This is the homepage for the UIUC Number Theory Seminar.
We will meet in Altgeld Hall, Room 141.
Seminars from 2012-2022 are archived here.
January 28: Alexandru Zaharescu and Amy Woodall (UIUC)
Title: Arithmetic polygons and fractional parts of polynomials
Abstract: We introduce a class of polygons with special arithmetic properties and discuss possible connections with some classical problems in number theory concerned with fractional parts of polynomials. The talk is based in part on joint work with Jack Anderson.
February 4: Aaron Landesman (Harvard)
Title: The Cohen-Lenstra moments over function fields
Abstract: The Cohen-Lenstra heuristics are influential conjectures in arithmetic statistics from 1984 which predict theaverage number of p-torsion elements in class groups of quadratic fields, for p an odd prime. So far, this average number has only been computed for p = 3. In joint work with Ishan Levy, we verify this prediction for arbitrary p over suitable function fields. The key input to the proof is a computation of the stable homology of Hurwitz spaces associated to dihedral groups. The algebraic geometry seminar later today will be, in some sense, a continuation of this talk.
February 18: Micah Milinovich (University of Mississippi)
Title: Fourier optimization, prime gaps, and the least quadratic non-residue
Abstract: There are many situations where one imposes certain conditions on a function and its Fourier transform and then attempts to optimize a certain quantity. I will describe how two such Fourier optimization frameworks can be used to study classical problems in number theory: bounding the maximum gap between consecutive primes assuming the Riemann hypothesis and bounding for the size of the least quadratic non-residue modulo a prime assuming the generalized Riemann hypothesis (GRH) for Dirichlet L-functions. The resulting extremal problems in analysis can be stated in accessible terms, but finding the exact answer appears to be rather subtle. However, we can experimentally find upper and lower bounds for our desired quantity that are numerically close. If time allows, I will discuss how a similar Fourier optimization framework can be used to bound the size of the least prime in an arithmetic progression on GRH. This is based upon joint works with E. Carneiro (ICTP), E. Quesada-Herrera (Lethbridge), A. Ramos (SISSA), and K. Soundararajan (Stanford).
March 4: Matilde Lalín (Université de Montréal)
Title: The shifted convolution problem in function fields
Abstract: We will discuss some results on the shifted convolution problem for the divisor function over function fields in the large degree limit, that is, the average value of d(f) d(f+h) where f runs over monic polynomials of given degree in Fq[T], and h is a given monic polynomial. We prove an asymptotic formula in the range deg(h) < (2-ε) deg(f). The central ingredient for this work is a Voronoi summation formula for the divisor function. The results also extend to various correlations of the convolution of 1 with a Dirichlet character mod ℓ, where ℓ is a monic irreducible polynomial. This is joint work with Alexandra Florea, Amita Malik, and Anurag Sahay.
March 11: Xiannan Li (Kansas State University)
Title: One level and n-level density for a large orthogonal family of L-functions
Abstract: We study a new orthogonal family of L-functions associated with holomorphic Hecke newforms of level q, averaged over q~Q. I will describe joint work with Baluyot and Chandee on a one level density result assuming GRH with the support of the Fourier transform of the test function being extended to be inside (-4, 4), which doubles the range from previous results. I will further describe ongoing work with Chandee and Lee on proving an n-level density result for the same family with a similar range.
March 25: Alexander Smith (UCLA)
Title: Faithful characters and the Chebotarev density theorem (jt. with Robert Lemke-Oliver)
Abstract: Choose a finite group G and a number field F. We show that, given any large family of G-extensions of F, almost all are subject to a strong effective form of the Chebotarev density theorem. As one consequence, given a prime p, we are able to give nontrivial upper bounds for the size of the p-torsion of the class group of most G-extensions of F.
To prove this result requires a tool we have developed in the character theory of finite groups. More specifically, it requires a strengthened form of Artin's induction theorem that applies to faithful irreducible characters.
April 1: Jaebum Sohn (Yonsei University)
Title: Core Partition, Numerical Semigroups, and Related Problems
Abstract: In this talk, we will introduce the concept of t-core partitions. We will discuss the generating function and modularity, along with some results and applications of t-core partitions. Recent results on simultaneous core partitions will also be presented. At the end of the talk, we will introduce numerical semigroups and explore connections between numerical semigroups (or numerical sets) and partitions. Additionally, we will present some problems related to these topics.
April 8: Bruce Reznick and David Altizio (UIUC)
Title: The Stern Sequence and Stern Polynomials
Abstract: In 1858, responding to a question or Eisenstein about denominators in Farey sequences, Stern defined a sequence s(n) by s(0) = 0, s(1) = 1, s(2n) = s(n), s(2n+1) = s(n) + s(n+1) and proved (decades before Cantor!) that every positive rational occurs exactly once as s(n)/s(n+1). One generalization is the Stern polynomial: s(0,x) = 0, s(1,x) = 1, s(2n,x) = xs(n,x), s(2n+1,x) = s(n,x) + s(n+1,x). We'll discuss some of the amazing properties of {s(n,x)}, especially factorization and the location of zeros.
April 22: Ananth Shankar (Northwestern University)
Title: The Andre-Pink Zannier conjecture in characteristic p
Abstract: The Andre-Pink Zannier conjecture (proved by Richard and Yafaev) addresses the distribution of Hecke orbits in Shimura varieties over number fields. In this talk, I will address a case of this conjecture in positive characteristic, and more generally, I will address the question of how Hecke orbits are distributed in characteristic p. This is joint work with Josh Lam.
April 29: Zhuo Zhang (UIUC)
Title: A uniform Chebotarev density theorem and Artin's holomorphy conjecture
Abstract: We improve the uniformity in the Chebotarev density theorem for Galois extensions of number fields satisfying Artin's holomorphy conjecture. Using nonabelian base change, this yields an unconditional improvement to the uniformity in the Chebotarev density theorem along with the first theoretical improvement over Weiss's bound for the least norm of an unramified prime ideal with given Frobenius class. I will also compare our results to other ones and talk about some concrete examples. This is joint work with Jesse Thorner.