Informal analytic number theory seminar

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Spring Semester 2015

Date / Time

Speaker

Title

Location

3 March 2015
10:15-11:15

Prof. Dr. Emmanuel Kowalski
ETH Zurich, Switzerland

Event Details

Informal Analytic Number Theory Seminar

Title	Introduction and complements to Fouvry-Katz-Laumon stratification results
Speaker, Affiliation	Prof. Dr. Emmanuel Kowalski, ETH Zurich, Switzerland
Date, Time	3 March 2015, 10:15-11:15
Location	CLV B 4
Abstract	I will first describe the stratification results for exponential sums over finite fields due to Katz-Laumon and Fouvry-Katz, and some of their applications to analytic number theory. I will then describe some complements to these results in the direction of uniformity in terms of various parameters.

Introduction and complements to Fouvry-Katz-Laumon stratification resultsread_more

CLV B 4

10 March 2015
10:00-11:00

Jens Marklof
Bristol University

Event Details

Informal Analytic Number Theory Seminar

Title	Quadratic Weyl Sums, Automorphic Functions, and Invariance Principles
Speaker, Affiliation	Jens Marklof, Bristol University
Date, Time	10 March 2015, 10:00-11:00
Location	CLV B 4
Abstract	Hardy and Littlewood's approximate functional equation for quadratic Weyl sums (theta sums) provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classical Jacobi theta function, on the other hand, satisfies an exact functional equation, and extends to an automorphic function on the Jacobi group. In the present study we construct a related, almost everywhere non-differentiable automorphic function, which approximates quadratic Weyl sums up to an error of order one, uniformly in the summation range. This not only implies the approximate functional equation, but allows us to replace Hardy and Littlewood's renormalization approach by the dynamics of a certain homogeneous flow. The great advantage of this construction is that the approximation is global, i.e., there is no need to keep track of the error terms accumulating in an iterative procedure. Our main application is a new functional limit theorem, or invariance principle, for theta sums. The interesting observation here is that the paths of the limiting process share a number of key features with Brownian motion (scale invariance, invariance under time inversion, non-differentiability), although time increments are not independent and the value distribution at each fixed time is distinctly different from a normal distribution. Joint work with Francesco Cellarosi.

Quadratic Weyl Sums, Automorphic Functions, and Invariance Principlesread_more

CLV B 4

17 March 2015
10:15-11:15

Benny Löffel
ETH Zurich, Switzerland

Event Details

Informal Analytic Number Theory Seminar

Title	Correlation of Hecke eigenvalues with some functions of algebraic origin
Speaker, Affiliation	Benny Löffel, ETH Zurich, Switzerland
Date, Time	17 March 2015, 10:15-11:15
Location	CLV B 4

Correlation of Hecke eigenvalues with some functions of algebraic origin

CLV B 4

24 March 2015
10:15-11:15

Brad Rodgers
Universität Zürich

Event Details

Informal Analytic Number Theory Seminar

Title	Tail bounds for counts of zeros and the mean value of ratios of the zeta function
Speaker, Affiliation	Brad Rodgers, Universität Zürich
Date, Time	24 March 2015, 10:15-11:15
Location	CLV B 4
Abstract	In the 90s, D. Farmer made a conjecture for the mean value of certain ratios of the Riemann zeta function, and in subsequent work he and other authors have shown that information from these averages is sufficient to determine the local statistics of zeros of the zeta function (the GUE Hypothesis). We will discuss the converse implication: that the GUE Hypothesis implies an asymptotic formula for the mean value of ratios of the zeta function. A critical role will be played by tail bounds for counts of zeros in a random `microscopic' interval. We give estimates for the decay of these counts. This talk is based on results in the paper, arXiv: 1502.05658

Tail bounds for counts of zeros and the mean value of ratios of the zeta functionread_more

CLV B 4

31 March 2015
10:15-11:15

Prof. Dr. Paul Nelson
ETH Zurich, Switzerland

Event Details

Informal Analytic Number Theory Seminar

Title	Sketches on representations of the group of unimodular two-by-two matrices
Speaker, Affiliation	Prof. Dr. Paul Nelson, ETH Zurich, Switzerland
Date, Time	31 March 2015, 10:15-11:15
Location	CLV B 4
Abstract	I will attempt an idiosyncratic introduction to the representation theory of SL(2,R), with emphasis on how to heuristically understand analytic aspects (such as those arising in analytic number theory) without any actual calculation. I will do this by drawing pictures (inspired by the orbit method).

