| Date: | August 28, 2020 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Catherine Yan, Texas A&M |
| Title: | Enumeration with Moon Polyominoes and Beyond |
| Abstract: | A polyomino is a shape made by connecting certain numbers of equal-sized
squares, each jointed together with at least one other square along an
edge. A combinatorial model, the model of fillings of polyominoes, is
obtained by assigning a non-negative integer to each square of the polyomino.
This model emerged from the study of maximal monotone chains in various
combinatorial structures, including permutations, words, matchings, set
partitions, integer sequences, graphs, and multi-graphs. It provides a
unified frame in enumerative combinatorics and allows us to apply different
algebraic tools and combinatorial transformations. In this talk I will
show how to use this model to analyze some basic combinatorial
statistics. |
|
| Date: | September 4, 2020 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Chun-Hung Liu, Texas A&M |
| Title: | Asymptotic dimension of minor-closed families and beyond |
| Abstract: | The asymptotic dimension of metric spaces is an important notion in geometric group theory. The metric spaces considered in this talk are the ones whose underlying spaces are the vertex-sets of (edge-)weighted graphs and whose metrics are the distance functions in weighted graphs. A standard compactness argument shows that it suffices to consider the asymptotic dimension of classes of finite weighted graphs.
We prove that the asymptotic dimension of any minor-closed family of weighted graphs, any class of weighted graphs of bounded tree-width, and any class of graphs of bounded layered tree-width are at most 2, 1, and 2, respectively.
The first result solves a question of Fujiwara and Papasoglu; the second and third results solve a number of questions of Bonamy, Bousquet, Esperet, Groenland, Pirot and Scott.
These bounds for asymptotic dimension are optimal and generalize and improve some results in the literature, including results for Riemann surfaces and Cayley graphs of groups with a forbidden minor.
|
|
| Date: | September 18, 2020 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Jacob White, UT Rio Grande Valley |
| Title: | Combinatorial Hopf monoids and flag f-vectors |
| Abstract: | A combinatorial Hopf monoid in species provides an algebraic framework for understanding many polynomial and quasisymmetric function invariants in combinatorics. In this talk, we will discuss the problem of determining when the quasisymmetric functions associated to a combinatorial Hopf monoid are related to the flag f-vector of a family of relative simplicial complexes. We also discuss inequalities we obtain for the quasisymmetric functions in this situation, and describe some new examples of quasisymmetric functions, and combinatorial Hopf monoids. If there is time, we will also discuss F-positivity. |
|
| Date: | September 25, 2020 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Erika Ordog, Texas A&M |
| Title: | Minimal resolutions of monomial ideals |
| Abstract: | The problem of finding minimal free resolutions of monomial
ideals in polynomial rings has been central to commutative
algebra ever since Kaplansky raised the problem in the 1960s and
his student, Diana Taylor, produced the first general
construction in 1966. The ultimate goal along these lines is a
construction of free resolutions that is universal -- that is,
valid for arbitrary monomial ideals -- canonical, combinatorial,
and minimal. This talk describes a solution to the problem
valid in characteristic 0 and almost all positive characteristics. |
|
| Date: | October 2, 2020 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Bridget Tenner, DePaul University |
| Title: | Odd diagram classes of permutations |
| Abstract: | Recently, Brenti and Carnevale introduced an odd analogue of the classical permutation diagram. This gives rise to odd analogues of other permutation aspects such as length and inversions. Unlike in the classical setting, multiple permutations might have the same odd diagram. This prompts the study of "odd diagram classes": sets of permutations having the same odd diagram. We will discuss the rich combinatorial structure of these classes, including connections to pattern avoiding permutations and the Bruhat order.
