| Date: | February 11, 2022 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Ryan Martin, Iowa State University |
| Title: | Counting paths, cycles, and other subgraphs in planar graphs |
| Abstract: | For a planar graph H, let N_P(n,H) denote the maximum number of copies of H in an n-vertex planar graph. The case where H is the path on 3 vertices, H=P_3, was established by Alon and Caro. The case of H=P_4 was determined, also exactly, by Gy\H{o}ri, Paulos, Salia, Tompkins, and Zamora. In this talk, we will give the asymptotic values for H equal to P_5 and P_7 as well as the cycles C_6, C_8, C_{10} and C_{12} and discuss the general approach which can be used to compute the asymptotic value for many other graphs H. |
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| Date: | March 4, 2022 |
| Time: | 00:00am |
| Location: | |
| Title: | CombinaTexas |
| Abstract: | Conference website: https://www.math.tamu.edu/conferences/combinatexas/ |
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| Date: | March 5, 2022 |
| Time: | 00:00am |
| Location: | |
| Title: | CombinaTexas |
| Abstract: | Conference website: https://www.math.tamu.edu/conferences/combinatexas/ |
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| Date: | March 11, 2022 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Harshit Yadav, Rice University |
| Title: | Filtered Frobenius algebras in monoidal categories |
| Abstract: | We develop filtered-graded techniques for algebras in monoidal categories with the goal of establishing a categorical version of Bongale's 1967 result: A filtered deformation of a Frobenius algebra over a field is Frobenius as well. Towards the goal, we construct a monoidal associated graded functor, building on prior works of Ardizzoni-Menini, of Galatius et al., and of Gwillian-Pavlov. We then produce equivalent conditions for an algebra in a rigid monoidal category to be Frobenius in terms of the existence of categorical Frobenius form. These two results of independent interest are used to achieve our goal. As an application of our main result, we show that any exact module category over a symmetric finite tensor category is represented by a Frobenius algebra in it. This is joint work with Dr. Chelsea Walton (Rice University). |
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| Date: | April 1, 2022 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Chun-Hung Liu, TAMU |
| Title: | A decomposition theorem for immersion-free graphs with no 3-edge-cut |
| Abstract: | Structural decomposition theorems for graphs with forbidden minors and topological minors have been proved and led to many applications. Graph immersions is a notion related to graph minors and topological minors, and many analogous open problems about immersions have been proposed. In this talk we address the fundamental problem about the structure of a graph with forbidden immersions. We prove that every graph with no edge-cut of size 3 that forbids a fixed graph H as an immersion can be decomposed into graphs that are "nearly simpler" than H. The condition for having no 3-edge-cut is necessary to have a clean theorem. |
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| Date: | April 22, 2022 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Anton Bernshteyn, Georgia Tech |
| Title: | Lower bounds for difference bases |
| Abstract: | A difference basis with respect to $n$ is a subset $A \subseteq \mathbb{Z}$ such that $A - A \supseteq [n]$. R\'{e}dei and R\'{e}nyi showed that the minimum size of a difference basis with respect to $n$ is $(c+o(1))\sqrt{n}$ for some positive constant $c$. The precise value of $c$ is not known, but some bounds are available, and I will discuss them in this talk. This is joint work with Michael Tait (Villanova University). |
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| Date: | May 3, 2022 |
| Time: | 3:00pm |
| Location: | BLOC 302 |
| Speaker: | Daniel Perales, University of Waterloo |
| Title: | On the anti-commutator of two free random variables |
| Abstract: | Let a, b be freely independent random variables in a non-commutative probability space. Based on some considerations on bipartite graphs, we provide a formula to express the n-th free cumulant of the anticommutator ab+ba as a sum indexed by subset Y_{2n} of non-crossing partitions of {1,2,...,2n}. Specifically, Y_{2n} is the set of partitions that separate odd elements and where blocks with only even elements have even size. We will study the sets Y_{2n} and obtain new results regarding the distribution of ab+ba. For instance, the size |Y_{2n}| is closely related to the case when a,b have a Marchenko-Pastur (free Poisson) distribution of parameter 1. We will also provide a formula with the sum indexed by cacti graphs, and discuss a natural generalization to study quadratic forms in $k$ free random variables. The talk is based on the preprint arXiv:2101.09444. |
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