Sketches on representations of the group of unimodular two-by-two matricesread_more

CLV B 4

14 April 2015
10:15-11:15

Prof. Dr. Özlem Imamoglu
ETH Zurich, Switzerland

Event Details

Informal Analytic Number Theory Seminar

Title	Modular integrals and Dirichlet series
Speaker, Affiliation	Prof. Dr. Özlem Imamoglu, ETH Zurich, Switzerland
Date, Time	14 April 2015, 10:15-11:15
Location	CLV B 4

Modular integrals and Dirichlet series

CLV B 4

21 April 2015
10:15-11:15

Prof. Dr. Özlem Imamoglu
ETH Zurich, Switzerland

Event Details

Informal Analytic Number Theory Seminar

Title	Modular integrals and Dirichlet series
Speaker, Affiliation	Prof. Dr. Özlem Imamoglu, ETH Zurich, Switzerland
Date, Time	21 April 2015, 10:15-11:15
Location	CLV B 4
Abstract	This is a continuation of the talk of the previous week.

Modular integrals and Dirichlet seriesread_more

CLV B 4

28 April 2015
10:15-11:15

Prof. Dr. Emmanuel Kowalski
ETH Zurich, Switzerland

Event Details

Informal Analytic Number Theory Seminar

Title	Equidistribution properties of Mellin transforms over finite fields, following N. Katz
Speaker, Affiliation	Prof. Dr. Emmanuel Kowalski, ETH Zurich, Switzerland
Date, Time	28 April 2015, 10:15-11:15
Location	CLV B 4
Abstract	I will survey the results of N. Katz concerning equidistribution properties of families of exponential sums parameterized by multiplicative characters over finite fields.

Equidistribution properties of Mellin transforms over finite fields, following N. Katzread_more

CLV B 4

5 May 2015
10:15-01:15

Jean-Marc Deshouillers
Université de Bordeaux

Event Details

Informal Analytic Number Theory Seminar

Title	Sum-free subsets of $\mathbb{Z}/p\mathbb{Z}$
Speaker, Affiliation	Jean-Marc Deshouillers, Université de Bordeaux
Date, Time	5 May 2015, 10:15-01:15
Location	CLV B 4
Abstract	A subset $A$ of an additive monoid is said to be sum-free if $A + A$ and $A$ are disjoint. A classical result of Cauchy leads to the fact that a sum-free subset of $\mathbb{Z}/p\mathbb{Z}$, where $p$ is a prime number, has cardinality at most $[(p+1)/3]$; this bounds is attained when $A$ is the interval $[p/3, 2p/3]$ modulo $p$. The description of the structure of a sum-free subset of $\mathbb{Z}/p\mathbb{Z}$ with cardinality $\alpha p$, where $\alpha$ is less than (but not too far from) $1/3$ has been studied by Vsevolod F. Lev, Gregory A. Freiman and the lecturer. We shall present a joint work in progress, with R. Balasubramanian, treating the cases when $\alpha \ge 0.3$, improving on the previous bound $0.318$.After recalling previous results, we shall present the new combinatorial and harmonic innovations.

Sum-free subsets of $\mathbb{Z}/p\mathbb{Z}$read_more

CLV B 4

12 May 2015
10:15-11:15

Prof. Dr. Javier Fresan
ETH Zurich, Switzerland

Event Details

Informal Analytic Number Theory Seminar

Title	Tannakian categories for the working number theorist
Speaker, Affiliation	Prof. Dr. Javier Fresan, ETH Zurich, Switzerland
Date, Time	12 May 2015, 10:15-11:15
Location	CLV B 4

Tannakian categories for the working number theorist

CLV B 4

26 May 2015
10:15-11:15

Philippe Michel
EPF Lausanne

Event Details

Informal Analytic Number Theory Seminar

Title	Polya-Vinogradov and beyond
Speaker, Affiliation	Philippe Michel, EPF Lausanne
Date, Time	26 May 2015, 10:15-11:15
Location	CLV B 4

Polya-Vinogradov and beyond

CLV B 4

2 June 2015
10:15-11:15

Prof. em. Dr. Gisbert Wüstholz
ETH Zurich, Switzerland

Event Details

Informal Analytic Number Theory Seminar

Title	On curvature lines and geodesics on the triaxial ellipsoid
Speaker, Affiliation	Prof. em. Dr. Gisbert Wüstholz, ETH Zurich, Switzerland
Date, Time	2 June 2015, 10:15-11:15
Location	CLV B 4

On curvature lines and geodesics on the triaxial ellipsoid

CLV B 4

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Organizers: Henryk Iwaniec, Emmanuel Kowalski

Archive: AS 17 SS 15 AS 14