Feb 06, 2017
Monday

09:00 AM  09:15 AM


Welcome

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
 Supplements



09:15 AM  10:00 AM


Introductory talk (Ph. Michel)  targeted in particular to members of the harmonic analysis program
Philippe Michel (École Polytechnique Fédérale de Lausanne (EPFL))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
 
 Supplements


10:00 AM  10:30 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



10:30 AM  11:30 AM


Minicourse on multiplicative functions
Kaisa Matomäki (University of Turku), Maksym Radziwill (McGill University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The minicourse will be an introduction to the theory of general multiplicative functions and in particular to the theorem of MatomakiRadziwill on multiplicative function in short intervals. The theorem says that, for any multiplicative function $f: \mathbb{N} \to [1, 1]$ and any $H \to \infty$ with $X \to \infty$, the average of $f$ in almost all short intervals $[x, x+H]$ with $X \leq x \leq 2X$ is close to the average of $f$ over $[X, 2X]$. In the first lecture we will cover briefly the "pretentious theory" developed by GranvilleSoundararajan and a selection of some of the key theorems: Halasz's theorem, the Lipschitz behaviour of multiplicative functions, Shiu's bound, ... We will also describe some consequences of the MatomakiRadziwill theorem. In the second lecture we will develop sufficient machinery to prove a simple case of the latter theorem for the Liouville function in intervals of length $x^{\varepsilon}$. In the third lecture we will explain the proof of the full result. Time permitting we will end by discussing some open challenges
 Supplements


11:30 AM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Geometric analytic number theory
Jordan Ellenberg (University of WisconsinMadison)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will give an overview of recent progress by many people in analytic number theory over function fields like F_q(t), focusing on the relation between arithmeticstatistical problems over function fields and questions about the topology and algebraic geometry of moduli spaces (over finite fields and even over the complex numbers.) The talk will not assume knowledge of algebraic geometry
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Moments of arithmetical sequences
Daniel Fiorilli (University of Ottawa)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
I will start with an introduction to equidistribution results for arithmetic sequences in progressions, of BombieriVinogradov, BarbanDavenportHalberstam and FouvryBombieriFriedlanderIwaniec type. I will then discuss some of my recent results (with Greg Martin, and with Régis de la Bretèche) on the first two moments with usual and major arcs approximations
 Supplements



04:45 PM  05:45 PM


The Kuznetsov Formula, Kloostermania and Applications
Ian Petrow (ETH Zürich)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
 
 Supplements



Feb 07, 2017
Tuesday

09:30 AM  10:30 AM


$\ell$adic trace functions in analytic number theory
Philippe Michel (École Polytechnique Fédérale de Lausanne (EPFL))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Trace functions are arithmetic functions defined modulo $q$ (some prime number) obtained as Frobenius trace function of $\ell$adic sheaves. The basic example is that of a Dirichlet character of modulus $q$ but there are many other examples of interest for instance (hyper)Kloosterman sums. In this series of lectures we will explain how they arise in classical problems of analytic number theory and how (basi) methods from $\ell$adic cohomology allow to extract a lot out of them. Most of these lectures are based on works of E. Fouvry, E. Kowalski, myself and W. Sawin.
 Supplements


10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


A glimpse at arithmetic quantum chaos
Gergely Harcos (Central European University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Maass forms are fundamental in number theory, but they also arise naturally in mathematical physics and harmonic analysis. Exploring this connection turned out to be very fruitful, and we attempt to give a smooth but informative introduction.
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Quadratic twists of elliptic curves with 3torsion
Robert Lemke Oliver (Tufts University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
 
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Minicourse on multiplicative functions
Kaisa Matomäki (University of Turku), Maksym Radziwill (McGill University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The minicourse will be an introduction to the theory of general multiplicative functions and in particular to the theorem of MatomakiRadziwill on multiplicative function in short intervals. The theorem says that, for any multiplicative function $f: \mathbb{N} \to [1, 1]$ and any $H \to \infty$ with $X \to \infty$, the average of $f$ in almost all short intervals $[x, x+H]$ with $X \leq x \leq 2X$ is close to the average of $f$ over $[X, 2X]$. In the first lecture we will cover briefly the "pretentious theory" developed by GranvilleSoundararajan and a selection of some of the key theorems: Halasz's theorem, the Lipschitz behaviour of multiplicative functions, Shiu's bound, ... We will also describe some consequences of the MatomakiRadziwill theorem. In the second lecture we will develop sufficient machinery to prove a simple case of the latter theorem for the Liouville function in intervals of length $x^{\varepsilon}$. In the third lecture we will explain the proof of the full result. Time permitting we will end by discussing some open challenges.
 Supplements


04:30 PM  06:20 PM


Reception

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements




Feb 08, 2017
Wednesday

09:30 AM  10:30 AM


The Kuznetsov Formula, Kloostermania and Applications
Ian Petrow (ETH Zürich)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
 
 Supplements


10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


$\ell$adic trace functions in analytic number theory
Philippe Michel (École Polytechnique Fédérale de Lausanne (EPFL))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Trace functions are arithmetic functions defined modulo $q$ (some prime number) obtained as Frobenius trace function of $\ell$adic sheaves. The basic example is that of a Dirichlet character of modulus $q$ but there are many other examples of interest for instance (hyper)Kloosterman sums. In this series of lectures we will explain how they arise in classical problems of analytic number theory and how (basi) methods from $\ell$adic cohomology allow to extract a lot out of them. Most of these lectures are based on works of E. Fouvry, E. Kowalski, myself and W. Sawin.
 Supplements



