Seminars

Prime number theorems for polynomials from homogeneous dynamics

Abstract: The Bateman-Horn conjecture gives a prediction for how often an irreducible polynomial takes on prime values. In this talk, I will discuss the proof of Bateman-Horn for two new polynomials -- the determinant polynomial on nxn matrices and the determinant polynomial on nxn symmetric matrices. A key tool in the proof is the input of homogeneous dynamics to count the number of integral points on level sets. This is based on joint work with Giorgos Kotsovolis.

On Littlewood's estimate for the modulus of the zeta function on the critical line

Abstract: Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we discuss a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and prime powers. This approach uses the Guinand-Weil explicit formula in conjunction with extremal one-sided bandlimited approximations for the Poisson kernel. As an application, by carefully estimating the Dirichlet polynomial, we revisit a 100-year-old estimate of Littlewood and discuss how we can re-derive the sharpest known upper bound for the modulus of the zeta function on the critical line assuming RH (the main term is due to Chandee and Soundararajan in 2009), but now providing explicit lower-order terms. This is joint work with M. Milinovich (Univ. of Mississippi).

Energy minimization, Fourier interpolation and modular forms

Abstract: I will talk about recent results on energy minimization and sphere packing problems in dimensions 8 and 24. In these  results a key role is played by a certain Fourier interpolation formula that allows to reconstruct any nice radial function from discrete samples of it and its Fourier transform. Surprisingly, this interpolation formula is rarher intricate and its construction involves modular forms. I will explain this construction and discuss some recent works on the related concept of Fourier uniqueness pairs.

Averages of Long Dirichlet Polynomial Approximations of Primitive Dirichlet L-functions 

Abstract: 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. This is joint work with Siegfred Baluyot. 

Infinitely many zeros of additively twisted L-functions on the critical line

Abstract: For f a cuspidal modular form of integral or half-integral weight, we study the zeros of the L-function attached to f twisted by an additive character e^{2\pi i n p/q}, where p/q is a rational number. We prove that for certain f and rational number p/q, the additively twisted L-function has infinitely many zeros on the critical line. We develop a variant of the Hardy-Littlewood method which uses distributions to prove the result.

Extending the unconditional support in an Iwaniec--Luo--Sarnak family

Abstract: We study the harmonically weighted one-level density of low-lying zeros of L-functions attached to holomorphic cusps forms of fixed even weight $k$ and prime level tending to infinity. This family was proved to be of orthogonal type by Iwaniec, Luo and Sarnak who obtained the predicted main term for test functions having Fourier transform supported in $(-\tfrac32,\tfrac32)$ unconditionally. Using zero-density estimates for Dirichlet L-function, we extend this admissible support to $(-\Theta_k;\Theta_k)$, where $\Theta_2 = 1.866\dots$ and $\Theta_k \rightarrow 2$ as $k$ grows.

This is joint work with Daniel Fiorilli and Anders Södergren.

Twisted moments of characteristic polynomials of random matrices

Abstract: It is now well-known that various statistics of L-functions can be predicted using random matrix theory. In particular, the work of Keating and Snaith from the late 90's used random matrix theory to predict the exact leading terms of conjectural asymptotic formulas for all integral moments of the Riemann zeta-function. Prior to their work, no number-theoretic argument or heuristic has led to such exact predictions for all integral moments. In 2015, Conrey and Keating revisited the approach of using divisor sum heuristics to predict asymptotic formulas for moments of zeta. Their key idea is essentially to estimate moments of zeta using lower twisted moments. In this talk, I will discuss a rigorous random matrix theory analogue of the Conrey-Keating heuristic. This is ongoing joint work with Brian Conrey.