This is joint work with Francesco Brenti and Angela Carnevale.
|
|
| Date: | October 9, 2020 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Cvetelina Hill, Georgia Tech |
| Title: | Tropical convex hulls of polyhedral sets |
| Abstract: | This talk is based on joint work with Sala Lamboglia and Faye Pasley Simon. During the first part of the talk we will focus on the tropical convex hull of convex sets and polyhedral complexes. We will introduce results on the tropical convex hull of a line segment and a ray, show that for sets in two dimensions tropical convex hull and ordinary convex hull commute, and give a characterization of tropically convex polyhedra. In the second part of the talk we will use these results to show that the dimension of a tropically convex fan depends on the coordinates of its rays and give a lower bound on the degree of a fan tropical curve using only tropical techniques.
The talk will be based on the work in this paper: https://arxiv.org/abs/1912.01253v2.
|
|
| Date: | October 23, 2020 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Kassie Archer, UT Tyler |
| Title: | Cycle structure of pattern-avoiding permutations |
| Abstract: | Given a permutation written in its one-line notation, we say that the permutation avoids a pattern if there is no subsequence in the same relative order as that pattern. Though much is known about pattern-avoiding permutations in general, relatively little is known about the cycle structure of these permutations. In the first half of the talk, I will discuss some things that are known about this topic, in what context some of these results appear, and some open questions. In the second half, I’ll discuss a recent example regarding 132-avoiding permutations comprised of only 3-cycles which has an interesting answer. |
|
| Date: | October 30, 2020 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Tekin Karadag, Texas A&M |
| Title: | Gerstenhaber bracket on Hopf algebra cohomology of a Taft algebra |
| Abstract: | We calculate the Gerstenhaber bracket on Hopf algebra and Hochschild cohomologies of the Taft algebra T_p for any integer p>2 which is a nonquasi-triangular Hopf algebra. We show that the bracket is indeed zero on Hopf algebra cohomology of T_p, as in all known quasi-triangular Hopf algebras. This example is the first known bracket computation for a nonquasi-triangular algebra. |
|
| Date: | November 6, 2020 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Lauren Snider, Texas A&M |
| Title: | On 2-dimensional parking functions |
| Abstract: | A 2-dimensional U-parking function is a pair of integer sequences whose order statistics are bounded by certain weights along lattice paths in the plane. U-parking functions are natural higher-dimensional generalizations of classical parking functions. Other interesting generalizations include (p,q)-parking functions (Cori and Poulalhon) and graphical parking functions (Postnikov and Shapiro) . In this talk, we will show that (p,q)-parking functions are in fact U-parking functions for a particular node-set U, as well as explicitly describe the overlap between U-parking functions and graphical parking functions when U is affine. Along the way, we will discuss some results regarding the enumeration of increasing U-parking functions. This is based on joint work with Catherine Yan. |
|
| Date: | November 13, 2020 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Jay Yang, U of Minnesota |
| Title: | Virtual Resolutions of Monomial Ideals |
| Abstract: | Virtual Resolutions a concept introduced by Berkesch, Erman, and Smith to provide a more meaningful theory of syzygies for modules over the Cox ring of a toric variety. I will discuss recent progress on applying the tools of monomial ideals to the problem of virtual resolutions, in particular a virtual analog of Hilbert's Syzygy Theorem, as well as recent progress on virtually Cohen-Macaulay modules and current work towards a virtual Auslander-Buchsbaum formula. This includes upcoming joint work with Christine Berkesch, Patricia Klein, and Michael C. Loper. |
|
| Date: | November 20, 2020 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Pablo Ocal, Texas A&M |
| Title: | Hochschild cohomology of general twisted tensor products |
| Abstract: | The Hochschild cohomology is a tool for studying associative algebras that has a lot of structure: it is a Gerstenhaber algebra. This structure is useful because of its applications in deformation and representation theory, and recently in quantum symmetries. Unfortunately, computing it remains a notoriously difficult task. In this talk we will present techniques that give explicit formulas of the Gerstenhaber algebra structure for general twisted tensor product algebras. This will include an unpretentious introduction to this cohomology and to our objects of interest, as well as the unexpected generality of the techniques. This is joint work with Tekin Karadag, Dustin McPhate, Tolulope Oke, and Sarah Witherspoon. |
|
Menu