Georgia Institute of Technology College of Sciences

A recent advance by Smith establishes a quantitative converse (conjectured by Smyth and Serre) to Fekete's celebrated theorem for compact subsets of $\mathbb{R}$. Answering a basic question raised by Smith, we formulate and prove a quantitative converse of Fekete for general symmetric compact subsets of $\mathbb{C}$. We highlight and exploit the algorithmic nature of our approach to give concrete applications to abelian varieties over finite fields and to extremal problems in algebraic number theory.

A fundamental conjecture in number theory is the Riemann hypothesis, which implies the prime number theorem with an optimally strong error term. While a proof remains elusive, many results in number theory can nonetheless be proved using weaker inputs. I will discuss how one such weaker input, subconvexity, can be used to prove strong results on the equidistribution of geometric objects such as lattice points on the sphere. If time permits, I will also discuss how various proofs of subconvexity reduce to understanding period integrals of automorphic forms.

Unicritical polynomials, typically written in the form $z^d+c$, have been widely studied in arithmetic and complex dynamics and are characterized by their one finite critical point. The behavior of a map's critical points under iteration often determines the dynamics of the entire map. Rational maps where the critical points have a finite forward orbit are called post-critically finite (PCF), and these are of great interest in arithmetic dynamics. They are viewed as a dynamical analogue of abelian varieties with complex multiplication and often display interesting dynamical behavior. The family of (single-cycle normalized) dynamical Belyi polynomials have two fixed critical points, so they are PCF by construction, and these maps provide a new testing ground for conjectures and ideas related to post-critically finite polynomials. Using this family, we can begin to explore properties of polynomial maps with two critical points. In this talk we will discuss applications of this family in arithmetic dynamics; in particular, how this family can be used to determine more general reduction properties of PCF polynomials. 

A fundamental problem in analytic number theory is to calculate the maximal value of L-functions at a given point. For L-functions associated to quadratic Dirichlet characters at s = 1, the upper bounds and Ω-results of Littlewood differ by a factor of 2, and it is a long-standing (and still unsolved) problem to find out which one is closer to the truth. The important work of Granville and Soundararajan, who model the distribution of L(1, χ) by the distribution of random Euler products L(1, X) for random variables X(p) attached to each prime, shed more light to the question. We use similar techniques to study the distribution of L(1, χ) for cubic Dirichlet characters. Unlike the quadratic case, there is an asymmetry between lower and upper bounds for the cubic case, and small values are less probable than large values. This is a joint work with P. Darbar, M. Lalin and A. Lumley.