# Browning Group

## Analytic Number Theory and its Interfaces

Number theory is concerned with studying properties of integers (or their generalizations), both in their own right or as solutions to Diophantine equations. It is often advantageous to encode properties of number-theoretic objects through analytical constructs, such as the Riemann zeta function for prime numbers. Tim Browning's research takes this point of view and draws on techniques from analysis to study problems across a spectrum of number theory and algebraic geometry.

The study of Diophantine equations is a subject that is both ancient and challenging, having captured our attention since the time of the ancient Greeks nearly 2000 years ago. It has profound interactions with a host of subject areas, ranging from geometry to complex analysis via mathematical logic and everything in between. Two fundamental questions that drive today's research are whether or not it is possible to decide the existence of rational or integer solutions, and whether or not the space of solutions admits any structure.

While tools from algebraic geometry have long been known to be effective in this area, recent discoveries on the arithmetic side have led to a reversal of flow, whereby advances in number theory have opened up new lines of enquiry in algebraic geometry. In the same spirit, a rich seam of interactions continue to be exploited between number theory and additive combinatorics.

Modern analytic number theory has its roots in classical problems in number theory, but it thrives on its interfaces with a broad swathe of topics in mathematics.

**Contact**

Tim Browning

Institute of Science and Technology Austria (IST Austria)

Am Campus 1

A – 3400 Klosterneuburg

Phone: +43 (0)2243 9000-2103

E-mail: timothy.browning@
ist.ac.at

»Browning Group Working Seminar

»Algebraic Geometry & Number Theory Seminar

**Assistant**Lena Marr

Phone: +43 (0)2243 9000-1073

E-mail: lena.marr@ ist.ac.at

**Team**

- Kevin Destagnol, Postdoc
- Maria-Adelina Manzateanu, Predoctoral Visiting Scientist

**Selected Publications**

- Browning, T., Vishe P.: Rational curves on smooth hypersurfaces of low degree, Algebra & Number Theory 11 (2017), 1657-1675. (arXiv:1611.00553)
- Browning T., Heath-Brown, D.R.: Forms in many variables and differing degrees, J. Eur. Math. Soc. 9 (2017), 357-394. (arXiv:1403.5937)
- Browning, T., Matthiesen, L.: Norm forms for arbitrary number fields as products of linear polynomials, Ann. Sci. École Norm. Sup. 50 (2017), 1375-1438. (arXiv:1307.7641)
- Browning, T., Matthiesen, L., Skorobogatov, A.: Rational points on pencils of conics and quadrics with many degenerate fibres, Annals of Math. 180 (2014), 381-402. (arXiv:1209.0207)
- Browning, T., Vishe, P.: Cubic hypersurfaces and a version of the circle method for number fields, Duke Math. J. 163 (2014), 1825-1883. (arXiv:1207.2385)
- de la Bretèche, R., Browning, T., Peyre, E.: On Manin's conjecture for a family of Châtelet surfaces, Annals of Math. 175 (2012), 297-343. (arXiv:1002.0255)

**Career**

2018- Professor, IST Austria

2012-2019 Professor, University of Bristol

2008-2012 Reader, University of Bristol

2005-2008 Lecturer, University of Bristol

2002-2005 Research Fellow, University of Oxford

2001-2002 Université de Paris-Sud, Orsay

2002 DPhil, Magdalen College, University of Oxford

S**elected Distinctions**

2017 Simons Visiting Professorship (MSRI)

2017 EPSRC Standard Grant

2012 ERC Starting Grant

2010 Phillip Leverhulme Prize

2009 Ferran Sunyer i Balaguer Prize

2008 Whitehead Prize

2007 EPSRC Advanced Research Fellowship