International Conference in Number Theory and Physics

This school and conference aims to bring together mathematicians (from number theory and algebraic geometry,…) and theoretical physicists (from Quantum Chaos, QFT and String Theory,…). The first week will be aimed at introductory minicourses and introductory lectures in several topics on Number Theory and Physics. And the second week will focus on research lectures and advanced level minicourses aimed at graduate students and researchers at all levels. 

The main purpose of the school and conference is to bring together leading researchers in number theory and physics, to foster further mutually beneficial interaction between Number Theory and Physics. 

The school will be made of a series of plenary lectures, short lectures and minicourses in different subjects on Number Theory and Physics. One of the aims of this school is to discuss the recent developments in a wide range of topics at the crossroads between Number Theory and Physics. The main topics to be covered in this conference are: 

- Number Theory and Random Matrices 
- Number Theory over Function Fields and Riemann Hypothesis for Curves 
- Modular Forms, Mock Modular Forms and Generalizations 
- String Theory, String dualities and automorphic forms 
- Quantum Field Theory for Mathematicians and Number Theorists 
- Quantum Chaos, Arithmetic Quantum Chaos and Number Theory 
- Noncommutative Geometry in Number Theory and Physics 
- Polylogarithms, Multiple Zeta Values and Pertubative Physics 
- Mirror Symmetry, Calabi-Yau Manifolds and Zeta Functions 
- Prime Numbers and recent developments
- AdS/CFT and arithmetic



Mini Courses

Prof. Steve Gonek (University of Rochester)

A Motivated Introduction to the Riemann Zeta-Function The Riemann zeta-function is important in mathematics both because it encodes valuable information about the prime numbers and because it is the prototype for all L-functions. The purpose of this course is to present a motivated overview of the classical and modern theory of the zeta-function. We will present the main questions of the subject, explain why they are important, what we do and do not know about them, and the tools used to approach them. Some of the topics we will cover are the distribution of zeros and other values of the zeta-function, large and small values of the zeta-function, and moments of the zeta-function. The focus throughout will be on key concepts and applications.

Class 1 - 15-06-2015 - download
Class 2 - 15-06-2015 - download
Class 3 - 16-06-2015 - download
Class 4 - 16-06-2015 - download
Class 5 - 17-06-2015 - download
Class 6 - 22-06-2015 - download

Prof. Sameer Murthy (King's College)

Quantum Black Holes, Modular Forms and Mock Modular Forms In the quantum theory of black holes in superstring theory, the physical problem of counting the number of quarter-BPS dyonic states of a given charge has led to the study of Fourier coefficients of certain meromorphic Siegel modular forms and to the question of the modular nature of the corresponding generating functions. These Fourier coefficients have a wall-crossing behavior which seems to destroy modularity. In this mini-course I shall explain that these generating functions belong to a class of functions called mock modular forms. I shall discuss the physical consequences of this statement, and interesting mathematical examples that arise from this construction. This is based on joint work with Atish Dabholkar and Don Zagier.

Class 1 - 15-06-2015 - download
Class 2 - 16-06-2015 - download
Class 3 - 18-06-2015 - download
Class 4- 19-06-2015 - download

Prof. Xenia de la Ossa (University of Oxford)

Class 1 - 15-06-2015 - download
Class 2 - 16-06-2015 - download
Class 3 - 18-06-2015 - download
Class 4- 19-06-2015 - download

Prof. Jon Keating, FRS - (University of Bristol)

L-functions and Random Matrix Theory I will give an overview of interplay between Random Matrices and the Theory of L-Functions. In particular, I will discuss the distribution of zeros and eigenvalues, moments, and extreme values. I will start with an introduction to the Riemann zeta-function and to the basic ideas underpinning Random Matrix Theory. The main aim then will be to explore the connections.

Class 1 - 16-06-2015 - download
Class 2 - 17-06-2015 - download
Class 3 - 18-06-2015 - download
Class 4 - 19-06-2015 - download
Class 5 - 23-06-2015 - download

Prof. Brian Conrey AIM and University of Bristol

Random Matrix Theory and statistics of families of L-functions An important subject in number theory today goes by the name of \arithmetic statistics." This course will introduce the notion of a "family" of L-functions and will develop the random matrix tools needed to express detailed theorems and conjectures about the distribution of zeros and values of the L-functions in a family. It will make use of a good mix of techniques from random matrix theory and from analytic number theory to develop the modern theory of family statistics, including the \recipe" and the \ratios conjecture" - which are theorems in random matrix theory and conjectures in number theory. Examples from all three symmetry types - unitary, orthogonal, and symplectic will be considered in detail.

Class 1 - 17-06-2015 - download
Class 2 - 17-06-2015 - download
Class 3 - 18-06-2015 - download
Class 4 - 19-06-2015 - download
Class 5 - 22-06-2015 - download

Prof. Ingmar Saberi Caltech, USA

The Physics and Number Theory of the A-polynomial The zero locus of the A-polynomial defines an algebraic curves that, via generalized volume conjecture, connects standard quantum group invariants with complex Chern-Simons theory. This connection predicts many new number theoretic properties of the A-polynomial and explains familiar ones (e.g. the quantizability condition and the K-theory constraint in the joint work with P.Sulkowski).

Class 1 - 17-06-2015 - download
Class 2 - 18-06-2015 - download
Class 3 - 19-06-2015 - download

Monday 22

Opening - Michael Berry - download

Kiran Kedlaya - download

Raimar Wulkenhaar - download

Alina Bucur download

Tuesday 23

German Sierra - download

Yuri Tschinkel - download

Jeffrey Lagarias - download

Gunther Cornelissen - download

John Duncan - download

Lior Bary-Soroker - download

Wednesday 24

Jordan Ellenberg - download

Fernando Rodriguez Villegas - download

Alex Kontorovich - download

Michael Rubinstein - download

Nina Snaith - download

Nicolas Templier - download

Thursday 25

Michael Rosen - download

Philip Candelas - download

Herman Verlinde - download

Solomon Friedberg - download

Chris Hughes - download

Mark Srednicki - download

Friday 26

Bruce Berndt - download

David Broadhurst - download

Jeffrey Hoffstein - download

Micah Milinovich - download

Christopher Sinclair - download

Emanuel Scheidegger - download