This is the homepage for the UIUC Number Theory Seminar.
We will meet in Room 145 Altgeld Hall.
Seminars from 2012-2022 are archived here.
September 16: James Maynard (University of Oxford), Colloquium talk
Title: Prime numbers and zero densities
Abstract: The Riemann Hypothesis, often considered one of the most important open problems in mathematics, would have a number of fantastic consequences for our understanding of the distribution of prime numbers. It claims that all the (non-trivial) zeros of a complex function (the Riemann zeta function) have real part equal to 1/2. However, it turns out that several of these consequences would actually follow from a weaker 'zero density' result that says 'most' zeros lie 'close' to the line with real part equal to 1/2. I'll describe this picture assuming no prior knowledge, as well as describing some recent work (joint with Larry Guth) that improves an 80-year-old estimate on the density of zeros, with corresponding improvements for the distribution of primes.
September 18: Bianca Viray (University of Washington), Colloquium talk
Title: Parameterized and isolated points on curves
Abstract: Let C be an algebraic curve over Q of genus at least 2, i.e., a 1-dimensional negatively curved complex manifold defined by polynomial equations with rational coefficients. A celebrated result of Faltings implies that, despite the hyperbolicity of C, all algebraic points on C are organized into families of nonnegative curvature. We explore how these families provide insight into the arithmetic of C and give applications to the study of elliptic curves. This talk is based in part on joint work with A. Bourdon, Ö. Ejder, Y. Liu, and F. Odumodu, with I. Vogt, and with I. Balçik, S. Chan, and Y. Liu. This talk will be suitable for a general audience.
September 23: Asif Zaman (University of Toronto)
Title: Effective Brauer-Siegel theorems for Artin L-functions
Abstract: Given a number field K (other than Q), in a now classic work, Stark pinpointed the possible source of a so-called Landau--Siegel zero of the Dedekind zeta function ζ_K(s) and used this to give effective upper and lower bounds on the residue of ζ_K(s) at s=1. I will discuss an extension of Stark's work to give effective upper and lower bounds for the leading term of the Laurent expansion of general Artin L-functions at s=1 that are, up to the value of implied constants, as strong as could reasonably be expected given current progress toward the generalized Riemann hypothesis. The bounds are completely unconditional, and rely on no unproven hypotheses about Artin L-functions. This is joint work with Peter Cho and Robert Lemke Oliver.
September 30: Kyle Pratt (Brigham Young University)
Title: Diophantine problems with perfect powers
Abstract: Many Diophantine equations feature perfect powers in an essential way. For instance, the famous Catalan conjecture (proved by Mihailescu) asserts that the only solution to x^m-y^n=1 in positive integers x,y,m,n with m,n ≥ 2 is (x,y,m,n) = (3,2,2,3). In this talk, I discuss some recent work on other Diophantine problems involving perfect powers. In particular, I will focus on Lebesgue--Nagell equations, and a conjecture of Erdos concerning when products of terms in an arithmetic progression can form a perfect power.
October 7: David Zureick-Brown (Amherst College)
Title: Angle ranks of Abelian varieties
Abstract: I will discuss an elementary notion – the rank of the multiplicative group generated by roots of a polynomial. For Weil polynomials, the roots lie on a circle and one calls this the angle rank. I'll present new results about angle ranks and give some applications to the Tate conjecture for Abelian varieties over finite fields and to arithmetic statistics.
October 9: William Banks (University of Missouri, Columbia)
Title: Interactions between the zero sets of different Dirichlet L-functions
Abstract: My talk will describe recent work in Linnik-Sprindžuk program, which seeks to connect the generalized Riemann hypothesis (GRH) for Dirichlet L-functions L(s,\chi) with properties of the zeros of the Riemann zeta function.
October 9: Jesse Thorner (UIUC), Number Theory Web Seminar
Details are here
October 14: Ken Willyard (UIUC)
Title: The imaginary case of the Nonabelian Cohen-Lenstra heuristics
Abstract: The Cohen-Lenstra heuristics predict the distribution of class groups of quadratic number fields. These conjectures are wide open, but nevertheless has been generalized in several different directions, including a non-abelian generalization by Liu, Wood, and Zureick-Brown replacing quadratic extensions with certain totally real Galois extensions and class groups with Galois groups of certain unramified extensions of global fields. In this talk, I will discuss an imaginary analogue of the work of Liu, Wood, and Zureick-Brown by providing the group presentation for these Galois groups that suggests the non-abelian Cohen-Lenstra heuristics in the imaginary case, and giving an idea of the proof of a weak version of these heuristics in the function field case. This work is joint with Yuan Liu.
October 21: Jiuya Wang (University of Georgia)
Title: On b-constant in Malle's Conjecture
Abstract: Malle’s Conjecture predicts how often number fields with a fixed Galois group occur. Klüners found a counterexample using extensions that contain “cyclotomic” subfields. Türkelli proposed a modified version based on the analogy between number fields and function fields. In this talk, we give counterexamples to Türkelli’s modification over number fields and discuss the differences in field enumeration between the two settings. If time permits, we will also propose a fix to Malle’s Conjecture over number fields.
November 4: Qihang Sun (University of Lille)
Title: Sphere packing, Fourier interpolation, and real-variable Kloosterman sums
Abstract: In 2017, Viazovska et al. proved that the E8 and Leech lattices are the densest sphere packings in dimensions 8 and 24, respectively. Later in 2022, the same authors proved the universal optimality of these lattices. For these breakthrough works, Viazovska was awarded the Fields Medal. One of the key ingredients in these works is an interpolation formula allowing a reconstruction of radial Schwartz Fourier eigen-functions from their values at the corresponding lattices. These interpolation formulas was constructed from integrals of vector-valued modular forms. Following these results, the connection between Fourier interpolation problems and modular forms has been extensively studied by Bondarenko, Radchenko, Ramos, Seip, Sousa, Stoller, and many others. In this talk, we present an explicit construction of Fourier interpolation in dimensions 3 and 4 using Maass-Poincaré type series and real-variable Kloosterman sums. This is joint work with Danylo Radchenko.
November 11: Mikhail Gabdullin (UIUC)
Title: Primes with small primitive roots
Abstract: For a prime p, the least primitive root g(p) (smallest generator of (Z/pZ)^*) has been studied since the earliest works of Vinogradov. The folklore conjecture states that g(p)=O(p^{ε}) for every fixed ε > 0. The strongest unconditional result so far is Burgess' estimate g(p)=O(p^{1/4+ε}), established in the 1960s, which has remained unbeaten since. Our main result is the following. Let δ(p) tend to zero arbitrarily slowly as p → ∞. We exhibit an explicit set S of primes p, defined in terms of simple functions of the prime factors of p-1, for which g(p) ≤ p^{1/4-δ(p)} for all p ∈ S, and |{p ≤ x: p ∈ S}| ~ π(x) as x → ∞. This is a joint work with Kevin Ford and Andrew Granville.
November 18: Abhishek Jha (UIUC)
Title: Linnik's problem for Euler's totient function
Abstract: Linnik's famous theorem states that there exists a positive constant C such that for any sufficiently large integer q and any integer a co-prime to q, there exists a prime number p ⩽ q^C such that p ≡ a (mod q). Thanks to works of Jutila, Heath-Brown, Xylouris and others, we now know that one may take C = 5. In this talk I will discuss a variant of Linnik's problem for the Euler's totient function.