A quadratic Vinogradov mean value theorem in finite fields

Abstract: A conjecture of Bourgain–Demeter concerns the number of solutions J(A) to the system of equations

x_1 + x_2 + x_3 = x_4 + x_5 + x_6  and  x_1^2 + x_2^2 + x_3^2 = x_4^2 + x_5^2 + x_6^2

with x_1, …, x_6 lying in some finite set A of N integers. In particular, they conjectured that 

J(A) << N^{3 + o(1)}

for every such A. In this talk, we will analyse this problem along with its variant where A is a sparse subset of some finite field. We will present some of our recent results in this direction as well as highlight some of the proof ideas involved therein, in particular, mentioning how one can use techniques from incidence geometry and additive combinatorics to study such questions.

Metaplectic Ice: Using Statistical Mechanics in Number Theory

Abstract: Local Whittaker functions for reductive groups play an integral role in number theory and representation theory, and many of their applications extend to the metaplectic case, where reductive groups are replaced by their metaplectic covering groups. We will examine these functions for covers of GLr through the lens of a solvable lattice model, or ice model: a construction from statistical mechanics that provides a surprising bridge between spaces of Whittaker functions and representations of quantum groups. This story has been well studied before for the case of one particularly nice cover of GLr, which eliminates all complications arising from the center of the group. In this talk, we will see that the same types of connections hold for any metaplectic cover of GLr, as well as examine how different choices of covering group interact with the center of GLr to change the story.

Dispersion and Littlewood’s conjecture

Abstract: I’ll discuss some problems related to Littlewood’s conjecture in diophantine approximation, and the role hitherto played by discrepancy theory. I’ll explain why our new dispersion-theoretic approach should, and does, deliver stronger results. Our dispersion estimate is proved using Poisson summation and diophantine inequalities. This is joint work with Niclas Technau.

Why I love Monovariants: From Zombies and Conway's Soldiers to Fibonacci Games

Reciprocity, non-vanishing, and subconvexity of central $L$-values

Abstract: A reciprocity formula usually relates certain moments of two different families of $L$-functions which apparently have no connections between them. The first such formula was due to Motohashi who related a fourth moment of Riemann zeta values on the central line with a cubic moment of certain automorphic central $L$-values for $\mathrm{GL}(2)$. In this talk, we describe some instances of reciprocity formulas both in low and high-rank groups and give certain applications to subconvexity and non-vanishing of central L-values. These are joint works with Nunes and Blomer--Nelson.

Prehomogeneous vector spaces and the Arthur-Selberg trace formula

Abstract: The Arthur-Selberg trace formula is a central tool in the

theory of automorphic forms, and can be viewed as a nonabelian Poisson summation formula. Langlands' original beyond endoscopy proposal suggested combining analytic number theory techniques with the trace formula as a way to prove Langlands' Functoriality Conjecture in general. I will survey results towards this in the setting of GL(2) since then, including early results of Venkatesh, Altug, and others.

Time permitting, I will explain some of the obstructions and ideas for

general reductive groups, some of which involve prehomogeneous vector spaces.

Imaginary quadratic fields with p-torsion-free class groups and specified split primes

Abstract: We use Zagier’s weight 3/2 Eisenstein series to prove results on the classification of Ramanujan-type congruences for Hurwitz class numbers. 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.

The Sixth Moment of Automorphic L-Functions

Abstract: In this talk I will discuss my recent paper on the sixth moment of a family GL_2 automorphic L-functions. Before discussing my results, I will introduce moments and families of L-functions in some generality, as well as give some background on the specific family of L-functions I study in my paper. As such, the talk should be accessible to both experts and graduate students.

Graviton scattering and differential equations in automorphic forms

 Abstract:  Green, Russo, and Vanhove have shown that the scattering amplitude for gravitons (hypothetical particles of gravity represented by massless string states) is closely related to automorphic forms through differential equations. Green, Miller, Russo, Vanhove, Pioline, and K-L have used a variety of methods to solve eigenvalue problems for the invariant Laplacian on different moduli spaces to compute the coefficients of the scattering amplitude of four gravitons. We will examine two methods for solving the most complicated of these differential equations on $SL_2(\mathbb{Z})\backslash\mathfrak{H}$. We will also discuss recent work with S. Miller to improve upon his original method for solving this equation. 

