Invited Speakers

We study Ramanujan-type congruences for Hurwitz class numbers using harmonic Maass forms. As an application, we show that for any odd prime p and finite set of odd primes S, there exists an imaginary quadratic field which splits at each prime in S and has class number indivisible by p. This result is in the spirit of results by Bruinier, Bhargava (when p=3) and Wiles, but the methods are completely different. This is joint work with Martin Raum and Olav Richter.

Alex Iosevich (University of Rochester)

Pretalk:

Fourier transform over integers modulo N and signal recovery

Imagine that we want to transmit a signal, given in the form of a sequence of s and s of length , and that we send this sequence after first encoding it as the discrete Fourier transform. Suppose that some of the Fourier coefficients are missing. Can we still recover the original signal? The answer turns out to be yes, and while many of the methods are elementary, this line of inquiry quickly leads to the heart of some very interesting problems in number theory, combinatorics, and harmonic analysis.

Conference Talk:

Uncertainty Principles, restriction theory, and Applications

We are going to discuss how the classical Heisenberg Uncertainty Principle can be improved in the presence of non-trivial restriction estimates. We are also going to describe some applications of uncertainty principles to exact signal recovery.

Robert Lemke Oliver (Tufts University)

Faithful induction theorems and the Chebotarev density theorem

The Chebotarev density theorem is a powerful and ubiquitous tool in number theory used to guarantee the existence of infinitely many primes satisfying splitting conditions in a Galois extension of number fields. In many applications, however, it is necessary to know not just that there are many such primes in the limit, but to know that there are many such primes up to a given finite point. This is the domain of so-called effective Chebotarev density theorems. In forthcoming joint work with Alex Smith that extends previous joint work of the author with Thorner and Zaman and earlier work of Pierce, Turnage-Butterbaugh, and Wood, we prove that in any family of irreducible complex Artin representations, almost all are subject to a very strong effective prime number theorem. This implies that almost all number fields with a fixed Galois group are subject to a similarly strong effective form of the Chebotarev density theorem. Under the hood, the key result is a new theorem in the character theory of finite groups that is similar in spirit to classical work of Artin and Brauer on inductions of one-dimensional characters.

Zane Li (NCSU)

Mixed norm decoupling for paraboloids

In this talk we discuss mixed norm decoupling estimates for the paraboloid. One motivation of considering such an estimate is a conjectured mixed norm Strichartz estimate on the torus which essentially is an estimate about exponential sums. This is joint work with Shival Dasu, Hongki Jung, and José Madrid.

Benjamin Linowitz (Oberlin College)

The systolic geometry of arithmetic locally symmetric spaces

The systole of a compact Riemannian manifold M is the least length of a non-contractible loop on M. In this talk I will survey some recent work with S. Lapan and J. Meyer on the systolic geometry of arithmetic locally symmetric spaces, emphasizing systole growth along congruence covers.

Cezar Lupu (Beijing Institute of Mathematical Sciences and Applications)

Multiple zeta values from arithmetic to geometry

In this talk, we give a survey on multiple zeta values focusing more on their applications ranging from arithmetic properties of odd zeta values to Dirichlet eigenvalues in a regular polygon. We explore various identities involving different families of multiple zeta values and we emphasize their importance in proving some very important conjectures.

Amita Malik (Penn State)

Pretalk:

Hardy-Ramanujan circle method

The circle method invented more than a century ago by Hardy and Ramanujan to study integer partitions is still one of the most powerful tools in number theory. This method has been modified over the years, but the key-idea is pretty much the same. In this talk, we give an overview of this method and discuss some of its exciting applications.

conference talk:

Zeros of L-functions attached to Maass forms

Though the study pertaining the zeros of the Riemann zeta functions goes back to the seminal paper of Riemann from 1850’s, the known results are very far from the expected truth. In theory, most of the analogous statements should be true for its cousins, generally known as L-functions, but not all is known about these functions and thus it remains an active area of research. In this talk, we focus on one such type of functions which arise from Maass forms. This talk is based on joint work with Rahul Kumar.

Alexandria Rose (IAS)

Lattice Covering Densities and Additive Combinatorics

The well-known Lattice Covering Problem asks for the most optimal way to cover the space ℝn, n ≥ 2, by using copies of an Euclidean ball centered at points of a given lattice. More precisely, consider a closed Euclidean ball B and a lattice L ⊂ ℝd, we say that L is a covering lattice for B if

(1) ℝn = L + B

The covering density Θ (L) of whole space ℝn is defined as the minimal volume of a closed Euclidean

Ball B for which (1) holds. Define

Θn := inf { Θ (L): L is a lattice in ℝn of covolume one }

to be the minimal density of lattice coverings of ℝn.

Where the covolume of L is the volume of its fundamental parallepipeds (sometimes refer as the determinant of L). Thus the Lattice Covering Problem asks for the best upper bound for Θn.

So far, this problem has only been studied geometrically using Kakeya-type methods to obtain results for convex bodies in place of balls. In this talk, we make a connection between lattice covering densities and additive combinatorics, and consider the more general setting of approximate groups and sets with low doubling or high additive energy. This is joint work with Francisco Romero Acosta.

Caroline Turnage-Butterbaugh (Carleton College)

Moments of Dirichlet L-functions

In recent decades there has been much interest and measured progress in the study of moments of the Riemann zeta-function and, more generally, of various L-functions. Despite a great deal of effort spanning over a century, asymptotic formulas for moments of L-functions remain stubbornly out of reach in all but a few cases. In this talk, we consider the problem for the family of all Dirichlet L-functions of even primitive characters of bounded conductor. I will outline how to harness the asymptotic large sieve to prove an asymptotic formula for the general 2kth moment of an approximation to this family. The result, which assumes the generalized Lindelöf hypothesis for large values of k, agrees with the prediction of Conrey, Farmer, Keating, Rubenstein, and Snaith. Moreover, it provides the first rigorous evidence beyond the so-called “diagonal terms” in their conjectured asymptotic formula for this family of L-functions.