Discussion Meeting

L-functions, Circle-Method and Applications (HYBRID)

ORGANIZERS:
Soumya Das (IISc, Bengaluru, India), Ritabrata Munshi (ISI-Kolkata, India) and Saurabh Kumar Singh (IIT - Kanpur, India)

DATE & TIME: 27 June 2022 to 01 July 2022

VENUE: Ramanujan Lecture Hall and Online

The circle method originated in a paper of S. Ramanujan and G. H. Hardy on the partition function. This method has evolved with time and has seen many interesting applications. The classical applications of the circle method are to the Waring’s problem, to the ternary Gold-bach problem, and to count rational points on varieties. The modern applications of this method are to the subconvexity problem on various L-functions and to the shifted convolution problem. Also, the circle method is a powerful analytical tool to study correlations between two arithmetical functions and it is very flexible to use. The analytic study of L-functions is a central theme in analytic number theory, and it has many arithmetical consequences. The growth of L-functions (few classes of L-functions) can be understood by studying a correlation problem using the circle method. We hope that this method will continue to have many more interesting applications. The aim of this programme is to explore this method and look into its future.

Eligibility criteria: Any researcher working in the area of Analytic Number Theory.

CONTACT US : circlemthd@icts.res.in
PROGRAM LINK: https://www.icts.res.in/discussion-meeting/circlemthd2022

Table of Contents (powered by https://videoken.com)
0:00:00 Start
0:00:10 Beyond the Spherical Sup-norm Problem
0:01:33 Maass forms on the modular surface
0:02:23 Maass forms with 1 ~10' on the modular surface
0:05:19 Classical mechanics vs. quantum mechanics
0:05:20 Classical Mechanics vs. Quantum Mechanics
0:10:53 Quantum ergodicity on the modular surface (1 of 2)
0:18:55 Quantum ergodicity on the modular surface (2 of 2)
0:23:20 The spherical sup-norm problem
0:31:46 Spectral decomposition of the automorphic L' space
0:35:36 The unitary dual of G = SL2(R)
0:38:29 The non-spherical sup-norm problem for SL2(Z) SL2(R)
0:39:55 The unitary dual of G = SL2(C)
0:43:39 New results
0:46:28 Bounding the sup-norm via an automorphic kernel
0:53:00 Generalized spherical transform
0:54:24 Spherical trace function
0:56:27 Paley-Wiener space and Schwartz space
0:57:47 Choice of test function
0:58:29 Amplification
1:02:12 Q&A
1:23:34 Thank You!