
01/20 3:00pm 
BLOC 628 
Shamgar Gurevich U. Wisc. 
Harmonic Analysis on GL(n) over Finite Fields.
Harmonic Analysis on GL(n) over Finite Fields.
Abstract: There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio:
trace(ρ(g)) / dim(ρ),
for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.
Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank. Rank suggests a new organization of representations based on the very few "small" ones. This stands in contrast to HarishChandra's "philosophy of cusp forms", which is (since the 60s) the main organization principle, and is based on the (huge collection) of "large" representations.
This talk will discuss the notion of rank for the group GL(n) over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for random walks.


02/07 4:00pm 
BLOC 628 
Changho Han University of Georgia 
Counting hyperelliptic curves over global fields of bounded height via hyperelliptic fibrations
A hyperelliptic curve y^2=f(x) with coefficients in a global field (such as Q) comes equipped with a natural invariant called the height of the discriminant. Then, it is a natural question to find the asymptotic behavior of a function counting the number of reasonably behaving hyperelliptic curves with bounded height B. However, this problem turns out to be wide open if the base field is a number field. By jointly working with Junyong Park, we instead consider this problem for function fields, namely F_q(t) the generic point of a projective line over F_q. In this case, we can reinterpret hyperelliptic curves as hyperelliptic fibrations over P^1, allowing us to use both geometry of algebraic surfaces and arithmetic of the space of maps from P^1 into moduli spaces of hyperelliptic curves. By using this observation, I will illustrate the counting function over F_q(t) (which heuristically expect to look similar to that of number fields) and related results along the way (such as motive/point count of relevant moduli spaces and birational geometry involved). 

02/14 4:00pm 
BLOC 628 
Vladimir Dragovich UT Dallas 
Periodic ellipsoidal billiards and Chebyshev polynomials on several intervals
A comprehensive study of periodic trajectories of the billiards within ellipsoids in the ddimensional Euclidean space is presented. The novelty of the approach is based on a relationship established between the periodic billiard trajectories and the extremal polynomials of the Chebyshev type on the systems of d intervals on the real line.
As a byproduct, for d = 2 a new proof of the monotonicity of the rotation number is obtained and the result is generalized for any d. The case study of trajectories of small periods T, d < T < 2d is given. In particular, it is proven that all dperiodic trajectories are contained in a coordinatehyperplane and that for a given ellipsoid, there is a unique set of caustics which generates d + 1periodic trajectories. A complete catalog of billiard trajectories with small periods is provided for d = 3. This talk is based on: V. Dragovic, M. Radnovic, Periodic ellipsoidal billiard trajectories and extremal poly
nomials, arXiv 1804.02515, Comm. Math. Physics. https://doi.org/10.1007/s00220
01903552y, 2019, Vol. 372, p. 183211.
V. Dragovic, M. Radnovic, Caustics of Poncelet polygons and classical extremal polynomials, arXiv 1812.02907, Regular and Chaotic Dynamics, (2019), Vol. 24, No. 1, p.
135. A. Adabrah, V. Dragovic, M. Radnovic, Periodic billiards within conics in the Min
kowski plane and Akhiezer polynomials, arXiv; 1906.0491, Regular and Chaotic Dy
namics, No. 5, Vol. 24, 2019, p. 464501. 