MATHEMATICS AND COMPUTER SCIENCE

Browning Group

Analytic Number Theory and Its Interfaces

What is the precise connection between adding and multiplying whole numbers? This is a surprisingly deep question that can be thought about in many different lights. One natural extension studies the sequence of integers that arise as solutions to a polynomial equation with integer coefficients, viz. a Diophantine equation. The Browning group works on understanding such sequences using a blend of analytic, geometric and algebraic methods.

Low-dimensional Diophantine equations have been heavily exploited in cryptography, but the properties of higher-dimensional Diophantine equations remain largely mysterious. Hilbert’s 10th problem asks for an algorithm to decide if a given Diophantine equation has integer solutions or not. Methods of mathematical logic have revealed this to be an impossible dream, but we would still like to know if such a procedure exists when we merely ask for solutions in rational numbers. Moreover, when solutions are known to exist, there are deep conjectures that connect their spacing to the intrinsic geometry of the equation. In recent years quantitative methods have been found to be remarkably effective at resolving these fundamental questions. The Browning group is involved in actively expanding the available toolkit for studying these problems and their generalizations.

Group Leader


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Team


Current Projects

Moduli space of rational curves on hypersurfaces of low degree |Hasse principle for random Fano hypersurfaces | Manin’s conjecture for orbifolds | Distribution of number fields with given Galois group via Manin’s conjecture


Publications

Browning TD, Hu LQ. Counting rational points on biquadratic hypersurfaces. Advances in Mathematics. View

Browning TD, Matthiesen L. 2017. Norm forms for arbitrary number fields as products of linear polynomials. Annales Scientifiques de l’Ecole Normale Superieure. 50(6), 1383–1446. View

Browning TD. 2017. Many cubic surfaces contain rational points. Mathematika. 63(3), 818–839. View

Browning TD, Sawin W. 2017. A geometric version of the circle method. Unknown., 1–47. View

Browning TD, Schindler D. 2017. Strong approximation and a conjecture of Harpaz and Wittenberg. International Mathematics Research Notices. View

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Career

since 2018 Professor, IST Austria
2012 – 2019 Professor, University of Bristol, UK
2008 – 2012 Reader, University of Bristol, UK
2005 – 2008 Lecturer, University of Bristol, UK
2002 – 2005 Postdoctoral Research Fellow, University of Oxford, UK
2001 – 2002 Postdoctoral Research Fellow, Université de Paris-Sud, Orsay, France
2002 PhD, Magdalen College, University of Oxford, UK


Selected 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


Additional Information

Download CV
Open Browning website
Browning Group Working Seminar
Analytic Number Theory Seminar
Algebraic Geometry & Number Theory Seminar



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