# Number fields and function fields: coalescences, contrasts and emerging applications

- 9:00 am on Thursday 29 May 2014 — 5:00 pm on Friday 30 May 2014
- at The Royal Society at Chicheley Hall, home of the Kavli Royal Society International Centre, Buckinghamshire

Theo Murphy international scientific meeting organised by Professor Jon Keating FRS, Professor Zeev Rudnick and Professor Trevor Wooley FRS

### Event details

Connections between problems in number fields and in function fields are of central importance in Number Theory. The past decade has seen fundamental new developments emerge in this direction, attracting some of the leading mathematicians. Our goal is to explore the interplay between these ideas, and their implications for other areas of mathematical science, including mathematical physics and computational complexity.

Download the meeting programme

Biographies of the organisers and speakers will be made available shortly, as well as the two-day programme. Recorded audio of the presentations will be available on this page after the event.

### Attending this event

This is a residential conference, which allows for increased discussion and networking. It is free to attend, however participants need to cover their accommodation and catering costs if required.

Enquiries: Contact the events team

## Organisers

**
Jon Keating FRS, University of Bristol, UK
**

### Biography

Jon Keating is based at the University of Bristol. He gained his first degree from Oxford University and his PhD from the University of Bristol, under the supervision of Professor Sir Michael Berry FRS. His research is centred in the areas of quantum chaos, random matrix theory and number theory. He has held a BRIMS Research Fellowship, funded by Hewlett-Packard, and an EPSRC Senior Research Fellowship.

**
Zeev Rudnick, Tel Aviv University, Israel
**

### Biography

Zeev Rudnick has been professor of Mathematics at Tel Aviv University since 1995, specializing in Number Theory and Quantum Chaos.

Prior to his appointment at Tel Aviv he was an Assistant Professor at Princeton and in Stanford.

**
Professor Trevor Wooley FRS, University of Bristol, UK
**

### Biography

Trevor D.Wooley gained his PhD from Imperial College in 1990, He was faculty member of the University of Michigan, Ann Arbor, 1991 - 2007 (serving as Chair 2002-2005), and has been based at the University of Bristol from 2007 - present. His research is centred on the Hardy-Littlewood (circle) method, a method based on the use of Fourier series that delivers asymptotic formulae for counting functions associated with arithmetic problems. In the 21st Century, this method has become immersed in a turbulent mix of ideas on the interface of Diophantine equations and inequalities, arithmetic geometry, harmonic analysis and ergodic theory, and arithmetic combinatorics. Wooley has been awarded the Salem Prize (1998), Frohlich Prize (2012), elected FRS (2007) and FAMS (2012). He was a 45-minute speaker at ICM in Beijing (2002) and Seoul (2014).

## Session 1

**
Professor Roger Heath-Brown FRS, University of Oxford, UK
**
Chair of Session 1

### Biography

Roger Heath-Brown was an undergraduate at Cambridge, where he did his PhD under Alan Baker. He became a Research Fellow at Trinity College Cambridge in 1977, but moved to Magdalen College Oxford in 1979. Since 1999 he has been Statutory Professor of Pure Mathematics. He has won the LMS Junior Berwick Prize, Senior Berwick Prize, and Polya prize. He was an invited ICM speaker both in 1983 (Warsaw) and 2010 (Hyderabad). In 1993 he was elected a Fellow of the Royal Society.

His work covers a substantial range within analytic number theory, focussing particularly on prime number theory and analyitc methods for Diophantine equations. He has written over 160 research articles, and has produced major revisions of Titchmarsh's "Theory of the Riemann Zeta-function" and Hardy & Wright's "Introduction to the Theory of Numbers".

**
Professor Trevor Wooley FRS, University of Bristol, UK
**
Robust estimates for exponential sums in function fields

### Biography

Trevor D.Wooley gained his PhD from Imperial College in 1990, He was faculty member of the University of Michigan, Ann Arbor, 1991 - 2007 (serving as Chair 2002-2005), and has been based at the University of Bristol from 2007 - present. His research is centred on the Hardy-Littlewood (circle) method, a method based on the use of Fourier series that delivers asymptotic formulae for counting functions associated with arithmetic problems. In the 21st Century, this method has become immersed in a turbulent mix of ideas on the interface of Diophantine equations and inequalities, arithmetic geometry, harmonic analysis and ergodic theory, and arithmetic combinatorics. Wooley has been awarded the Salem Prize (1998), Frohlich Prize (2012), elected FRS (2007) and FAMS (2012). He was a 45-minute speaker at ICM in Beijing (2002) and Seoul (2014).

**
Professor Kannan Soundararajan, Stanford University, USA
**
Moments of L-functions and a one-sided central limit theorem

### Biography

Kannan Soundararajan received his undergraduate degree from the University of Michigan in 1995, and his PhD from Princeton University in 1998. After postdoctoral positions at Princeton and the Institute for Advanced Studies, he was a professor at the University of Michigan until 2006. Since then he has been a professor at Stanford University.

