Algebra Seminar [Binghamton University Department of Mathematics and Statistics]

Problem of the Week
BUGCAT
Zassenhaus Conference
Hilton Memorial Lecture
BingAWM
Math Club

Fall 2023

August 22
Organizational Meeting

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

August 29
Chris Schroeder (Binghamton University)
Finite groups with many $p$-regular conjugacy classes

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.

September 12
Dikran Karagueuzian (Binghamton University)
Polynomial and Random Maps of Finite Fields

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.

September 19
Nguyen N. Hung (University of Akron)
A conjectural link between the field of values and degree of an irreducible character

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 $\chi(1).$

September 26
Daniel Studenmund (Binghamton University)
Hidden symmetries of lattices in solvable Lie groups

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.

October 31
Chris Schroeder (Binghamton University)
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November 14
Andrew Manuel Velasquez-Berroteran (Binghamton University)
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Fall 2023