organization Meeting Date: September 6, 2019 Time: 3:00pm Location: BLOC 628 Speaker: Dustin McPhate, Texas A&M University Title: Resolutions for truncated Ore extensions Abstract: We begin by introducing the notions of a twisted tensor product of algebras and the class of algebras known as Ore extensions. We will then develop a method for constructing projective resolutions for modules over a certain class of twisted tensor products. We do this by first taking note of the conditions necessary to think of these algebras as a type of Ore extension and then use this parallel to extend recent results. Date: September 13, 2019 Time: 3:00pm Location: BLOC 628 Speaker: Alexander Ruys de Perez, Texas A&M University Title: Max Intersection Complete Codes and the Factor Complex Abstract: A place cell is a neuron corresponding to a subset of Euclidean space known as a place field, that will fire if and only if the individual to which the neuron belongs is within that place field. The firing patterns of a collection of n place fields can be represented by a neural code C on n neurons, which is a subset of 2n . Determining whether C is convex, meaning that there is an arrangement of convex place fields for which C is the code, remains an open problem. A sufficient condition for convexity is being max intersection complete: any intersection of maximal codewords is also a codeword. Currently, the only way to determine this property is to evaluate all such intersections. We present a new method to determine max intersection completeness by introducing a simplicial complex for a code C called the factor complex ∆O(C) of C. We show how to construct ∆O(C) using Stanley-Reisner theory, describe how ∆O(C) encodes information about C, and give an algorithm to check whether C is max intersection complete using the factor complex of a closely related code. Date: September 20, 2019 Time: 3:00pm Location: BLOC 628 Speaker: Gillian Grindstaff, University of Texas Title: Geometric comparison of phylogenetic trees with different leaf sets Abstract: The metric space of phylogenetic trees defined by Billera, Holmes, and Vogtmann, which we refer to as BHV space, provides a natural geometric setting for describing collections of trees on the same set of taxa. However, it is sometimes necessary to analyze collections of trees on non-identical taxa sets (i.e., with different numbers of leaves), and in this context it is not evident how to apply BHV space. Davidson et al. recently approached this problem by describing a combinatorial algorithm extending tree topologies to regions in higher dimensional tree spaces, so that one can quickly compute which topologies contain a given tree as partial data. In this talk, based on joint work with Megan Owen, I'll present an extension algorithm for metric trees inspired by their approach, which inverts a natural projection on tree space called tree dimensionality reduction (TDR). This algorithm can be used to define and search a space of possible supertrees and, for a collection of tree fragments with different leaf sets, to measure their compatibility. Date: September 27, 2019 Time: 3:00pm Location: BLOC 506A ** Speaker: Amy Huang, Texas A&M University Title: Syzygies of determinantal thickenings and gl(m|n) representations Abstract: The coordinate ring $S = \mathbb{C}[x_{i,j}]$ of space of $m \times n$ matrices carries an action of the group $\mathrm{GL}_m \times \mathrm{GL}_n$ via row and column operations on the matrix entries. If we consider any $\mathrm{GL}_m \times \mathrm{GL}_n$-invariant ideal $I$ in $S$, the syzygy modules $\mathrm{Tor}_i(I,\mathbb{C})$ will carry a natural action of $\mathrm{GL}_m \times \mathrm{GL}_n$. Via BGG correspondence, they also carry an action of $\bigwedge^{\bullet} (\mathbb{C}^m \otimes \mathbb{C}^n)$. It is a recent result by Raicu and Weyman that we can combine these actions together and make them modules over the general linear Lie superalgebra $\mathfrak{gl}(m|n)$. We will explain how this works and how it enables us to compute all Betti numbers of any $\mathrm{GL}_m \times \mathrm{GL}_n$-invariant ideal $I$. The latter part will involve combinatorics of Dyck paths Date: October 4, 2019 Time: 3:00pm Location: BLOC 628 Speaker: Daniel Eman, Wisconsin Title: Asymptotic syzygies Abstract: Asymptotic syzygies refers to the study of the syzygies of a variety under increasingly ample embeddings; the canonical example is to study the syzygies of projective space under the d-uple embedding as d goes to infinity. I’ll discuss some open questions related to asymptotic syzygies, and some recent work of myself and Jay Yang which uses a combinatorial model to produce new heuristics about this topic. Date: October 11, 2019 Time: 3:00pm Location: BLOC 628 Speaker: Chun-Hung Liu, Texas A&M University Title: Length of cycles in non-sparse graphs Abstract: Intuitively, "dense" graphs contain any "desired substructure". In this talk, we