FOUVRY73

Brandon Alberts (Eastern Michigan University)

Number Field Counting via Multiple Dirichlet Series

Abstract: We show how to use multiple Dirichlet series techniques to prove new asymptotics for the number of G-extensions with bounded discriminant, inspired by their use in the study of moments of $L$-functions. In particular, assuming the generalized Lindelof Hypothesis we prove the existence of an asymptotic whenever $G$ has nilpotency class $2$. This work is joint with Alina Bucur.

Dante Bonolis (Graz University of Technology)

Counting integral points in thin sets of type II, Part II

Abstract: In this second talk, we focus on the proof of the upper bounds presented in Part I. The central input is a uniform version of the Fouvry--Katz--Laumon stratification theorem for exponential sums. We will explain how this tool is combined with the polynomial sieve and outline the main ideas behind establishing the required uniformity.
This talk is based on joint work with Emmanuel Kowalski, Lillian B. Pierce, Tim Santens, and Katy Woo.

Martin Čech (Charles University)

Quadratic number fields with large class numbers in arithmetic progressions

Abstract: Recently, Cherubini et al. Proved, for any $k\geq 1$ the existence of surprisingly many positive integers $N$ such that the fields $\mathbb{Q}(\sqrt N), \mathbb{Q}(\sqrt {N+1}),\dots ,\mathbb{Q}(\sqrt {N+k}),$ have as large class numbers as possible. For fixed $a_1, \dots , a_k$, $b_1 , \dots ,b_l,$ we consider the question whether there exist integers $n$ such that all the number fields $\mathbb{Q}(\sqrt{a_1 n+b_1}), \dots, \mathbb{Q}(\sqrt{a_1 n+b_l}), \mathbb{Q}(\sqrt{a_2 n+b_1}),\dots,\mathbb{Q}(\sqrt{a_k n+b_l})$ have large class numbers. We provide data and heuristics showing that the answer depends on the number of times that $a_i / a_j$ is a square and prove them in some cases.

Alexander Dunn (Georgia Tech)

Recent developments in non-vanishing for higher order Hecke L-functions and CM elliptic curves

Abstract: In this talk I will describe recent progress on the non-vanishing problem for cubic and quartic Hecke L-functions over number fields. Important ingredients include the large sieve and inputs from the theory of metaplectic forms. The cubic case is based on a joint work with A. de Faveri, C.David, and J.Stucky. The quartic case is joint work with C.Castillo and A. de Faveri. We also will discuss some recent unconditional results on the non-orthogonality of the cubic and quartic large sieve. This is a joint work with A.de Faveri and J.Hoffstein.

Peter Koymans (Utrecht University)

Value sets of binary forms.

Abstract: Given a binary form $F$ with integer coefficients, we define its value set $\mathrm{Val}(F)$ to be the set $\{F(x, y) : x, y \in \mathbb{Z}\}$. For two binary forms $F$ and $G$, what can we say if $\mathrm{Val}(F) = \mathrm{Val}(G)$? The goal of this talk is to show how one may give a complete answer to this question. This is joint work with Étienne Fouvry.

Junxian Li (University of California Davis)

Shifted convolution problems

Abstract: Shifted convolution problems concern correlations of arithmetic functions evaluated at shifted integers. We will discuss methods for studying these problems, including dispersion estimates and delta symbol methods. This is based on joint work with Valentin Blomer.

Kaisa Matomäki (University of Turku)

On the mollified second moment of the Riemann zeta function

Abstract: Evaluating the main terms of mollified moments is often a very cumbersome task, especially for complicated mollifiers. In this talk, I will present a more convenient method, based on utilizing the Laurent expansion of an associated multiple Dirichlet series. The talk is based on joint work with Martin Čech.

Jori Merikoski (University of Helsinki)

Primes and multiplicative functions in arithmetic progressions to large moduli

Abstract: I will discuss work-in-progress on primes and multiplicative functions in arithmetic progressions to large moduli. For primes, we establish partial uniformity in the residue class for factorable moduli. For multiplicative functions, we extend the range of moduli by proving a new Type II estimate with one small factor. These improvements rely on automorphic kernel methods in place of sums of Kloosterman sums. This is joint work with Lasse Grimmelt.

Philippe Michel (EPFL)

Applied $\ell$-adic cohomology: a survey through the ages.

Abstract: The term ''applied $\ell$-adic cohomology'' was coined to characterise various efforts to harness the powerful methods from $\ell$-adic cohomology (especially the works of Deligne, Katz, Laumon and others) towards application in analytic number theory. This talk will survey some of these developments and applications and in particular highlight the contributions of Etienne Fouvry.

Alexandru Pascadi (University of Bonn)

Bilinear forms with Kloosterman sums via quadratic characters

Abstract: We discuss recent work, joint with Valentin Blomer, on bilinear forms with $S(m, n; c)$. Our results apply for general moduli c and save $c^{-1/32}$ for sums of square-root length, improving the savings of Kowalski--Michel--Sawin for prime moduli. We rely on a new connection to sums of quadratic Dirichlet characters, as well as on the representation theory of $\mathrm{SL}_2(\mathbb{Z}/c\mathbb{Z})$ (building on a ''non-abelian amplification'' argument from previous work). Time permitting, we briefly mention applications to moments of twisted $L$-functions and the $\mathrm{GL}_2$ exceptional-spectrum large sieve.

