MenuAlgebra and Combinatorics Seminar
Spring 2019
| Date: | January 25, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Aleksandra Sobieska, TAMU |
| Title: | Minimal Free Resolutions over Rational Normal Scrolls |
| Abstract: | Free resolutions of monomial ideals over the polynomial ring are well-studied and reasonably well-understood, though they are still an active area of research in commutative algebra. However, resolutions over quotients of the polynomial ring are much more mysterious, and even simple examples can violate the nicer properties that the polynomial ring provides. Starting in the 1990's, there is some work on resolutions over toric rings, a particular (and well-behaved) quotient of the polynomial ring. In this talk, we will present a minimal free resolution of the ground field over a specific toric ring that arises from rational normal scrolls. We also provide a computation of the Betti numbers for the resolution of the ground field for all rational normal $k$-scrolls. |
| Date: | February 1, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Qing Zhang, TAMU |
| Title: | Classification of Super-modular Categories |
| Abstract: | I will explain the connection between topological quantum computing and tensor
categories. The most promising proposal for topological quantum computation is
anyon braiding in 2D topological phases of matter. The algebraic theory of 2D
topological phases corresponds to tensor category theory in a very precise way.
I will focus on super-modular categories, a type of tensor category related to
fermionic topological phases similarly to the way that modular categories are
related to bosonic phases.
Super-modular categories are interesting from a purely mathematical standpoint as well. For example, any unitary pre-modular category is the equivariantization of a modular or super-modular category. In this talk, I will discuss a number of properties of super-modular categories parallel to those of modular categories. Time permitting, I will also discuss the classification of super-modular categories of rank 8. |
| Date: | February 8, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Sarah Witherspoon, TAMU |
| Title: | Hopf algebras and the cohomological finite generation conjecture |
| Abstract: |
A powerful tool for understanding representations of finite groups
is group cohomology. One reason why it is so powerful is that the
group cohomology ring is finitely generated and graded commutative,
thus pointing to geometric methods. Hopf algebras generalize groups
and include many important classes of algebras such as Lie algebras
and quantum groups. Their cohomology rings are known to be graded
commutative, and it is conjectured that they are finitely generated
whenever the Hopf algebra is finite dimensional.
In this introductory talk, we will define Hopf algebras, their cohomology rings, and mention their uses in representation theory. We will discuss Hopf algebras for which the conjecture has been proven and those for which it has not, including recent and ongoing research. |
| Date: | February 15, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Galen Dorpalen-Barry, University of Minnesota |
| Title: | Whitney Numbers for Cones |
| Abstract: | An arrangement of hyperplanes dissects space into connected components called chambers. A nonempty intersection of halfspaces from the arrangement will be called a cone. The number of chambers of the arrangement lying within the cone is counted by a theorem of Zaslavsky, as a sum of certain nonnegative integers that we will call the cone's "Whitney numbers of the 1st kind". For cones inside the reflection arrangement of type A (the braid arrangement), cones correspond to posets, chambers in the cone correspond to linear extensions of the poset, and these Whitney numbers refine the number of linear extensions. We present some basic facts about these Whitney numbers, and interpret them for two families of posets. |
| Date: | February 22, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Westin King, TAMU |
| Title: | Decompositions of Parking Functions on Trees |
| Abstract: | Parking functions describe a sequence of drivers attempting to find a place to park in a one-way linear parking lot. We can introduce a more complicated "parking lot" by considering directed trees and again ask if all the drivers can park. In this talk, I will give enumerative results concerning certain types of these generalized parking functions by considering decompositions based on the movement of drivers while they attempt to park. Additionally, I will discuss the number of "lucky drivers," those who park in the first parking space they encounter, and will find they are related to the Narayana numbers, which refine the ubiquitous Catalan numbers. |
| Date: | March 1, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Benjamin Briggs, University of Utah |
| Title: | Reflection Groups and Derivations |
| Abstract: | If you start with a polynomial ring (say over the complex numbers) and you factor out by the ideal generated by symmetric polynomials (of positive degree), then you get a very interesting ring. For example, it is isomorphic to the cohomology ring of a flag manifold.
