# Algebra & Number Theory Seminar Spring 2015

Fridays from 12:00-1:00pm

McHenry Library room 1257

Friday, April 3, 2015

"Broués abelian defect group conjecture and auto-equivalences in the cyclic defect group case."

Robert Boltje, University of California, Santa Cruz

We recall Broué's abelian defect group conjecture for a block $A$ of a group algebra $kG$ in positive characteristic $p$. It states that if $A$ has abelian defect group $D$ then the homotopy categories of $A$ and of its Brauer correspondent block $B$ of $kN_G(D)$ are equivalent via tensoring with a chain complex of bimodules with particular structure. Although the existence of such chain complexes is known in many examples, there is no general method of construction or a general candidate for such a chain complex. Even in the case when $D$ is cyclic, the construction is inductive and very involved. We present a result on auto-equivalences for cyclic $D$ which sheds light on how unique such chain complexes are. This is based on joint work with Philipp Perepelitsky.

Friday, April 10, 2015

"Introduction to vertex operator algebras"

Chongying Dong, University of California, Santa Cruz

In this introductory talk, I will discuss the origin of the vertex operator algebras and their connections with the monstrous moonshine. I will also present some recent developments in the field.

Friday, April 17, 2015

"Modularity in vertex operator algebras"

Chongying Dong, University of California, Santa Cruz

This talk is about the trace functions in vertex operator algebras. We will present a recent result that the kernel of the representation of the modular group on the conformal blocks of any rational, C2-cofinite vertex operator algebra is a congruence subgroup. In particular, the q-character of each irreducible module is a modular function on the same congruence subgroup.

Friday, April 24, 2015

"Modular forms on $\Gamma_0(2)$"

Geoff Mason, University of California, Santa Cruz

I will explain how to prove the unbounded denominator conjecture of Atkin-Swinnerton-Dyer for modular forms (with nebentypus) on $\Gamma_0(2)$.
This is joint work with Cameron Franc.

Friday, May 1, 2015

"More on vector-valued modular forms and noncongruence modular forms"

Cameron Franc, University of Michigan

In this talk we will present ongoing work with Geoff Mason on vector-valued modular forms. More specifically, we will recall how one may use vector-valued modular forms to give explicit formulae for certain noncongruence modular forms. The expressions are such that one can use them to evaluate these modular forms to high precision. We will report on computer experiments studying the CM-values of these noncongruence modular forms. Background on the congruence case will also be provided.

Friday, May 8, 2015

"Galois representations for general symplectic groups"

Sug Woo Shin, UC Berkeley

We prove the existence of GSpin-valued Galois representations corresponding to regular algebraic cuspidal automoprhic representations of general symplectic groups under simplifying local hypotheses. This is joint work with Arno Kret.

Friday, May 15, 2015

"On the occurrence of Hecke eigenvalues."

Nahid Walji, University of California, Berkeley

For a cuspidal automorphic representation of GL(2), we determine upper bounds on the number of Hecke eigenvalues with absolute value equal to a fixed number \gamma; in the case \gamma = 0, this answers a question of Serre. These bounds are then strengthened by restricting to non-monomial representations. We also obtain analogous bounds for a family of cuspidal automorphic representations for GL(3)

Friday, May 22, 2015

"Averaged Colmez Conjecture"

Xinyi Yuan, University of California, Berkeley

The Colmez conjecture expresses the Faltings height of a CM abelian variety in terms of the logarithmic derivatives of certain Artin L-functions. In this talk, I will focus on an averaged version of the conjecture, which is proved in my recent joint work with Shou-Wu Zhang. Combining with the recent work of Jacob Tsimerman, the Andre-Oort conjecture for Siegel modular varieties is proved to be true.

Friday, May 29, 2015

"Rationality, regularity and C2-cofiniteness"

Nina Yu, University of California, Riverside

Rationality, regularity and C2-cofiniteness are three most important concepts in representation theory of vertex operator algebras. In this talk I will talk about connections among these three notions and recent progress in proving the conjecture that rationality implies C2-cofinitenss.

Thursday, June 4, 2015 3-4pm McHenry Room 4130 * PLEASE NOTE THE DIFFERENT DAY, TIME AND LOCATION*

"Connecting invariants associated to generalized Grothendieck-Springer resolutions to homotopy theory"

Mee Seong Im, University of Illinois at Urbana-Champaign and United States Military Academy

Grothendieck-Springer resolutions are fundamental and important objects in representation theory, with connections to quiver Hecke algebras, four dimensional TQFT via categorification of certain algebras, Nakajima quiver varieties, and quiver flag varieties, to name a few. I will describe the geometry of generalized Grothendieck-Springer resolutions using quivers and then discuss certain parabolic group equivariant invariants arising from them. It has been discovered that for graphs that are homotopic to a point or a circle, certain polynomial invariants could completely be understood. I will discuss such connections, ending with several conjectures.