Feb 06, 2017
Monday
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09:00 AM - 09:15 AM
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Welcome
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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09:15 AM - 10:00 AM
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Introductory talk (Ph. Michel) -- targeted in particular to members of the harmonic analysis program
Philippe Michel (École Polytechnique Fédérale de Lausanne (EPFL))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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10:00 AM - 10:30 AM
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Break
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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--
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10:30 AM - 11:30 AM
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Mini-course on multiplicative functions
Kaisa Matomäki (University of Turku), Maksym Radziwill (McGill University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
-
- Abstract
The mini-course will be an introduction to the theory of general multiplicative functions and in particular to the theorem of Matomaki-Radziwill 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 Granville-Soundararajan 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 Matomaki-Radziwill 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
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11:30 AM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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--
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02:00 PM - 03:00 PM
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Geometric analytic number theory
Jordan Ellenberg (University of Wisconsin-Madison)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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 arithmetic-statistical 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
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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--
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03:30 PM - 04:30 PM
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Moments of arithmetical sequences
Daniel Fiorilli (University of Ottawa)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
I will start with an introduction to equidistribution results for arithmetic sequences in progressions, of Bombieri-Vinogradov, Barban-Davenport-Halberstam and Fouvry-Bombieri-Friedlander-Iwaniec 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
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--
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04:45 PM - 05:45 PM
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The Kuznetsov Formula, Kloostermania and Applications
Ian Petrow (ETH Zürich)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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Feb 07, 2017
Tuesday
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09:30 AM - 10:30 AM
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$\ell$-adic trace functions in analytic number theory
Philippe Michel (École Polytechnique Fédérale de Lausanne (EPFL))
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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--
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11:00 AM - 12:00 PM
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A glimpse at arithmetic quantum chaos
Gergely Harcos (Central European University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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--
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02:00 PM - 03:00 PM
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Quadratic twists of elliptic curves with 3-torsion
Robert Lemke Oliver (Tufts University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
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--
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03:30 PM - 04:30 PM
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Mini-course on multiplicative functions
Kaisa Matomäki (University of Turku), Maksym Radziwill (McGill University)
|
- Location
- SLMath: Eisenbud Auditorium
- Video
-
- Abstract
The mini-course will be an introduction to the theory of general multiplicative functions and in particular to the theorem of Matomaki-Radziwill 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 Granville-Soundararajan 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 Matomaki-Radziwill 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
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04:30 PM - 06:20 PM
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Reception
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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--
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Feb 08, 2017
Wednesday
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09:30 AM - 10:30 AM
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The Kuznetsov Formula, Kloostermania and Applications
Ian Petrow (ETH Zürich)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
- --
- Supplements
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
-
--
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11:00 AM - 12:00 PM
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$\ell$-adic trace functions in analytic number theory
Philippe Michel (École Polytechnique Fédérale de Lausanne (EPFL))
|
- Location
- SLMath: Eisenbud 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
-
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Feb 09, 2017
Thursday
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09:30 AM - 10:30 AM
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Large fixed order character sums
Youness Lamzouri (York University)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
For a non-principal Dirichlet character $\chi$ modulo $q$, the classical P\'olya-Vinogradov 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\'olya-Vinogradov and Montgomery-Vaughan 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
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
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--
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11:00 AM - 12:00 PM
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On Epstein's zeta function and related results in the geometry of numbers
Anders Sodergren (Chalmers University of Technology)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
-
--
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02:00 PM - 03:00 PM
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|
Mini-course on multiplicative functions
Kaisa Matomäki (University of Turku), Maksym Radziwill (McGill University)
|
- Location
- SLMath: Eisenbud Auditorium
- Video
-
- Abstract
The mini-course will be an introduction to the theory of general multiplicative functions and in particular to the theorem of Matomaki-Radziwill 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 Granville-Soundararajan 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 Matomaki-Radziwill 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
-
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03:00 PM - 03:30 PM
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|
Tea
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- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
-
--
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03:30 PM - 04:30 PM
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$\ell$-adic trace functions in analytic number theory
Philippe Michel (École Polytechnique Fédérale de Lausanne (EPFL))
|
- Location
- SLMath: Eisenbud 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
-
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Feb 10, 2017
Friday
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09:30 AM - 10:30 AM
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Trace functions and special functions
Will Sawin (ETH Zürich)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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
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10:30 AM - 11:00 AM
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Break
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- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
-
--
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11:00 AM - 12:00 PM
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The Kuznetsov Formula, Kloostermania and Applications
Ian Petrow (ETH Zürich)
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- Location
- SLMath: Eisenbud Auditorium
- Video
-
- Abstract
- --
- Supplements
-
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12:00 PM - 02:00 PM
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Lunch
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- Location
- SLMath: Atrium
- Video
-
--
- Abstract
- --
- Supplements
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--
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02:00 PM - 03:00 PM
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Variations on the Chebychev bias phenomenon
Florent Jouve (Université de Bordeaux I)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- 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 L-functions 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
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03:00 PM - 03:30 PM
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Tea
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- Location
- SLMath: Atrium
- Video
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--
- Abstract
- --
- Supplements
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03:30 PM - 04:30 PM
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L-functions and spectral summation formulae for the symplectic group
Valentin Blomer (Georg-August-Universität zu Göttingen)
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- Location
- SLMath: Eisenbud Auditorium
- Video
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- Abstract
We present an introduction to L-functions, spectral summation formulae and Siegel modular forms of degree 2. We show how to compute moments of spinor L-functions and give some applications
- Supplements
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