| Date: | September 6, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Daniel Perales Anaya, TAMU |
| Title: | A Hopf algebra on non-crossing partitions |
| Abstract: | In non-commutative probability there are different notions of independence (free, Boolean and monotone), each with a notion of cumulants (analogue of classic cumulants) that linearize the addition of independent random variables. Formulas relating moments and cumulants can be expressed as a sum indexed by set partitions.
Our goal is to construct a Hopf algebra T on non-crossing partitions NC that allows us to systematically study the transitions between distinct brands of cumulants in non-commutative probability. The Hopf algebra T is such that its character group can be identified with a group of 'semi-multiplicative' functions on the incidence algebra of NC, used to encode the formulas. While a basic tool of the Hopf Algebra, such as the antipode of T, helps in inverting such formulas.
We will explain how T relates to other (more famous) Hopf algebras and explain some extensions we have worked on.
This is a joint work with Celestino, Ebrahimi-Fard, Nica and Witzman. |
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| Date: | September 20, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Nick Veldt, Iowa State University |
| Title: | Chain Saturation on the Boolean Lattice |
| Abstract: | Given a set X, a collection F ⊂ P(X) is said to be k-Sperner if it does not contain a chain of length k + 1 under set inclusion, and it is said to be saturated if it is maximal with respect to this property. Gerbner et al. conjectured that, if |X| is sufficiently large compared to k, then the minimum size of a saturated k-Sperner system is 2k−1. Noel, Morrison, and Scott disproved this conjecture later by proving that there exists ε such that for every k and |X| > n_0(k), there exists a saturated k-Sperner system of cardinality at most 2(1−ε)k .
In particular, Noel, Morrison, and Scott proved this for ε = 1 − 14 log_2 (15) ≈ 0.023277. We find an improvement to ε= 1 − 15 log2 28 ≈ 0.038529. We also prove that, for k sufficiently large, the minimum size of a saturated k-Sperner
family is at least √k 2^(k/2), improving on the previous Gerbner, et al. bound of 2^(k/2−0.5) |
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| Date: | September 27, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Youngho Yoo, TAMU |
| Title: | Erdos-Posa property of A-paths in group-labelled graphs |
| Abstract: | An A-path is a non-trivial path that intersects a vertex set A exactly at its endpoints. Beginning with a classical result of Gallai from 1961, several families of A-paths have been shown to satisfy an approximate packing-covering duality known as the Erdos-Posa property. However, there is very little known about the structures of graphs where this property fails for A-paths, which is in contrast to many similar situations where one can salvage a half-integral version of the Erdos-Posa property. In this talk, we prove a structure theorem that characterizes the obstructions to the Erdos-Posa property of A-paths in group-labelled graphs. This gives a general half-integral Erdos-Posa result as well as a characterization of the full Erdos-Posa property for A-paths in group-labelled graphs. Joint work with O-joung Kwon. |
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| Date: | October 11, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Trevor Karn, University of Minnesota |
| Title: | Equivariant Kazhdan–Lusztig theory of paving matroids |
| Abstract: | We study the way in which equivariant Kazhdan–Lusztig polynomials change under the operation of relaxation of a collection of stressed hyperplanes. This allows us to compute these polynomials for arbitrary paving matroids, which we do in a number of examples. We focus on the combinatorial consequences of the general theory. This is joint work with George Nasr, Nick Proudfoot, and Lorenzo Vecchi. |
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| Date: | October 18, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Galen Dorpalen-Barry, TAMU |
| Title: | The Poincare-Extended ab-Index |
| Abstract: | Motivated by a conjecture of Maglione—Voll concerning Igusa zeta
functions, we introduce and study the Poincaré-extended ab-index. This
polynomial generalizes both the ab-index and the Poincaré polynomial.
For posets admitting R-labelings, we prove that the coefficients are
nonnegative and give a combinatorial description of the coefficients.
This proves Maglione—Voll’s conjecture as well as a conjecture of the
Kühne—Maglione. We also recover, generalize, and unify results from
Billera—Ehrenborg—Readdy, Ehrenborg, and Saliola—Thomas. This is joint
work with Joshua Maglione and Christian Stump. |
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| Date: | October 25, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Chelsea Walton, Rice University |
| Title: | On extended Frobenius structures |
| Abstract: | A classical result in quantum topology is that oriented 2-dimensional topological quantum field theories (2-TQFTs) are fully classified by commutative Frobenius algebras. This is useful as TQFTs are categorical gadgets that produce topological invariants. In 2006, Turaev and Turner introduced additional structure on Frobenius algebras, forming what are called extended Frobenius algebras, to classify 2-TQFTs in the unoriented case. In this talk, I will stay on the algebraic side of this story and provide an introduction to extended Frobenius algebras in various settings. Numerous examples, classification results, and general constructions of extended Frobenius algebras will be presented, and I'll aim to keep this down-to-earth. This is joint work with Agustina Czenky, Jacob Kesten, and Abiel Quinonez. |
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| Date: | November 1, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Oeyvind Solberg, Norwegian University of Science and Technology (NTNU) |
| Title: | Decomposing modules of topological data analysis |
| Abstract: | The relative new and evolving Topological Data Analysis = TDA is using algebraic topology to analyze the geometric structure of data, ranging from tissue in the brain to pieces of text. One end product for TDA is a module over a ring. How this module breaks into indecomposable ones describes/characterizes the stability of the topological structures associated to the underlying data. In this talk I discuss one approach to this via the structure theorem for finitely generated modules over a principal ideal domain. |
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| Date: | November 15, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Moxuan (Jasper) Liu, UCSD |
| Title: | Matrix Loci and Orbit Harmonics |
| Abstract: | Consider the affine space of n by n complex matrices with coordinate ring C[x_{n*n}]. We define graded quotients of C[x_{n*n}] where each quotient ring carries a group action. These quotient rings are obtained by applying the orbit harmonics method to matrix loci corresponding to the permutation matrix group S_n, the colored permutation matrix group S_{n,r}, the collection of all involutions in S_n, and the conjugacy classes of involutions in S_n with a given number of fixed points. In each case, we explore how the algebraic properties of these quotient rings are governed by the combinatorial properties of the matrix loci. Based on joint work with Yichen Ma, Brendon Rhoades, and Hai Zhu.
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| Date: | November 22, 2024 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Chun-Hung Liu, TAMU |
| Title: | Disjoint paths problem with group-expressable constraints |
| Abstract: | We study an extension of the k-Disjoint Paths Problem where, in addition to finding k disjoint paths joining k given pairs of vertices in a graph, we ask that those paths satisfy certain constraints expressable by abelian groups. We give an O(n^8) time algorithm to solve this problem under the assumption that the constraint can be expressed as avoiding a bounded number of group elements; moreover, our O(n^8) algorithm allows any bounded number of such constraints to be combined. Group-expressable constraints include, but not limited to: (1) paths of length r modulo m for any fixed r and m, (2) paths passing through any bounded number of prescribed sets of edges and/or vertices, and (3) paths that are long detours (paths of length at least r more than the distance between their ends for fixed r). The k=1 case with the modularity constraint solves problems of Arkin, Papadimitriou and Yannakakis from 1991. Our work also implies a polynomial time algorithm for testing the existence of a subgraph isomorphic to a subdivision of a fixed graph, where each path of the subdivision between branch vertices satisfies any combination of a bounded number of group-expressable constraints. In addition, our work implies similar results addressing edge-disjointness. It is joint work with Youngho Yoo. |
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