Bounding self-dual L-functions: the Conrey-Iwaniec method revisited

Abstract: Let chi_q be real characters of large conductor q. Let f be a fixed GL(2) cusp form with trivial central character. In 2000, by using a moment method, Conrey and Iwaniec obtained the best available subconvexity bounds for self-dual L-functions L(\chi_q,1/2) and L(f\times\chi_q,1/2) in the large q-aspect. This approach was later adapted by Xiaoqing Li to give the first subconvexity bounds for GL(3)xGL(2) self-dual L-functions in the large GL(2) spectral aspect. In this talk, we revisit Li's work and update her bounds to the limit of this method. Some Motohashi-type formulae surrounding these problems will also be discussed. This is joint work with Ramon Nunes and Zhi Qi.

Bounds for subsets of $\mathbb{F}_p^n \times \mathbb{F}_p^n$ without L-shaped configurations

Abstract: I will discuss the difficult problem of proving reasonable bounds in the multidimensional generalization of Szemeredi’s theorem and describe a proof of such bounds for sets lacking nontrivial configurations of the form (x,y), (x,y+z), (x,y+2z), (x+z,y) in the finite field model setting.

Longer gaps between values of binary quadratic forms

Modular forms of half-integral weight on G_2

Abstract: Classical holomorphic modular forms are number-theoretic objects that have been intensely studied.  The split exceptional group G_2 does not support a theory of holomorphic modular forms, but it does possess so-called quaternionic modular forms.  These are a special class of automorphic forms that appear to behave similarly to holomorphic modular forms.  In the talk, I will describe a theory of modular forms of half-integral weight on G_2 and other exceptional groups.  In particular, we prove the existence of a modular form of weight 1/2 on G_2 whose Fourier coefficients are related to the 2-torsion in the narrow class groups of totally real cubic fields.  This is joint work with Spencer Leslie.

Partitions into primes with a Chebotarev condition

Abstract: In this talk, we discuss the asymptotic behavior of the number of partitions into primes concerning a Chebotarev condition. In special cases, this reduces to the study of partitions into primes in arithmetic progressions. While the study for ordinary partitions goes back to Hardy and Ramanujan, partitions into primes were recently re-visited by Vaughan. Our error term is sharp and improves on previous known estimates in the special case of primes as parts of the partition. As an application, monotonicity of this partition function is established explicitly via an asymptotic formula in connection to a result of Bateman and Erd\H{o}s.

L^4 norm bounds for automorphic forms

Abstract: Conjectures in the field of Quantum Chaos predict that waves of high energy on negatively curved manifolds must be spread out evenly and randomly. Number theorists are interested in the modular surface and their waves, which are SL(2,Z) invariant. Some headway can be made in this situation by connecting some of these problems to the theory of L-functions. This is what I will describe in my talk, which is based on ongoing joint work with Peter Humphries. 

Solutions to Diagonal Congruences

Abstract: I will discuss solutions to general diagonal congruences and some questions we have been exploring concerning them. I will also give a brief history of this topic. This talk is based on joint work with Todd Cochrane and Craig Spencer.

The sixth moment of Dirichlet L-functions without the average over critical lines

Abstract: Sixth and higher moments of L-functions are important and challenging problems in analytic number theory. There are several models to produce good conjectures for these moments, but none of them gives a good understanding how to approach them. I will discuss my recent joint work with Xiannan Li, Kaisa Matomaki and Maksym Radziwill on an asymptotic formula of the sixth moment of Dirichlet L-functions averaged over primitive characters mod q over all moduli q \leq Q. Unlike the previous work of Conrey, Iwaniec and Soundararajan, we do not need to include an average on the critical line, thus requiring analysis on the "unbalanced" sums. At least half of the talk will be an overview of the problem, and it should be accessible to graduate students.

t-aspect subconvexity for GL(3) L-functions

 Abstract: Let $\pi$ be a Hecke cusp form for $\rm SL_3(\BZ)$. We will start a brief overview of the best known t-aspect subconvexity estimate for $L(1/2+it,\pi)$ due to us. This will be followed by an improvement where we bound the second moment average of $L(1/2+it,\pi)$ over a short interval. This is a work in progress, joint with Wing Hong Leung and Ritabrata Munshi.

