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)
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October 9: Jesse Thorner (UIUC), Number Theory Web Seminar
Details are here
October 14: Ken Willyard (UIUC)
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October 21: Jiuya Wang (University of Georgia)
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November 11: Mikhail Gabdullin (UIUC)
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November 18: Abhishek Jha (UIUC)
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