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

Abstract: Here’s a fun way to build a group by cutting and pasting: Start with a Euclidean, spherical, or hyperbolic model geometry $X$ carrying a collection $\mathcal{H}$ of totally geodesic codimension-1 submanifolds determining a regular tessellation $\Delta$ of $X$. A piecewise isometry of $\Delta$ is defined by cutting out finitely many subspaces $S_1,\dotsc, S_k \in \mathcal{H}$ and isometrically mapping the components of what remains to the components obtained by cutting out another finite collection of subspaces $T_1,\dotsc, T_k \in \mathcal{H}$. The collection of all piecewise isometries is a group $PI(\Delta)$. When $\Delta$ is a tessellation of $\mathbb{R}$ by isometric line segments, $PI(\Delta)$ is an extension of Houghton’s group $H_2$. When $\Delta$ is a tessellation of the hyperbolic plane by ideal triangles, $PI(\Delta)$ naturally extends Thompson’s group $V$. Bieri and Sach studied $PI(\mathbb{Z}^n)$, where $\mathbb{Z}^n$ is the standard tessellation of Euclidean space by isometric cubes, obtaining lower bounds on their finiteness lengths and presenting a careful analysis of their normal subgroup structure.

Our story will start with the piecewise isometry group of the tessellation of the Euclidean plane by equilateral triangles, and generalize to piecewise isometry groups of Euclidean tessellations associated with affine Weyl groups of type $A_n$. Pictures will be drawn and preliminary results on algebraic structure and finiteness properties will be discussed. Time permitting, we will connect our discussion to the tessellation of hyperbolic 3-space by regular ideal tetrahedra. This talk covers work in progress with Robert Bieri and Alex Feingold.

Abstract: The Yoneda Lemma is widely regarded as the most-commonly-quoted result of category theory. This (expository) talk will discuss instances of the lemma appearing in the undergraduate mathematics curriculum, particularly linear algebra.

Abstract: In this talk, I will discuss some problems concerning the orders of some commutators in finite groups and how they affect the structure of the group.

Abstract: Let $G$ be a finite group and $K$ be a conjugacy class of $G$. Then $K^2$ consists of the products of any two elements in $K$. In this talk, we consider some equivalent conditions for $K^2$ to be a conjugacy class of $G$. This talk is based on the paper by Guralnick and Navarro in 2015.

Abstract: I will give the definition of Chabauty's space of marked groups and use it to give a nicer proof of a result from my thesis. I will then discuss joint work with Collin Bleak, Casey Donoven, Scott Harper on stronger notions of small generating sets for groups of homeomorphisms of the Cantor set.

Abstract: A recurring theme in finite group theory is understanding how the structure of a finite group is determined by the arithmetic properties of group invariants. There are results in the literature determining the structure of finite groups whose irreducible character degrees, conjugacy class sizes or indices of maximal subgroups are odd. These results have been extended to include those finite groups whose character degrees or conjugacy class sizes are not divisible by 4. In this paper, we determine the structure of finite groups whose maximal subgroups have index not divisible by 4. As a consequence, we obtain some new 2-nilpotency criteria. This is joint work with Prof. Hung Tong-Viet.

Abstract: Given a group G, a covering of G is a collection of proper subgroups of G whose set-theoretic union is G. The first part of my talk will be dedicated to some history of coverings of groups and providing some results on which finite groups have an equal covering, which is a type of covering where each subgroup is of the same order. The second part of my talk will be dedicated to extending the notion of coverings of groups to that of rings. One result of this extension is determining necessary conditions for a ring $R$ so that the ring of polynomials R[X] has a special type of covering.

Abstract: In general, the classification of finitely generated subgroups of a given group is intractable. Even restricting to two-generator subgroups is not enough. However, in a geometric setting classification is possible. For example, a two-generator subgroup of a right-angled Artin group is either free or free abelian. Jaco and Shalen proved that a two-generator subgroup of the fundamental group of an orientable atoroidal irreducible 3-manifold is either free, free-abelian, or finite-index. In this talk I will present recent work proving a similar classification theorem for two generator mapping-torus groups of free group endomorphisms: every two generator subgroup is either free or conjugate to a sub-mapping-torus group. As an application we obtain an analog of the Jaco-Shalen result for free-by-cyclic groups with fully irreducible atoroidal monodromy. While the statement is algebraic, the proof technique uses the topology of finite graphs, a la Stallings. This is joint work with Naomi Andrew, Ilya Kapovich, and Stefano Vidussi.

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