Feb 09, 2017
Thursday

09:30 AM  10:30 AM


Large fixed order character sums
Youness Lamzouri (York University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
For a nonprincipal Dirichlet character $\chi$ modulo $q$, the classical P\'olyaVinogradov inequality asserts that $M(\chi):=\max_{x}\sum_{n\leq x} \chi(n)=O\left(\sqrt{q}\log q\right)$. This was improved to $\sqrt{q}\log\log q$ by Montgomery and Vaughan, assuming the Generalized Riemann hypothesis GRH. For quadratic characters, this is known to be optimal, owing to an unconditional omega result due to Paley. In this talk, we shall present recent results on higher order character sums. In the first part, we discuss even order characters, in which case we obtain optimal omega results for $M(\chi)$, extending and refining Paley's construction. The second part, joint with Sasha Mangerel, will be devoted to the more interesting case of odd order characters, where we build on previous works of Granville and Soundararajan and of Goldmakher to provide further improvements of the P\'olyaVinogradov and MontgomeryVaughan bounds in this case. In particular, assuming GRH, we are able to determine the order of magnitude of the maximum of $M(\chi)$, when $\chi$ has odd order $g\geq 3$ and conductor $q$, up to a power of $\log_4 q$ (where $\log_4$ is the fourth iterated logarithm).
 Supplements


10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


On Epstein's zeta function and related results in the geometry of numbers
Anders Sodergren (Chalmers University of Technology)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In this talk I will discuss certain questions concerning the asymptotic behavior of the Epstein zeta function E_n(L,s) in the limit of large dimension n. In particular I will describe the value distribution of E_n(L,s) for a random lattice L of large dimension n, giving partial answers to questions raised by Sarnak and Strömbergsson in their study of the minima of E_n(L,s). Many of the key ingredients in our discussion will come from the rich interplay between the value distribution of the Epstein zeta function and classical problems in the geometry of numbers
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Minicourse on multiplicative functions
Kaisa Matomäki (University of Turku), Maksym Radziwill (McGill University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The minicourse will be an introduction to the theory of general multiplicative functions and in particular to the theorem of MatomakiRadziwill on multiplicative function in short intervals. The theorem says that, for any multiplicative function $f: \mathbb{N} \to [1, 1]$ and any $H \to \infty$ with $X \to \infty$, the average of $f$ in almost all short intervals $[x, x+H]$ with $X \leq x \leq 2X$ is close to the average of $f$ over $[X, 2X]$. In the first lecture we will cover briefly the "pretentious theory" developed by GranvilleSoundararajan and a selection of some of the key theorems: Halasz's theorem, the Lipschitz behaviour of multiplicative functions, Shiu's bound, ... We will also describe some consequences of the MatomakiRadziwill theorem. In the second lecture we will develop sufficient machinery to prove a simple case of the latter theorem for the Liouville function in intervals of length $x^{\varepsilon}$. In the third lecture we will explain the proof of the full result. Time permitting we will end by discussing some open challenges
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


$\ell$adic trace functions in analytic number theory
Philippe Michel (École Polytechnique Fédérale de Lausanne (EPFL))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Trace functions are arithmetic functions defined modulo $q$ (some prime number) obtained as Frobenius trace function of $\ell$adic sheaves. The basic example is that of a Dirichlet character of modulus $q$ but there are many other examples of interest for instance (hyper)Kloosterman sums. In this series of lectures we will explain how they arise in classical problems of analytic number theory and how (basi) methods from $\ell$adic cohomology allow to extract a lot out of them. Most of these lectures are based on works of E. Fouvry, E. Kowalski, myself and W. Sawin.
 Supplements



Feb 10, 2017
Friday

09:30 AM  10:30 AM


Trace functions and special functions
Will Sawin (ETH Zürich)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will explain the analogy between trace functions over finite fields defined by exponential sums and certain classical functions on the complex numbers defined by integrals of exponentials. There are close analogies, largely due to Katz, that sometimes allow one to guess results in one domain from results in the other. For instance, many important properties of Kloosterman sums are related to facts about Bessel functions. I will explain some of these correspondences, and how to use them to understand exponential sums
 Supplements


10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


The Kuznetsov Formula, Kloostermania and Applications
Ian Petrow (ETH Zürich)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
 
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Variations on the Chebychev bias phenomenon
Florent Jouve (Université de Bordeaux I)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Chebychev's bias, in its classical form, is the preponderance in ``most'' intervals [2,x] of primes that are 3 modulo 4 over primes that are 1 modulo 4. Recently many generalizations and variations on this phenomenon have been explored. We will highlight the role played by some wide open conjectures on Lfunctions in the study of Chebychev's bias. Our focus will be on analogues of Chebychev's question to elliptic curves. In the case where the base field is a function field (of a curve over a finite field) we will report on joint work with Cha and Fiorilli and explain how unconditional results can be obtained
 Supplements


03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Lfunctions and spectral summation formulae for the symplectic group
Valentin Blomer (GeorgAugustUniversität zu Göttingen)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We present an introduction to Lfunctions, spectral summation formulae and Siegel modular forms of degree 2. We show how to compute moments of spinor Lfunctions and give some applications
 Supplements