An approximate form of Artin's holomorphy conjecture and nonvanishing of Artin $L$-functions

Abstract:  (Joint with Robert Lemke Oliver and Asif Zaman)  Let $p$ be a prime, and let $\mathscr{F}_p(Q)$ be the set of number fields $F$ with $[F:\mathbb{Q}]=p$ with absolute discriminant $D_F\leq Q$.  Let $\zeta(s)$ be the Riemann zeta function, and for $F\in\mathscr{F}_p(Q)$, let $\zeta_F(s)$ be the Dedekind zeta function of $F$.  The Artin $L$-function $\zeta_F(s)/\zeta(s)$ is expected to be automorphic and satisfy GRH, but in general, it is not known to exhibit an analytic continuation to a half-plane of the form $\mathrm{Re}(s) \geq 1-\delta$, where $\delta>0$ is fixed.  I will describe new work that unconditionally shows that for all $\epsilon>0$ and all except $O_{p,\epsilon}(Q^{\epsilon})$ of the $F\in\mathscr{F}_p(Q)$, $\zeta_F(s)/\zeta(s)$ analytically continues to a region in the critical strip containing the box $[1-\epsilon/(20(p!)),1]\times [-D_F,D_F]$ and is nonvanishing in this region.  This result is a special case of something more general.  I will describe some applications to class groups (extremal size, $\ell$-torsion) and the distribution of periodic torus orbits (in the spirit of Einsiedler, Lindenstrauss, Michel, and Venkatesh).

Weighted central limit theorem for central values of L-functions.

Abstract: A classical result of Selberg says that \log|\zeta(1/2 + it)| has a Gaussian limit distribution. We expect the same thing holds for \log|L(1/2, \chi)| for \chi being over the primitive Dirichlet characters modulo q, as q tends to infinity. Proving such a result remains completely out of reach, as it would imply 100% of these central L-values are non-zero, which is a well-known open conjecture. In this talk, I will describe how one can establish a weighted central limit theorem for the central values of Dirichlet $L$-functions. Under the Generalized Riemann Hypothesis, one can also obtain a weighted central limit theorem for the joint distribution of the central L-values corresponding to twists of two distinct primitive Hecke eigenforms. This is joint work with Natalie Evans, Stephen Lester and Kyle Pratt.

Ellipsephic efficient congruencing for the Vinogradov system

Abstract: Ellipsephic sets are subsets of the natural numbers defined by digital restrictions in a given base---such sets have a fractal-like structure which can be seen as a p-adic analogue of generalised real Cantor sets. The recent work of Maynard on primes with missing digits can be seen as an ellipsephic problem, although in this talk we focus on smaller sets of permitted digits, one motivating example being the set of natural numbers whose digits are squares. I will present an upper bound for the number of ellipsephic solutions to the Vinogradov system of diagonal equations, and highlight the key features of the proof, which uses Wooley's efficient congruencing method.

 Computational aspects of Bianchi modular forms

Abstract: In the classical case, modular forms are defined over rationals. We can generalize this to other fields. In particular, we can define Modular forms over imaginary quadratic fields which give us Bianchi Modular Forms. We can define a Bianchi Modular form as an analytic function on the hyperbolic 3-space. But for the purposes of computing, we will focus more on how to view them as classes of homology of certain congruence subgroups. 

In this talk, I will go over some of these techniques. With great fields comes great subtleties. We will discuss some of these and how we can "fix" them. 

Nonlinear problems in arithmetic Ramsey theory

Abstract: Richard Rado characterised those systems of linear Diophantine equations which are ‘unbreakable' with respect to finite partitions, so that any partition of the positive integers yields a set containing a solution. Similarly, a celebrated theorem of Endre Szemerédi gives rise to a characterisation of linear systems possessing solutions in any ‘dense' set of integers. We discuss variants of these results for certain nonlinear equations/configurations.