Cédric Pilatte (University of Oxford)

Improved unconditional bounds for the Fourier uniformity conjecture

Abstract: Motivated by its close connection to Chowla's conjecture, we study exponential sums of the Liouville function in short intervals. Improving a result of Walsh, we show that the Liouville function has negligible correlations with all linear phases in the interval $[x, x+H]$ for almost all $x\asymp X$, provided $H \geq \exp((\log X)^{2/5+\varepsilon})$.

Damaris Schindler (Goettingen University)

Quantitative weak approximation and quantitative strong approximation for certain quadratic forms

Abstract: In this talk we discuss recent results on optimal quantitative weak approximation for certain projective quadrics over the rational numbers as well as quantitative results on strong approximation for quaternary quadratic forms over the integers. We will also discuss how a Brauer-Manin obstruction can affect the growth of coprime integer points of bounded height on a quaternary quadratic form. This is joint work with Zhizhong Huang and Alec Shute.

K. Soundararajan (Stanford University)

TBC

Abstract: tbc

Julia Stadlmann (University of Illinois Urbana-Champaign)

Improvements on the Johnsen-Selberg prime power sieve and sums of squares

Abstract: Bounding the size of a set which is missing various residue classes modulo primes is one of the most fundamental and well-studied problems in sieve theory. On the other hand, the size of sets which are missing residue classes modulo prime squares is comparatively less well understood. One important tool available is the Johnsen-Selberg prime power sieve. In this talk, I will discuss how careful manipulations of the large sieve inequality can give big improvements over the Johnsen-Selberg sieve for certain interesting residue class configurations modulo prime squares. As an application of these new sieve estimates, we will show that any non-trivial sumset decomposition of the set of sums of squares must consist of two sets of roughly equal size. This result is motivated by and closely related to Ostmann's problem. This is joint work with Christian Elsholtz.

Cathy Swaenepoel (Université Paris Cité)

Prime numbers with an almost prime reverse

Abstract: Let $b\geq 2$ be an integer. For any integer $n\geq 0$, we call reverse of $n$ in base $b$ the integer obtained by reversing the digits of $n$. The existence of infinitely many prime numbers whose reverse is also prime remains an open problem. In a joint work with Cécile Dartyge and Joël Rivat, we show that there are infinitely many primes with an almost prime reverse. More precisely, we show that there exist an explicit $\Omega_b\in \mathbb{N}$ and $c_b>0$ such that, for at least $c_b b^{\lambda} \lambda^{-2}$ primes $p \in [b^{\lambda-1},b^{\lambda}[$, the reverse of $p$ has at most $\Omega_b$ prime factors. Our proof is based on sieve methods and on establishing a result in the spirit of the Bombieri-Vinogradov theorem concerning the distribution in arithmetic progressions of the reverse of prime numbers.

Gerald Tenenbaum (Institut Elie Cartan de Lorraine)

New estimates in the theory of the Erdős-Hooley Delta function

Abstract: Defined by Erdős and further developed by Hooley, the Delta-function $\Delta(n)$ reflects the concentration of the numbers $\log d$ as $d$ runs through the divisors of a natural integer $n$. Estimates for this function has many applications in number theory. Despite many efforts made in recent years and significant progress achieved, the average and normal orders of $\Delta(n)$ have not yet been fully elucidated. The aim of this lecture is to provide an overview of new approaches and corresponding results.

Maryna Viazovska (EPFL)

TBA

Abstract: tba

Victor Wang (Institute of Mathematics, Academia Sinica)

Some Diophantine problems

Abstract: I will discuss some Diophantine counting problems with connections to Professor Fouvry's work.

Katharine Woo (Stanford University)

Counting integral points in thin sets of type II, Part I

Abstract: For $n>1$, consider the absolutely irreducible polynomial $F(Y,X_1,...,X_n)$ that is monic in $Y$. Let $N(F,B)$ count the number of integral vectors x of height at most $B$. In this talk, we will show how to prove nontrivial upper bounds on $N(F,B)$ using the polynomial sieve and estimates on exponential sums. We will further discuss how to weaken and remove certain conditions on GRH by modifying the sieve weights.
This talk is based on joint work with Dante Bonolis, Emmanuel Kowalski, Lillian B. Pierce, and Tim Santens.

Max Xu (Courant Institute)

Random Multiplicative functions in short intervals

Abstract: We completely determine the limiting distribution of a Steinhaus random multiplicative function in short intervals. An interesting feature of the result is that the limiting distribution is always Gaussian but could with a non-obvious normalization when the interval is not very short. This is based on joint work with Adam Harper and Kannan Soundararajan.

Asif Zaman (University of Toronto)

TBA

Abstract: tba