Algebra and Combinatorics Seminar
The current seminar's organizers are
Chun-Hung Liu and
Catherine Yan.
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Date Time |
Location | Speaker |
Title – click for abstract |
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01/29 3:00pm |
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Eric Rowell TAMU |
Representations of Braid Groups and Motion Groups
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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02/05 3:00pm |
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Jurij Volcic TAMU |
Freely noncommutative Hilbert's 17th problem
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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02/12 3:00pm |
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Alexander Ruys de Perez TAMU |
Wheels of Neural Codes: A New Criterion for Nonconvexity
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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02/20 3:00pm |
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CombinaTexas |
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02/21 3:00pm |
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CombinaTexas |
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02/26 3:00pm |
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He Guo Georgia Tech |
Prague dimension of random graphs
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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03/05 3:00pm |
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Chun-Hung Liu TAMU |
Well-quasi-ordering digraphs by the strong immersion relation
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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03/12 3:00pm |
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Youngho Yoo Georgia Tech |
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03/26 3:00pm |
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Zixia Song University of Central Florida |
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04/09 3:00pm |
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Anton Dochtermann Texas State University |
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04/16 3:00pm |
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Songling Shan Illinois State University |
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04/23 3:00pm |
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