An approximate form of Artin's holomorphy conjecture and nonvanishing of Artin L-functions

Abstract: Let $k$ be a number field and $G$ be a finite group, and let $\mathfrak{F}_{k}^{G}$ be a family of number fields $K$ such that $K/k$ is normal with Galois group isomorphic to $G$.  Together with  Robert Lemke Oliver and Jesse Thorner, we prove for many families that for almost all $K \in \mathfrak{F}_k^G$, all of the $L$-functions associated to Artin representations whose kernel does not contain a fixed normal subgroup are holomorphic and non-vanishing in a wide region. 

These results have several arithmetic applications. For example, we prove a strong effective prime ideal theorem that holds for almost all fields in several natural large degree families, including the family of degree $n$ $S_n$-extensions for any $n \geq 2$ and the family of prime degree $p$ extensions (with any Galois structure) for any prime $p \geq 2$. I will discuss this result, describe the main ideas of the proof, and share some applications to bounds on $\ell$-torsion subgroups of class groups, to the extremal order of class numbers, and to the subconvexity problem for Dedekind zeta functions.

Abstract: A remarkable theorem due to Khovanskii asserts that for any finite subset A of an abelian semigroup, the cardinality of the h-fold sumset hA grows like a polynomial for all sufficiently large h. However, neither the polynomial nor what sufficiently large means are understood in general. In joint work with Michael Curran (Oxford), we obtain an effective version of Khovanskii's theorem for any subset of $\mathbb{Z}^d$ whose convex hull is a simplex; previously such results were only available for d=1. Our approach also gives information about the structure of hA, answering a recent question posed by Granville and Shakan.

Functions

functions and what is known in the case of primes, $k$-th powers, and $k$-th powers of primes.

Counting problems: open questions in number theory, from the perspective of moments

Abstract: Many questions in number theory can be phrased as counting problems. How many number fields are there? How many elliptic curves are there? How many integral solutions to this system of Diophantine equations are there? If the answer is “infinitely many,” we want to understand the order of growth for the number of objects we are counting in the “family." But in many settings we are also interested in finer-grained questions, like: how many number fields are there, with fixed degree and fixed discriminant? We know the answer is “finitely many,” but it would have important consequences if we could show the answer is always “very few indeed.” In this accessible talk, we will describe a way that these finer-grained questions can be related to the bigger infinite-family questions. Then we will use this perspective to survey interconnections between several big open conjectures in number theory, related in particular to class groups and number fields.

Diophantine Problems in Function Fields

Abstract: Let $\mathbb{Z}$ be the ring of integers, and let $\mathbb{F}_q[t]$ be the ring of polynomials in one variable defined over the finite field $\mathbb{F}_q$ of $q$ elements.  Since the characteristic of $\mathbb{Z}$ is $0$, while that of $\mathbb{F}_q[t]$ is the positive prime number $p$, it is an interesting phenomenon in arithmetic that these two rings resemble one another so faithfully. The study of the similarity and difference between $\mathbb{Z}$ and $\mathbb{F}_q[t]$ lies in the field that relates number fields to function fields. In this talk, we will investigate some Diophantine problems in the settings of  $\mathbb{Z}$ and $\mathbb{F}_q[t]$, including Waring's problem about representations of elements with fixed powers.

Gaps between zeros of the Riemann zeta-function

Abstract: Let $0 < \gamma_1 \le \gamma_2 \le \cdots $ denote the ordinates of the complex zeros of the Riemann zeta-function function in the upper half-plane. The average distance between $\gamma_n$ and $\gamma_{n+1)$ is $2\pi / \log \gamma_n$ as $n\to \infty$. An important goal is to prove unconditionally that these distances between consecutive zeros can much, much smaller than the average for a positive proportion of zeros. We will discuss the motivation behind this endeavor, progress made assuming the Riemann Hypothesis, and recent work with A. Simonič and T. Trudgian to obtain an unconditional result that holds for a positive proportion of zeros. 