How many derivations does this ring have (i.e. what is the dimension of the space of C-linear derivations)? The ring is also graded: how many derivations does it have in each degree? These are tricky to count, but it turns out there is a surprisingly nice formula. You get this formula by writing down a free resolution of the module of derivations, which for some reason turns out to be periodic. You can do all this by messing around with symmetric polynomials (but the combinatorics get quite complicated). It turns out though that this all works for certain reflection groups (all the real reflection groups included, and some complex reflection groups). I'll talk about this too, mainly focusing on the symmetric group example. |
| Date: | March 8, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Title: | |
| Date: | March 22, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Title: | |
| Date: | March 23, 2019 |
| Time: | 09:00am |
| Location: | Blocker building |
| Title: | CombinaTexas 2019 |
| Date: | March 24, 2019 |
| Time: | 09:00am |
| Location: | Blocker Building |
| Title: | CombinaTexas 2019 |
| Date: | March 29, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Ayo Adeniran, TAMU |
| Title: | Combinatorial Theory of Goncarov polynomials |
| Abstract: | The concept of Exponential families deals with the question of counting structures which are built out of connected pieces. Polynomials of binomial type are sequences of polynomials which obey a binomial-type relation. These types of polynomials are well studied and they show up in the enumeration of exponential families. On the other hand, Goncarov polynomials arise in interpolation theory and this concept was generalized by the work of Lorentz, Tringali and Yan(2018). Given any binomial-type sequence associated with an exponential family, there is a unique Goncarov polynomial sequence associated with this family. It turns out that this Goncarov sequence helps serve as a basis for counting u-parking functions. This is joint work with Catherine Yan. |
| Date: | April 5, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Title: | |
| Date: | April 6, 2019 |
| Time: | 08:00am |
| Location: | Rice University |
| Title: | AWM Research Symposium April 6-7 |
| Date: | April 12, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Michael Brannan, Texas A&M University |
| Title: | Hopf algebras and non-local games |
| Abstract: | A non-local game consists of two players, who are each provided questions from a referee and then supply answers. The game comes with rules which determine if the answers supplied by the players are correct or not. The players cooperate to win each round of the game, but the ``non-locality'' of the game means that the players cannot communicate by classical means during each round of the game. They can, however, agree upon a shared strategy for producing satisfactory answers. Non-local games are of interest in quantum information theory because quite often the only winning strategy is a so-called quantum strategy - i.e., one which utilizes some shared resource of quantum entanglement between the players. In this talk, I will focus on a particular class of non-local games, called synchronous games. For these games one can associate to a game an associative algebra A whose structure completely characterizes the existence of winning deterministic and probabilistic (quantum) strategies for these games. As a particular example, I will focus on the graph isomorphism game, which takes as inputs two graphs, and is devised so that a winning deterministic strategy requires that the two graphs be isomorphic. On the other hand, a probabilistic winning strategy relaxes this condition to the two graphs being what quantum information theorists call ``quantum isomorphic''. I will explain how the notion of quantum isomorphism mentioned above is intimately connected to the theory of Hopf bi-Galois objects: Two graphs are quantum isomorphic if and only if the game algebra A is a Hopf bi-Galois extension over the universal Hopf algebras coacting on the function algebras of the two graphs. I will explain how this Hopf-algebraic interpretation of the graph isomorphism game provides some fundamental new insights. **NOTE**: There will be a sequel to this talk given by Kari Eifler (TAMU) at 4pm in the Linear Analysis Seminar. Both talks will be complementary, yet self-contained. |
| Date: | April 19, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Anton Dochtermann, Texas State University |
| Title: | Exposed circuits, linear quotients, and chordal clutters |
| Abstract: | Chordal graphs are widely studied combinatorial objects, with various characterizations and applications. They also appear in commutative algebra in the context of Froberg’s theorem, which says that a graph G is chordal if and only if the edge ideal of its complement has a linear resolution. Recently Culbertson, Guralnik, and Stiller give a new characterization of chordal graphs in terms of what they call ‘edge-erasures’. We show that these moves are in fact equivalent to a linear quotient ordering on the ideal, leading to an algebraic proof of their result. We consider higher-dimensional analogues and show that linear quotients for more general circuit ideals of d-clutters can be characterized in terms of removing `exposed circuits' in the complement clutter. Here a circuit is exposed if it is uniquely contained in a maximal clique, reminiscent of the free faces of simple homotopy theory. This leads to a notion of a higher-dimensional ‘chordal clutter’ which borrows from commutative algebra, related to other constructions from the literature. We discuss some applications including a proof of a special case of (a generalization of) Simon’s conjecture, which posits that the k-skeleta of a simplex are extendably shellable. |
| Date: | April 26, 2019 |
| Time: | 3:00pm |
| Location: | BLOC 628 |
| Speaker: | Yue Cai, Texas A&M University |
| Title: | Rational parking functions |
| Abstract: | Vector parking functions are sequences of non-negative integers whose order statistics are bounded by a given integer sequence $(a_0, a_1, \dots, a_n)$. In this talk, we will generalize the notion to parking functions with a linear boundary of rational slope. Using the theory of fractional power series and the Newton-Puiseux Theorem, we convert an Appell relation of Goncarov polynomials to the exponential generating function for rational parking functions in terms of elementary symmetric functions. Time permits, we will also discuss the case of vector parking functions with periodic boundaries. This is joint work with Catherine Yan. |
