Representation Theory / Number Theory Seminar, Spring 2021


All talks online (Zoom)*. Seminar times will vary.

*For security reasons, Zoom links will not be posted here. If you would like to attend a talk, but do not have the link, please contact Gil Moss or Petar Bakic (last name at math.utah.edu).


Wednesday, January 20, 3:00-4:00pm

Speaker: Joseph Hundley, SUNY Buffalo
Title: Functorial Descent in the Exceptional Groups
Abstract: In this talk, I will discuss the method of functorial descent. This method, which was discovered by Ginzbug, Rallis and Soudry, uses Fourier coefficients of residues of Eisenstein series to attack the problem of characterizing the image of Langlands functorial liftings. We'll discuss the general structure, the results of Ginzburg, Rallis and Soudry in the classical groups, some recent attempts to extend the same ideas to the exceptional group, and challenges and new phenomena which emerge in these attempts. The new work discussed will mainly be joint with Baiying Liu. Time permitting I may also comment on unpublished work of Ginzburg and joint work with Ginzburg.


Wednesday, January 27, 3:00-4:00pm

Speaker: Lucas Mason-Brown, Oxford
Title: What is a unipotent representation?
Abstract: The concept of a unipotent representation has its origins in the representation theory of finite Chevalley groups. Let G(Fq) be the group of Fq-rational points of a connected reductive algebraic group G. In 1984, Lusztig completed the classification of irreducible representations of G(Fq). He showed:
1) All irreducible representations of G(Fq) can be constructed from a finite set of building blocks -- called `unipotent representations.'
2) Unipotent representations can be classified by certain geometric parameters related to nilpotent orbits for a complex group associated to G(Fq).
Now, replace Fq with C, the field of complex numbers, and replace G(Fq) with G(C). There is a striking analogy between the finite-dimensional representation theory of G(Fq) and the unitary representation theory of G(C). This analogy suggests that all unitary representations of G(C) can be constructed from a finite set of building blocks -- called `unipotent representations' -- and that these building blocks are classified by geometric parameters related to nilpotent orbits. In this talk I will propose a definition of unipotent representations, generalizing the Barbasch-Vogan notion of `special unipotent'. The definition I propose is geometric and case-free. After giving some examples, I will state a geometric classification of unipotent representations, generalizing the well-known result of Barbasch-Vogan for special unipotents.
This talk is based on forthcoming joint work with Ivan Loseu and Dmitryo Matvieievskyi.


Wednesday, February 17, 4:00-5:00pm

Speaker: Hiraku Atobe, Hokkaido University
Title: The Zelevinsky-Aubert duality for classical groups
Abstract: In 1980, Zelevinsky studied representation theory for p-adic general linear groups. He gave an involution on the set of irreducible representations, which exchanges the trivial representation with the Steinberg representation. Aubert extended this involution to p-adic reductive groups, which is now called the Zelevinsky-Aubert duality. It is expected that this duality preserves unitarity. In this talk, we explain an algorithm to compute the Zelevinsky-Aubert duality for odd special orthogonal groups or symplectic groups. This is a joint work with Alberto Minguez in University of Vienna.


Wednesday, March 3, 3:00-4:00pm

Speaker: Allechar Serrano López, University of Utah
Title: Counting elliptic curves with prescribed torsion over imaginary quadratic fields
Abstract: A generalization of Mazur's theorem states that there are 26 possibilities for the torsion subgroup of an elliptic curve over a quadratic extension of $\mathbb{Q}$. If $G$ is one of these groups, we count the number of elliptic curves of bounded naive height whose torsion subgroup is isomorphic to $G$ in the case of imaginary quadratic fields.


Wednesday, March 10, time TBD

Speaker: Justin Trias, University of East Anglia
Title: TBA
Abstract: TBA


Wednesday, March 17, 3:00-4:00pm

Speaker: Ila Varma, University of Toronto
Title: TBA
Abstract: TBA


Wednesday, March 24, time TBD

Speaker: Baiying Liu, Purdue University
Title: TBA
Abstract: TBA


Wednesday, March 31, time TBD

Speaker: Michael Griffin, BYU
Title: TBA
Abstract: TBA


Thursday, April 8, time TBD (Colloquium talk)

Speaker: Aaron Pollack, UCSD
Title: TBA
Abstract: TBA


Wednesday, April 14, 3:00-4:00pm

Speaker: Martin Weissman, UC Santa Cruz
Title: The compact induction theorem for rank-one p-adic groups
Abstract: A folklore conjecture predicts that when G is a p-adic group, every irreducible supercuspidal representation of G is induced from a compact-mod-center open subgroup. This was proven for GL(n) by Bushnell and Kutzko. For other groups, e.g., classical groups, tame groups, etc., the conjecture is proven for sufficiently large p thanks to hard work by many people. In this talk, I will describe a recent proof of the conjecture which applies to all groups G of relative rank one, with no assumptions about p. The method is to use the work of Schneider and Stuhler to connect supercuspidal representations to sheaves on the Bruhat-Tits tree of G, and "refine" these sheaves until the induction theorem becomes obvious.