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

Abstract: It is a classical topic in finite group theory to understand a finite group through its simple composition factors. To this end, one would like to construct group invariants that distinguish the nonabelian finite simple groups. One heuristic for simple groups is that they have “few” inequivalent irreducible linear representations, as they have few normal subgroups. In this talk, we will construct invariants from this observation and show how they can be used to determine the structure of finite groups. Our talk aspires to be accessible and interesting to a wide mathematical audience.

Abstract: We study the relations between polynomial and random maps by computing the moments of the random variable of inverse image sizes. In the polynomial case, these moments are connected to the Galois Theory of the polynomial over a function field. For random maps, the moments can sometimes be computed using generating function techniques. These computations show both similarities and differences between the two cases.

Abstract: For an irreducible character $\chi$ of a finite group $G$, we define $f(\chi)$ as the `cyclotomic deficiency' of $\chi$. This deficiency is the degree of the field extension from the field of values of $\chi$ to its cyclotomic closure. Over thirty years ago, Cram proved that when $G$ is solvable, $f(\chi)$ is always a divisor of the character degree $\chi(1)$. In this talk, I will present strong evidence suggesting that for all finite groups, $f(\chi)$ is bounded above by $\chi(1).$

Abstract: The abstract commensurator of a group $G$, the group of isomorphisms between finite-index subgroups modulo equivalence, encodes symmetries of $G$ that may be ``hidden.'' When $G$ is a lattice in a simple Lie group incommensurable with $\operatorname{PSL}(2,\mathbb{R})$, work of Mostow, Prasad, Borel, and Margulis shows that the abstract commensurator of $G$ detects whether the group arises through arithmetic constructions. In this talk, I will discuss results on abstract commensurators of the other significant class of lattice in Lie groups, the solvable groups. This builds on classical work of Malcev for nilpotent groups and more recent rigidity results of Mostow, Morris, and Baues–Grunewald.

Abstract: The abstract commensurator of a group $G$, the group of isomorphisms between finite-index subgroups modulo equivalence, encodes symmetries of $G$ that may be `hidden'. When $G$ is a lattice in a simple Lie group incommensurable with $\operatorname{PSL}(2,\mathbb{R})$, work of Mostow, Prasad, Borel, and Margulis shows that the abstract commensurator of $G$ detects whether the group arises through arithmetic constructions. In this talk, I will discuss results on abstract commensurators of the other significant class of lattice in Lie groups, the solvable groups. This builds on classical work of Malcev for nilpotent groups and more recent rigidity results of Mostow, Morris, and Baues–Grunewald.

Abstract: In this talk we describe what a g-digroup is, and some of its group-type properties.

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Abstract: Totally unimodular matrices have very nice properties with respect to solutions of linear equations, linear programming and combinatorial optimization. I will introduce them and some of the reasons they get attention.

Abstract: The state of a quantum system is described by a vector in a complex vector space. Therefore, whenever a group of transformations acts on a quantum system, we obtain a complex representation of the group. In this talk, we will work out a concrete example – the representation theory of physical rotations acting on a two-state quantum system – and discover the spookiness of quantum theory.

Abstract: We will define localization and mention some results of Bousfield on localization in the stable homotopy category. If time permits we might explain the difference between localization of spectra and spaces.

Abstract: First, we will define the wreath product of finite groups. Then we will define a polynomial called `indicatrix of a group' that captures the fixed points of the action of the group on some set. It turns out that the indicatrix behaves `nicely' upon taking the wreath product. If time permits, we shall go through specific examples; we will compute the indicatrix of the symmetric group on k letters, S_k, acting naturally on {1,2,…,k}.

Abstract: It is about the theme of “Recovering Objects with Morphisms.” It will begin by introducing basic concepts of category and related functors, as well as natural transformations. Following that, I will use universal properties and the Yoneda Lemma to illustrate the theme, and I will outline the general approach for proving the Yoneda Lemma. During the talk, I will use some examples from the book “Abstract Algebra” by Dummit and other relevant sources, such as “Category Theory: A Gentle Introduction” by Peter Smith and “The Rising Sea: Foundations of Algebraic Geometry” by Ravi Vakil.

Abstract: Given a group G and a ring R we can formulate a new ring, called the group ring RG, whereby elements in this ring can be thought of as an R-linear combination of elements of G. We will look at examples of groups rings, theorems and propositions of groups rings as well as look at some basic module theory.

This talk will be accessible to students who have taken a first semester undergraduate course in modern algebra.