This is the homepage for the UIUC Number Theory Seminar.
We will meet in Altgeld Hall, Room 347 (for Tuesday meetings) and Room 345 (for Thursday meetings).
Seminars from 2012-2022 are archived here.
February 6: Chantal David (Concordia University)
Title: Moment of cubic L-functions over Fq(t) at s=1/3
Abstract: We compute the first moment of Dirichlet L-functions over Fq[t] attached to cubic characters, evaluated at an arbitrary s ∈ (0, 1). We find a transition term at the point s = 1/3 , using the deep connections between Dirichlet series of cubic Gauss sums and metaplectic Eisenstein series first introduced by Kubota to obtain cancellation between the principal sum and the dual sum at s = 1/3 . We also explain how at s = 1/3 , the first moment matches corresponding statistics of the group of unitary matrices multiplied by a weight function. This is a joint work with P. Meisner (Gothenburg).
February 13: Kevin Ford (UIUC)
Title: Toward a theory of prime detecting sieves
Abstract: I will give a gentle introduction to the "Type-I/Type-II" sieves that have in recent years been successful at detecting primes in difficult sequences, and recent work with James Maynard to develop a systematic theory for such sieves.
February 20: Shizhang Li (Institute for Advanced Study)
Title: On Cohomology of BG
Abstract: Cohomology of classifying space/stack of a group G is the home which resides all characteristic classes of G-bundles/torsors. In this talk, we will try to explain some results on Hodge/de Rham cohomology of BG where G is a p-power order commutative group scheme over a perfect field of characteristic p, in terms of its Dieudonné module. This is a joint work in progress with Dmitry Kubrak and Shubhodip Mondal.
February 27: Alexander Dunn (Georgia Institute of Technology)
Title: Quartic Gauss sums over primes and metaplectic theta functions
Abstract: We improve 1987 estimates of Patterson for sums of quartic Gauss sums over primes. Our Type-I and Type-II estimates feature new ideas, including use of the quadratic large sieve over the Gaussian quadratic field, and Suzuki's evaluation of the Fourier-Whittaker coefficients of quartic theta functions at squares. We also conjecture asymptotics for certain moments of quartic Gauss sums over primes. This is a joint work with C. David, A. Hamieh and H. Lin.
March 5: Mikhail Gabdullin (UIUC)
Title: Long strings of consecutive composite values of polynomials
Abstract: There is a famous and still open conjecture of Bunyakovsky that for any irreducible polynomial f over the integers (with a positive leading coefficient and no common factor for its values) there are infinitely many n for which f(n) is prime. Since we do not know how to attack it, one can try to think in the opposite direction: how many consecutive numbers n, n+1,...,n+m up to some threshold X we are able to construct so that all the values f(n), f(n+1),...,f(n+m) are composite? Some simple argument implies that this is possible for m of order log X. In a recent work of Ford, Konyagin, Tao, Maynard and Pomerance it was shown that one can take m to be as large as (log X)(log log X)^{C(f)}, where the exponent C(f) is exponentially small in the degree of f. We make this bound independent on f and improve it to (log X)(log log X)^{1/835}. This is a joint work with Kevin Ford.
March 12: Spring Break
Title: None
Abstract: None
March 19: Stelios Sachpazis (University of Turku)
Title: Primes in arithmetic progressions under Siegel zeroes
Abstract: Let x ≥ 1 and let q and a be two coprime positive integers. As usual,
ψ(x;q,a) := sum_{n ≤ x: n = a (mod q)} Λ(n),
where Λ(n) is the von Mangoldt function. In 2003, Friedlander and Iwaniec assumed the existence of ''extreme'' Siegel zeroes and established an asymptotic formula for ψ(x;q,a) beyond the limitations of GRH, with moduli q beyond √x yielding non-trivial information. In particular, they obtained a meaningful asymptotic for q ≤ x^{1/2+1/231}. We will see how one can relax the ''extremity'' of the exceptional zeroes and replace it by simply the definition of a Siegel zero. We will also discuss an idea to improve the Friedlander-Iwaniec regime and reach the range q ≤ x^{1/2 + 1/82 - ε}. This talk is based on on-going work.
March 26: Ploy Wattanawanichkul (UIUC)
Title: Holomorphic quantum unique ergodicity and weak subconvexity for L-functions
Abstract: Quantum unique ergodicity (QUE) describes the equidistribution of the L2-mass of eigenfunctions of the Laplacian as their eigenvalues approach infinity. My focus lies on a specific variant known as holomorphic QUE, which concerns the distribution of the L2-mass of normalized Hecke eigenforms of even weight k (where k ≥ 2). In 2010, Soundararajan and Holowinsky proved the equidistribution of normalized Hecke eigenforms as k tends to infinity. In my talk, I will discuss their proof ideas, explore the connection with the subconvexity problem, and present my new results on the topic.
April 9: William Banks (University of Missouri, Columbia)
Title: Recent results on zeros of the zeta function
Abstract: My talk will focus on some of my recent work on zeros of the Riemann zeta function and Dirichlet L-functions. In particular, I plan to announce a new result and state some related open problems.