| Date: | January 29, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Eric Rowell, TAMU |
| Title: | Representations of Braid Groups and Motion Groups |
| Abstract: | Representations of braid groups appear in many (related) guises, as sources of knot and link invariants, transfer matrices in statistical mechanics, quantum gates in topological quantum computers and commutativity morphisms in braided fusion categories. Regarded as trajectories of points in the plane, the natural generalization of braid groups are groups of motions of links in 3 manifolds. While much of the representation theory of braid groups and motions groups remains mysterious, we are starting to see hints that suggest a few conjectures. I will describe a few of these conjectures and some of the progress towards verification. |
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| Date: | February 5, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Jurij Volcic, TAMU |
| Title: | Freely noncommutative Hilbert's 17th problem |
| Abstract: | One of the problems on Hilbert's 1900 list asked whether every positive rational function can be written as a sum of squares of rational functions. Its affirmative resolution by Artin in 1927 was a breakthrough for real algebraic geometry. The talk addresses the analog of this problem for positive semidefinite noncommutative rational functions. More generally, a rational Positivstellensatz on matricial sets given by linear matrix inequalities will be presented; a crucial intermediate step is an extension theorem on invertible evaluations of linear matrix pencils, which has less to do with positivity and ostensibly more to do with representation theory. One of the consequences of the Positivstellensatz is an algorithm for eigenvalue optimization of noncommutative rational functions. Finally, some contrast between the polynomial and the rational Positivstellensatz in the noncommutative setting will be discussed. |
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| Date: | February 12, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Alexander Ruys de Perez, TAMU |
| Title: | Wheels of Neural Codes: A New Criterion for Nonconvexity |
| Abstract: | A neural code C on n neurons is a collection of subsets of the set of integers {1,2,...,n}. Usually, C is paired with a collection of n open subsets of some Euclidean space, with C encoding how those open sets intersect. A central problem concerning neural codes is determining convexity; that is, if the code can encode the intersections of n convex open subsets.
In this talk, I will generalize an example of Lienkaemper, Shiu, and Woodstock (2017) into a new criterion for nonconvexity called a 'wheel'. I will show why it forbids convexity, explain how one can find it combinatorially, and provide examples of previously unclassified codes that we now know to be nonconvex due to containing a wheel. |
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| Date: | February 20, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Title: | CombinaTexas |
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| Date: | February 21, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Title: | CombinaTexas |
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| Date: | February 26, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | He Guo, Georgia Tech |
| Title: | Prague dimension of random graphs |
| Abstract: | The Prague dimension of graphs was introduced by Nesetril, Pultr and Rodl in the 1970s. Proving a conjecture of Furedi and Kantor, we show that the Prague dimension of the binomial random graph is typically of order n/log n for constant edge-probabilities. The main new proof ingredient is a Pippenger-Spencer type edge-coloring result for random hypergraphs with large uniformities, i.e., edges of size O(log n).
Based on joint work with Kalen Patton and Lutz Warnke. |
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| Date: | March 5, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Chun-Hung Liu, TAMU |
| Title: | Well-quasi-ordering digraphs by the strong immersion relation |
| Abstract: | A well-quasi-ordering is a reflexive and transitive binary relation such that every infinite sequence has a non-trivial increasing subsequence. The study of well-quasi-ordering was stimulated by two conjectures of Vazsonyi in 1940s: trees and subcubic graphs are well-quasi-ordered by the topological minor relation. It is known that the topological minor relation does not well-quasi-order all graphs. Nash-Williams in 1960s introduced the notion of strong immersion and conjectured that the strong immersion relation well-quasi-orders all graphs, which is a common generalization of both conjectures of Vazsonyi. In this talk we consider strong immersion on digraphs. Paths that change direction arbitrarily many times form an infinite antichain with respect to the strong immersion relation. In this talk, we will prove that it is the only obstruction. Namely, for any integer k, digraphs with no paths that change direction at least k times are well-quasi-ordered by the strong immersion relation. Joint work with Irene Muzi. |
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| Date: | March 12, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Youngho Yoo, Georgia Tech |
| Title: | Packing A-paths and cycles with modularity constraints |
| Abstract: | A classical result of Erdős and Pósa states that a graph without many disjoint cycles contains a small vertex set intersecting every cycle. The analogous statement fails however for odd cycles, as can be shown by large projective planar grids. In 1987, Dejter and Neumann-Lara raised the question of when this approximate packing-covering duality holds for cycles of length L mod M, and there had been minimal progress on this problem until recently. There are similar questions on packing A-paths (paths meeting a vertex set A at exactly its endpoints) with modularity constraints. Casting these problems in the more general setting of group-labelled graphs and studying their structure, we obtain a complete characterization of the integer pairs L and M for which the above approximate duality holds for A-paths and cycles of length L mod M. Joint work with Robin Thomas and with Pascal Gollin, Kevin Hendrey, O-joung Kwon, and Sang-il Oum. |
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| Date: | March 26, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Zixia Song, University of Central Florida |
| Title: | Hadwiger’s Conjecture |
| Abstract: | Hadwiger's conjecture from 1943 states that for every integer t, every graph either can be t-colored or has a subgraph that can be contracted to the complete graph on t+1 vertices. This is a far-reaching generalization of the Four-Color Theorem and perhaps the most famous conjecture in graph theory. In this talk we will survey the history of Hadwiger's conjecture and the main ideas of recent results. |
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| Date: | April 9, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Anton Dochtermann, Texas State University |
| Title: | Betti numbers of random edge ideals |
| Abstract: | We study asymptotic homological properties of random quadratic monomial ideals in a polynomial ring R = k[x_1, . . . , x_n], utilizing methods from the Erd\"os-R\'enyi model of random graphs. Here we consider a graph on n vertices and exclude an edge (corresponding to a quadratic generator of the ideal I) with probability p, and consider algebraic properties as n tends to infinity. Our main results involve fixing the edge parameter p = p(n) so that asymptotically almost surely the Krull dimension of R/I is fixed. Under these conditions we establish various properties regarding the Betti table of R/I, including sharp bounds on regularity and projective dimension and distribution of nonzero Betti numbers. These results extend work of Erman-Yang, who studied such ideals in the context of conjectured phenomena in the nonvanishing of asymptotic syzygies. Our results use collapsibility properties of random clique complexes and Garland's method regarding spectral gaps of graphs, and in particular rely on the underlying field in some cases. This is joint work with Andrew Newman. |
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| Date: | April 16, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Songling Shan, Illinois State University |
| Title: | Chromatic index of dense quasirandom graphs |
| Abstract: | Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ on $n$ vertices with $\Delta(G)>n/3$ has chromatic index $\Delta(G)$ if and only if $G$ contains no overfull subgraph. Glock, K\"{u}hn, and Osthus in 2016 showed that the conjecture is true for dense quasirandom graphs with even order, and they conjectured that the same should hold for such graphs with odd order. We show that the conjecture of Glock, K\"{u}hn, and Osthus is affirmative. |
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| Date: | April 23, 2021 |
| Time: | 3:00pm |
| Location: | Zoom |
| Speaker: | Byeongsu Yu, TAMU |
| Title: | When is the quotient of a semigroup ring by a monomial ideal Cohen-Macaulay? |
| Abstract: | We give a new combinatorial criterion for quotients of affine semigroup rings by monomial ideals to be Cohen-Macaulay, by computing the homology of finitely many polyhedral complexes. This provides a common generalization of well-known criteria for affine semigroup rings and monomial ideals in polynomial rings. This is joint work with Laura Matusevich. |
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