will use a unified tool to prove few conjectures and open questions proposed since 1980s with this flavor, where the "desired substructure" is a set of cycles whose lengths satisfy certain conditions. They include two conjectures of Thomassen about minimum degree, a conjecture of Dean about connectivity, a conjecture of Sudakov and Verstraete about chromatic number, and an optimal answer of a question of Bondy and Vince about minimum degree. Joint work with Jun Gao, Qingyi Huo and Jie Ma. Date: October 18, 2019 Time: 3:00pm Location: BLOC 628 Speaker: Tolulope Oke, Texas A&M University Title: Cup products on Hochschild cohomology of a family of quiver algebras Abstract: Let k be a field, q\in k. We derive a cup product formula on the Hochschild cohomology HH^*(A_q) of a family $A_q$ of quiver algebras. Using this formula, we determine a subalgebra of k[x,y] isomorphic to HH^*(A_q)/N, where N is the ideal generated by homogeneous nilpotent elements. We discuss a finite generation conjecture in relation to this family. Date: October 25, 2019 Time: 3:00pm Location: BLOC 506A ** Speaker: Frank Sottile, Texas A&M University Title: Composed Schubert Problems Abstract: A composition of Schubert problems is a construction that takes two Schubert problems on possibly different Grassmannians and gives a Schubert problem on a larger Grassmannian whose number of solutions is the product of the numbers of solutions of the original problems. This generalizes a construction that was discovered while classifying Schubert problems with imprimitive Galois groups.I will explain this construction and the product formula, which has both an algebraic and a bijective proof. I will also discuss how this construction is related to Galois groups of Schubert problems. This is joint work with Li Ying and Robert Williams. Date: November 1, 2019 Time: 3:00pm Location: BLOC 628 Speaker: Byeongsu Yu, Texas A&M University Title: Generalized Ishida Complex Abstract: Today, we will discuss the generalized Ishida complex. Masa-nori Ishida devised the Ishida complex to calculate local cohomology over the maximal ideal of a normal affine semigroup ring. We generalized this to calculate the local cohomology over all monomial supporting ideal. First of all, we will recall the definition of local cohomology and Čech Complex method. Then, we will investigate the properties of an affine monoid. Actually, an affine monoid can be viewed as a ring or as a polyhedral complex. A combination of these viewpoints allows us to have a cochain complex. This cochain complex comes from the polyhedral cone structure of the monomial ideal. Lastly, we will sketch to prove a statement that Generalized Ishida's complex calculates the local cohomology on affine semigroup ring over any monomial supporting ideal. Date: November 8, 2019 Time: 3:00pm Location: BLOC 628 Speaker: Peter Stiller, Texas A&M University Title: Edge Erasures and Chordal Graphs with Applications to Data Clustering Abstract: We prove several results about chordal graphs and weighted chordal graphs by focusing on exposed edges. These are edges that are properly contained in a single maximal complete subgraph. This leads to a characterization of chordal graphs via deletions of a sequence of exposed edges from a complete graph. Most interesting is that in this context the connected components of the edge-induced subgraph of exposed edges are 2-edge connected. We use this latter fact in the weighted case to give a modified version of Kruskal’s second algorithm for finding a minimum spanning tree in a weighted chordal graph. This modified algorithm benefits from being local in an important sense. In recent work with Culbertson, Dochtermann and Guralnik these results have been generalized, leading to a new result on Simon's conjecture concerning the extendable shellability of certain complexes. Date: November 15, 2019 Time: 3:00pm Location: BLOC 628 Speaker: Jurij Volcic, Texas A&M University Title: The Procesi-Schacher conjecture and positive trace polynomials Abstract: Hilbert’s 17th problem asked whether every positive polynomial can be written as a sum of squares of rational functions. An affirmative answer by Artin is one of the cornerstones of real algebraic geometry. Procesi and Schacher in 1976 developed a theory of orderings and positivity on central simple algebras with involution and posed a H17 problem for a universal central simple $*$-algebra of degree $n$. It has a positive answer for $n=2$. Recently we proved that the answer for $n=3$ is negative. Nevertheless, we obtained several positivity certificates (Positivstellensätze) for trace polynomials on semialgebraic sets of $n\times n$ matrices. The talk will be a gentle introduction to this mix of central simple algebras, invariant theory and real algebraic geometry. Date: November 22, 2019 Time: 3:00pm Location: BLOC 628 Title: Date: November 29, 2019 Time: 3:00pm Location: Title: Thanksgiving Holiday