Multiplicative orders mod p

Abstract: I will survey what is known about the distribution of the orders of integers mod p, as p varies. Particular attention will be paid to problems of the following sort: For fixed a and b, how do the order of a mod p and the order of b mod p compare, as p varies? The proofs will draw from the elementary, algebraic, and analytic strands of number theory. (So hopefully something for everyone!)

Zeroes of Rankin-Selberg L-Functions and Nonsplit Quantum Ergodicity

Abstract: Rudnick and Sarnak have conjectured that the L^2-mass of Laplacian eigenfunctions of a negatively curved surface should equidistribute in the large Laplacian eigenvalue limit. This is known as the quantum unique ergodicity conjecture. When this surface is the modular surface, these eigenfunctions are a type of automorphic form called Maass forms, and this conjecture is implied by nontrivial bounds for special values of certain Rankin-Selberg L-functions associated to these automorphic forms. I will discuss a generalisation of this conjecture involving the restriction to the modular surface of automorphic forms associated to quadratic number fields, and how progress towards this conjecture is dependent on nontrivial bounds for certain Rankin-Selberg L-functions. This is joint work with Jesse Thorner.

Dicyclic Group Determinant

Abstract: For a finite group $G={g_1, g_2,…,g_n}$  we assign a variable $x_{g_i}$ for each group element $g_i$ and define the group determinant to be the determinant of $n x n$ matrix whose ij^th entry is $x_{g_i*g_j^{-1}}$. Let $\lambda(G)$ be the smallest non-trivial value taken by the group determinant when the $x_{g_i}$ are all integers.  In this talk, we will see the $\lambda(G)$ for every dicyclic group $G=Q_{4n}$ of order $4n$ when $n$ is odd. Time Permitting, I will prove some lemmas which help us to get $\lambda(G)$.

Summing $\mu(n)$: an even faster elementary algorithm 

Abstract: We present a new elementary algorithm for computing $M(x) = \sum_{n \leq x} \mu(n),$ where $\mu(n)$ is the M\"{o}bius function. Our algorithm

takes  

\[\begin{aligned}

\mathrm{time} \ \ O_\epsilon\left(x^{\frac{3}{5}} (\log x)^{\frac{3}{5}+\epsilon} \right)

\ \ \mathrm{and}\ \ \mathrm{space} \ \ O\left(x^{\frac{3}{10}} (\log x)^{\frac{13}{10}}

\right)\end{aligned},\] 

which improves on existing combinatorial algorithms. While there is an analytic algorithm due to Lagarias-Odlyzko with computations based on the integrals of $\zeta(s)$ that only takes time $O(x^{1/2 + \epsilon})$, our algorithm has the advantage of being easier to implement. The new approach roughly amounts to analyzing the difference between a model that we obtain via Diophantine approximation and reality, and showing that it has a simple description in terms of congruence classes and segments. This simple description allows us to compute the difference quickly by means of a table lookup. This talk is based on joint work with Harald Andr\'{e}s Helfgott.

What does string theory teach us about number theory?

Abstract: Certain models of string theory predict correction terms to Einstein’s theory of general relativity, with coefficients that are automorphic functions.  Some of these are Eisenstein series with very delicate special properties.  I’ll give an overview of this topic and its recent application to Arthur’s conjectures from the 1980s about unitary representations.

The sixth moment of Dirichlet L-functions without the average over critical lines

Abstract: Sixth and higher moments of L-functions are important and challenging problems in analytic number theory. There are several models to produce good conjectures for these moments, but none of them gives a good understanding how to approach them. I will discuss my recent joint work with Xiannan Li, Kaisa Matomaki and Maksym Radziwill on an asymptotic formula of the sixth moment of Dirichlet L-functions averaged over primitive characters mod q over all moduli q \leq Q. Unlike the previous work of Conrey, Iwaniec and Soundararajan, we do not need to include an average on the critical line, thus requiring analysis on the "unbalanced" sums. At least half of the talk will be an overview of the problem, and it should be accessible to graduate students.