Please think about giving a talk in the Algebra Seminar, or inviting an outside speaker.

Abstract: Weyl groups are Coxeter groups generated by reflections determined by a Cartan matrix associated with a Kac-Moody Lie algebra. We will discuss examples where the Weyl group acts on a Euclidean space (affine case) as well as examples where it acts on a hyperbolic space (hyperbolic case). In either case, the action is properly discontinuous so it has a fundamental domain whose reflections tessellate the space. In joint work with Robert Bieri and Daniel Studenmund, we are are investigating the geometry of such tessellations in order to define and obtain generators for groups of piecewise isometries of the tessellations which generalize the Thompson group PPSL(2,Z).

Abstract: A theorem on the random variable of inverse image sizes for a polynomial over a finite field of order q computes the moments of the random variable of inverse images sizes up to an error term. This error term decreases with the inverse of the square root of q. Standard results in the theory of elliptic curves will be used to show that this error term cannot generally be improved. No familiarity with the extensive theory of elliptic curves will be assumed.

Abstract: Digroups, and generalized digroups, g-digroups for short, have been considered as a generalization of continuous groups whose tangent space is a Leibniz algebra. This structure has been seen as a generalization of groups, therefore, efforts have been done to study properties and results that come from group theory, to explore if they hold in this new setting. In this talk, we'll discuss the isomorphism theorems for g-digroups, and show that the results for groups do extend to g-digroups. This is joint work with Olga Patricia Salazar-Diaz.

Abstract: Given a group $G$, define an equivalence relation on the elements of $G$ by $x\sim y$ iff $C_G(x) = C_G(y)$, and let $X$ denote a fixed set of representative elements. The set $X$ can be used to define any centralizer in $G$, since for every centralizer $H$, there is a unique maximal subset $S$ of $X$ so that $H = C_G(S)$. We observe a Galois connection between the lattice of centralizer subgroups of $G$ and this poset of maximal subsets of $X$ corresponding to centralizers. We show other interesting properties related to this relationship.

This is joint work with Wil Cocke and Mark Lewis.

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Abstract: The topic of this lecture is the algebraic Lefschetz properties, which are abstractions of the important Hard Lefschetz theorem from geometry. I will introduce what are Lefschetz properties and show the relation between Lefschetz properties and commutative algebra.

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Abstract: In this talk, I will first explain how a notion of positivity in algebraic geometry / number theory could be captured in terms of semirings by providing an example of the narrow class group of a number field as a reflexive Picard group. Then, I will introduce a notion of equivariant vector bundles over the natural numbers, and prove a version of Klyachko classification theorem of toric vector bundles in this setting.

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Abstract: One often first encounters the representation theory of groups through their actions on vector spaces. It is also natural to consider group actions on sets with no additional structure, so-called permutation representations. Concretely, whereas a linear representation is a group homomorphism to a general linear group, a permutation representation is a group homomorphism to a symmetric group. In this talk, we will introduce the basic definitions and examples of permutation representations of finite groups, and discuss the analogies with linear representations.

Abstract: Given an irreducible root system, the Worpitzky-compatible subsets are defined by a geometric property of the alcoves inside the fundamental parallelepiped of the root system. This concept is motivated and mainly understood through a lattice point counting formula concerning the characteristic and Ehrhart quasi-polynomials. In this talk, we show that the Worpitzky-compatibility has a simple combinatorial characterization in terms of roots. As a byproduct, we obtain a complete characterization by means of Worpitzky-compatibility for the freeness of the arrangements interpolating between the extended Shi and Catalan arrangements. This completes the earlier result by Yoshinaga in 2010 which was done for simply-laced root systems. This is joint work (arXiv:2403.17274) with Takuro Abe (Tokyo).

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