**
Zeev Rudnick, Tel Aviv University, Israel
**
Some problems in analytic number theory for polynomials over a finite field

### Biography

Zeev Rudnick has been professor of Mathematics at Tel Aviv University since 1995, specializing in Number Theory and Quantum Chaos.

Prior to his appointment at Tel Aviv he was an Assistant Professor at Princeton and in Stanford.

**
Professor Ben Green, University of Oxford, UK
**
The inverse large sieve problem

### Biography

Ben Green is the Waynflete Professor of Mathematics at the University of Oxford. He grew up in Bristol and has previously held permanent positions at Bristol and Cambridge, as well as short-term positions at UBC Vancouver, MIT and Harvard.

## Session 2

**
Alina Bucur, University of California - San Diego, USA
**
Chair of Session 2

### Biography

Alina Bucur is currently at University of California at San Diego. She received her PhD from Brown University in 2006, and before joining the Department of Mathematics at UCSD, she held postdoctoral fellowships at the Institute for Advanced Study and a Moore Instructorship at MIT. Her research interests lie in the areas of analytic number theory and arithmetic statistics.

**
Professor Lior Bary-soroker, Tel Aviv University, Israel
**
Using Galois theory in analytic number theory

### Biography

I finished my PhD in 2009 under the supervision of Professor Dan Haran at Tel Aviv University. I spend about 3 years as a postdoc, half of the time in the Hebrew University of Jerusalem and the other half at Essen University in Germany. In 2011 I started my current position as a senior lecturer back in Tel Aviv University.

**
Professor Chantal David, Université de Montréal, Canada
**
Statistics for cyclic trigonal curves over finite fields

### Biography

Chantal David got her PhD at McGill University in 1993, and she is currently a professor at Concordia University in Montreal. She was also a member of the Institute for Advanced Studies in the academic year 2009-2010. She is interested in analytic number theory, and distribution questions related to arithmetic objects, as elliptic curves, or curves over finite fields. She was awarded the Krieger-Nelson Prize of the Canadian Mathematical Society in 2013.

**
Professor Emmanuel Kowalski, ETH Zurich, Switzerland
**
Sums of products of trace functions over finite fields

### Biography

Emmanuel Kowalski studied mathematics in France at the E.N.S Lyon and Institut Fourier (Grenoble) then in the U.S.A at Rutgers University where he obtained his PhD in 1998. After a post-doc position at Princeton University and the Institute for Advanced Study, he was a professor in Bordeaux, France from 2000 to 2007, and he is currently professor of mathematics at the Swiss Federal Institute of Technology in Zurich. His research is devoted to number theory, in particular to analytic number theory, taken in a very broad sense, with a focus on sieve methods and their applications, automorphic forms and sums over finite fields.

**
Kiran S Kedlaya, University of California, San Diego
**
How many points on a random curve over a finite field?

### Biography

Kiran S Kedlaya is a Professor of Mathematics at the University of California, San Diego; he was previously an Associate Professor at MIT, where he also received his PhD in 2000. He also held postdoctoral positions at UC Berkeley, MSRI, and IAS. His research interests are in number theory and algebraic geometry, including p-adic analysis and geometry, p-adic differential equations, zeta functions of algebraic varieties, and computational and algorithmic questions. He received the Presidential Early Career Award for Scientists and Engineers in 2006 and was a 45-minute speaker at the ICM in Hyderabad (2010).

## Session 3

**
Professor Bryan Birch FRS, Oxford University Mathematical Institute, UK
**
Chair of Session 3

### Biography

As a doctoral student at the University of Cambridge, Birch was officially working under J W S Cassels. More influenced by Harold Davenport, he proved Birch's theorem, one of the definitive results to come out of the Hardy–Littlewood circle method; it shows that odd-degree rational forms in a *large enough* set of variables must have zeroes.

In later work he contributed to algebraic K-theory (Birch–Tate conjecture). He then formulated ideas on the role of Heegner points (he had been one of those reconsidering Kurt Heegner's original work, on the class number one problem, which had not initially regained acceptance). Birch put together the context in which the Gross–Zagier theorem was proved; the correspondence is now published.

Birch was a visiting scholar at the Institute for Advanced Study in the fall of 1983. He was elected a Fellow of the Royal Society in 1972; was awarded the Senior Whitehead Prize in 1993 and the De Morgan Medal in 2007 both of the London Mathematical Society. In 2012 he became a fellow of the American Mathematical Society.

**
Professor Brian Conrey, AIM and University of Bristol, UK
**
Questions about moments

### Biography

Brian Conrey is the Executive Director of the American Institute of Mathematics and is a Professor at Bristol University. He works in Number Theory and is especially interested in L-functions and their values, especially zeros, and in models for L-functions inspired by Random Matrix Theory. He enjoys working with middle school teachers in the Math Teachers Circle program.

