Scheme: University Research Fellowship
Organisation: University of Warwick
Dates: Oct 2013-Sep 2018
Summary: Number theory is concerned with questions about the most fundamental mathematical concepts, such as prime numbers and solutions of polynomial equations. It is connected to virtually every other area of mathematics, and has been responsible for the development of some of the most sophisticated mathematical theories. It has also found applications outside themathematical realm, most notably becoming the backbone of modern information security: whether one is using a credit card or the internet, the secrecy of the information is now provided by number theory rather than by physical safeguards or code books.
A central focus of today's number theory research is so-called L-functions. These are abstract, almost mysterious, mathematical tools that have unexplained predictive powers. Understanding their special properties has led to some of the most powerful mathematical results throughout the past two centuries, from the Prime Number Theorem (1896), to Fermat's Last Theorem (1994). However, our knowledge of L-functions remains fairly poor, and the lack of a concise and conceptual theory has prevented them from being added to the mathematical toolkits of other scientists. Their development is a major topic in mathematics, to the extent that among the seven Clay Millennium problems (these represent challenges to 21st century mathematicians, with a $1000000 prize attached to each) two directly address L-functions: the Riemann Hypothesis and the Birch-Swinnerton-Dyer conjecture.
My research on L-functions is closely related to the Birch-Swinnerton-Dyer conjecture. One goal is to develop the L-function machinery for higher degree polynomial equations, beyond the standard classes of conics and elliptic curves. Another, more experimental strand, is to interpret the special values of L-functions, when these happen to be irrational numbers. Here the hope is to uncover new underlying structures and reveal new arithmetic phenomena.