### Schedule

### 9:00-9:45 - Prof. Thomas Hales

Title: Endoscopic transfer of Orbital integrals in Large Residual characteristic.

Abstract: This talk will indicate how identities of integrals that occur in the Arthur-Selberg trace formula can be transported from one field to another, using methods of motivic integration. In particular, I will describe identities that were obtained by Waldspurger using the a trace formula based on the Poisson summation formula. I will also give background about the theory of descent of orbital integrals, as developed by Harish-Chandra and others.

### 10:00-10:45 - Prof. Julia Gordon

Title: Product formulas for the size of an isogeny class of elliptic curves

Abstract: Consider the question: how likely is a random elliptic curve over the finite field F_p to have exactly N rational points, where N is a given integer in the appropriate range? In 2003, Gekeler gave an explicit answer based on a heuristic that was too strong to be literally true, thus the answer appeared somewhat mysterious. We provide an explanation for this formula by making an explicit and very natural connection with a formula of Langlands and Kottwitz which expresses the size of an isogeny class of principally polarized abelian varieties in terms of an adelic orbital integral. Then we discuss a possible extension of Gekeler's computations from elliptic curves to abelian varieties. (This is joint work with Jeff Achter.)

#### (coffee break)

### 11:00-11:45 - Prof. Loren Spice

Title: Asymptotic expansions of characters

Abstract: Harish-Chandra made the analogy that characters (of irreducible representations) are to a reductive group as Fourier transforms of orbital integrals are to its Lie algebra. This was formalised by the Harish-Chandra--Howe local character expansion (about arbitrary semisimple elements) in terms of Fourier transforms of nilpotent orbital integrals, and later by the Kim--Murnaghan--Kirillov asymptotic expansion (about the identity) in terms of Fourier transforms of orbital integrals with a fixed semisimple part. In this talk, we discuss an analogue of the Kim--Murnaghan--Kirillov expansion (for characters and related distributions) about arbitrary points, and how to compute it effectively for supercuspidal characters.

### 12:00-12:50 - Prof. Frederica Fanoni

Title: The maximum injectivity radius of hyperbolic 2-orbifolds

Abstract: Given a hyperbolic surface, its maximum injectivity radius is the radius of the biggest disk we can isometrically embed in the surface. There are known upper and lower bounds for this radius which depend only on the topology of the surface. In this talk I will present some of these results and I will discuss the same problem in the case of surfaces with a hyperbolic metric and singular points (hyperbolic 2-orbifolds).

###### This event is sponsored by the Math Research Center

View the event poster here.