**
Professor Nick Katz, Princeton University, USA
**
Equidistribution questions raised by Entin, Keating, and Rudnick

### Biography

Education

BA 1964, Johns Hopkins University

MA 1965, Princeton University

PhD 1966, Princeton University

Employment

1966-67, Instructor, Princeton University Mathematics Department

1967-68, Lecturer, Princeton University Mathematics Department

1968-71, Assistant Professor, Princeton University Mathematics Department

1971-74, Associate Professor, Princeton University Mathematics Department

1974-present, Professor, Princeton University Mathematics Department

Visiting Positions

University of Minnesota

I.H.E.S., Orsay, Paris VI

University of Tokyo, Nagoya University

I.A.S.

Honors

Editor, Annals of Mathematics, 2004-present

Chair, Princeton University Mathematics Department, 2002-2005

National Academy of Sciences, elected 2004

American Academy of Arts and Sciences, elected 2003

Levi L. Conant AMS prize (joint with P. Sarnak), 2003

Visiting Miller Professor, Spring 1993

Guggenheim Fellowship, 1987-88 and 1975-76

JSPS Fellowship, 1983

Sloan Fellowship, 1971-72

Nato Postdoctoral Fellowship, 1968-69

**
Professor Nina Snaith, University of Bristol, UK
**
Elliptic curves and random matrices

### Biography

Nina Snaith is a member of the mathematical physics group at the University of Bristol. She works in random matrix theory, with applications to number theory.

**
Dr Andrew Granville, Université de Montréal, Canada
**
The anatomy and pretentions of function fields

### Biography

Andrew Granville is the Canadian Research Chair in number theory at the Université de Montréal. He is interested in various areas of mathematics, and of number theory, as well as popularizations of mathematics. He specializes in analytic number theory. He is also a co-organizer of the 2014-2015 thematic year in number theory at the CRM in Montreal, the first semester of which is focused on analytic areas, including analytic aspects of the work of Bhargava. He has been a visitor at Cambridge for parts of the 2013-2014 academic year.

## Session 4

**
Paul Pollack, University of Georgia, USA
**
Chair of Session 4

### Biography

Paul Pollack received his Bachelor's degree from the University of Georgia (2003) and his Master's degree (2007) and PhD (2008) from Dartmouth College. His thesis work, conducted under the supervision of Carl Pomerance, concerns the distribution of irreducible polynomials over finite fields, with particular attention to problems motivated by analogies with rational prime number theory. After earning his doctorate, Pollack went on to postdoctoral positions with Kevin Ford at the University of Illinois and with Greg Martin at the University of British Columbia. In 2012, he returned to the mathematics department at the University of Georgia to assume a tenure-track position. His first book, Not always buried deep: A second course in elementary number theory, was published by the American Mathematical Society in 2009.

**
Professor Michael Rubinstein, University of Waterloo, Canada
**
Conjectures, theorems, and experiments concerning the moments of zeta functions associated to quadratic function fields

### Biography

Michael Rubinstein received his PhD in 1998 under the supervision of Peter Sarnak. He presently works at the University of Waterloo.

**
Professor Philip Candelas, University of Oxford, UK
**
A physicists take on the conjecture of Birch and Swinnerton-Dyer

### Biography

Candelas read Maths at Cambridge and then moved to Oxford to do a DPhil, on quantum fuel theory in curved space-time and the Hawking radiation of black holes, under the astrophysicist Dennis Sciama. Candelas went from Oxford to the Physics Department of the University of Texas at Austin, initially for a year, but stayed on as an Assistant, then Associate and Full Professor. During this time Candelas turned to a study of Kaluza-Klein theories and then string theory, an area in which he has worked since. In 1999 Candelas moved back to Oxford to take up the Rouse-Ball Chair of Mathematics. Candelas has published 85 articles in theoretical and mathematical physics. The majority in string theory and the physics and mathematics of Calabi-Yau manifolds.

### Abstract

We will show that the BDS conjecture can be recast as the problem of estimating a "path integral” of a form that is at least superficially familiar to physicists.

**
Dr Xenia de la Ossa, University of Oxford, UK
**
Arithmetic of Calabi-Yau manifolds

### Biography

Xenia de la Ossa is a Reader at the Mathematical Institute, University of Oxford. Her research interests are in geometry and string theory, specifically in the mathematical structures arising in string theory for instance the quantum moduli spaces of bundles over riemannian manifolds. Also, she is interested in the interactions between number theory and physics. In particular, she research is related to the arithmetic of Calabi-Yau manifolds, and also the interpretation of the conjecture of Birch and Schinnerton-Dyer in terms of path integrals.

### Abstract

This talk will discuss the arithmetic of Calabi-Yau 3-folds considered as a manifold over finite fields. De la Ossa will be concerned with the computation of these numbers and their dependence on the complex structure parameters. We will see that these numbers are given by expressions which involve the periods of the manifold. The talk will discuss the form of zeta function and the L-function of the Dwork pencil of quintic 3-folds with attention as to what happens at singularities.