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Number theory is the study of whole numbers. It is one of the oldest academic disciplines.

Some of the questions in number theory are relatively easy to understand.

• Can every even integer greater than two be expressed as the sum of two primes?
• Which integers can be the area of a right-angled triangle with integer length sides?
• Are there infinitely many prime numbers p such that p+2 is also prime?

People are sometimes surprised that there is anything left to answer in mathematics. Yet, even seemingly simple questions, such as those above, have remained unanswered for centuries.

One of the most appealing features of number theory is that this apparent simplicity gives rise to lots of different and beautiful ideas. Number theory is not all equations; it is full of symmetry, structure, and unforeseen connections to other branches of mathematics. It also plays a crucial role in the real world: numbers are the language of computers, and how we store, transmit, and encrypt those numbers is of fundamental importance to modern life.

Research in number theory at the ºÚ¹Ï³ÔÁÏÍø is led by Dr Chris Daw. Dr Daw’s research focuses on geometric objects, known as Shimura varieties, that possess a rich arithmetic structure and are central to several areas of current research, perhaps most notably the Langlands Programme. Dr Daw’s research aims to understand fundamental arithmetic and geometric properties of Shimura varieties, as part of an international area of research known as Unlikely Intersections.

The other permanent members of the number theory group are Professor Michael Levitin and Dr Sugata Mondal, who specialise foremost in analysis.

Professor Levitin writes,

“Many questions in spectral geometry are closely linked to number theory. In a simple example, the eigenvalues of the Laplacian on a flat square torus of period 2π can be identified with the squared distance to the origin of integer lattice points lying in the first quadrant. Other more involved cases also exhibit the links between the eigenvalue counting functions and lattice counts in particular regions. Other problems, such as the study of eigenvalues and resonances of hyperbolic surfaces, have further close links with analytic number theory.”

Dr Mondal explains,

“One of my research interests is the spectral geometry of hyperbolic surfaces. More precisely, I am interested in small or exceptional eigenvalues of the Laplacian on hyperbolic surfaces, and their connection with explicit geometry of the hyperbolic surface. Arithmetic hyperbolic surfaces form a special class of hyperbolic surfaces, and they have very interesting spectral geometric properties. For example, by a result of Selberg the first non-zero eigenvalue of a congruent arithmetic hyperbolic surface is at least 3/16. Selberg conjectured that this bound is 1/4, which is unsolved to-date. In a way, this conjecture served as the motivation to study small or exceptional eigenvalues (eigenvalues below 1/4) of hyperbolic surfaces.”

• Dr Chris Daw
• Professor Michael Levitin
• Dr Sugata Mondal