Inverse monoids are an important generalization of groups, and they form a variety of algebras. We focus on the class of inverse monoids called F-inverse: this is not closed under taking inverse subsemigroups or homomorphic images, but does form a variety in the signature enriched with an extra unary operation. Recently, Auinger, Kudryavtseva and M. Szendrei gave an elegant model for free F-inverse monoids in this extended signature. More generally, they describe the initial object in the category of X-generated F-inverse monoids with greatest group image G. We show that this category is equivalent to the category of finitary (i.e. algebraic) G-invariant closure operators on subgraphs of the Cayley graph of G, and adapt Stephen's approach to study the word problem in presentations of inverse monoids for F-inverse monoids in the enriched signature.
Closure operators on group Cayley graphs, and presentations of F-inverse monoids
Nov. 14, 2023 2:30pm (MATH 3…
Lie Theory
Karola Meszaros (Cornell)
X
The central question of knot theory is that of distinguishing links up to isotopy. The first polynomial invariant of links devised to help answer this question was the Alexander polynomial (1928). Almost a century after its introduction, it still presents us with tantalizing questions, such as Fox’s conjecture (1962) that the absolute values of the coefficients of the Alexander polynomial of an alternating link are unimodal. Fox’s conjecture remains open in general with special cases settled by Hartley (1979) for two-bridged knots, by Murasugi (1985) for a family of alternating algebraic links, and by Ozsv\’ath and Szab\’o (2003) for the case of genus 2 alternating knots, among others. We settle Fox’s conjecture for special alternating links. We do so by proving that a certain multivariate generalization of the Alexander polynomial of special alternating links is Lorentzian. As a consequence, we obtain that the absolute values of the coefficients of , where is a special alternating link, form a log-concave sequence with no internal zeros. In particular, they are unimodal. This talk is based on joint work with Elena Hafner and Alexander Vidinas.
Log-concavity of the Alexander polynomial Sponsored by the Meyer Fund
In this training, participants will examine how their own mathematical backgrounds can impact their teaching choices in the classroom. The session will go beyond individual perspectives to promote a community mindset that fosters inclusivity and student belonging. Participants will explore the relationship between deficit language and its effects on students, learn about the pedagogical concepts of "warm demander and the hidden curriculum", and acquire practical strategies to transform the student experience in mathematics classrooms. This session welcomes anyone interested in learning more about how to promote student belonging in mathematical spaces.
Promoting Student Belonging in Mathematics Classrooms through Equity-Driven Pedagogies
At the intersection of Topological Data Analysis and machine learning, the field of cellular signal processing has advanced rapidly in recent years. In this context, each signal on the cells of a complex is processed using the combinatorial Laplacian and the resulting Hodge decomposition. Meanwhile, discrete Morse theory has been widely used to speed up computations by reducing the size of complexes while preserving their global topological properties. In this talk, we introduce an approach to signal compression and reconstruction on complexes that leverages the tools of discrete Morse theory. The main goal is to reduce and reconstruct a cell complex together with a set of signals on its cells while preserving their global topological structure as much as possible. This is joint work with Stefania Ebli and Kelly Maggs.
Signal processing on cell complexes using discrete Morse theory
Closure operators on group Cayley graphs, and presentations of F-inverse monoids