Mathematics Department

Number Theory Research Group

Gauss famously described Mathematics as the "Queen of the Sciences" and Number Theory as the "Queen of Mathematics". Humanity has probably been fascinated by numbers since the discovery of counting. The ancient Greeks established it as a subject (which they called arithmetic): Books 7--10 of Euclid's Elements are devoted to number theory. These books contain a treatment of the Euclidean algorithm, a proof that there are infinitely many primes, a theory of incommensurables (or irrationals -- the original proof that the square root of 2 is irrational is attributed to Pythagoras). The sieve of Eratosthenes is the basis of the modern sieve and Diophantus of Alexandria began the systematic study of equations with integer coefficients. The subject has flourished since the Renaissance and today is as vigorous as ever: the proof of Fermat's Last Theorem sought for over 300 years was solved a few years ago. Until the middle of the 20th century, Number Theory was considered to be the purest area of Mathematics and least likely to be sullied by applications. This has all changed over the last 50 years with the advent of digital computing, the growth of cryptography and the emergence of links with dynamical systems and physical phenomena. Now the subject is an exciting mix of ideas pursued for their own interest and a rather exotic variety of applications.

People in the Number Theory Group at York have varied interests, which include analytic number theory, Diophantine approximation, geometry of numbers and connections with dynamical systems. As well as participating in the departmental GANT seminars , members of the group enjoy contacts with mathematicians from all over the world. Within the department, we overlap with the algebra group, the analysis group and the networks and nonlinear dynamics group.

We have close links with Edinburgh, Liverpool, Southampton, Queen Mary and Westfield College, London and Warwick; and abroad with Brisbane, Maynooth, Ostrava, Strasbourg and through an INTAS programme, with Minsk, Marburg and St. Petersburg.

Current research areas:

 

  • Analytic number theory Concerned with the distribution of prime numbers, the Riemann zeta function and Diophantine equations, it owes its name from the techniques it uses being drawn from analysis. Currently Richard Hall is studying the distribution of the zeros of the Riemann zeta function.

    •  

  • Diophantine approximation A quantitative and more general study of the qualitative fact that the rationals are dense in the reals. We are interested in the metrical theory, in which exceptional sets of points of measure zero for which Diophantine inequalities do not hold are ignored -- this has the advantage of often giving results of a striking simplicity. Analytic concepts play an equal role with ideas from measure and ergodic theory. Detta Dickinson (Maynooth), Maurice Dodson and Simon Kristensen (Edinburgh) are investigating applications to the theory of dynamical systems through the phenomenom of `small divisors'. Deeper asymptotic formulae results are being investigated with Jason Levesley and with Victor Beresnevich (an EPSRC RA) and Sanju Velani (a RSUF), who have both recently joined the department. They are also looking at how well points in various sets of number theoretic and geometric interest can be approximated. A major goal is to establish a `non-linear' metrical theory, in which the variables lie on manifolds; Bernik (Minsk) and his co-workers are also involved in this programme. General measure theory aspects of the theory are being considered with Yann Bugeaud (Strasbourg).

    •  

  • Geometry of Numbers Its origins lie in Minkowski's observation at the end of the 19th century that some easily accessible geometric results having interesting consequences in number theory. A typical problem is to find the circumstances under which a given n-dimensional body contains a non-zero lattice point. For some kinds of bodies this problem corresponds to questions about the values of quadratic forms. Terence Jackson is studying representation questions for definite forms and lattice constants of bodies associated with indefinite forms.

    •  

  • Ergodic theory and dynamical systems Number theoretic ideas have applications in ergodic theory and dynamical systems and vice-versa. Recently Zaq Coehlo and Bill Parry (Warwick) have found connections between p-adic multiplication and Fibonacci numbers.

    • Return to Mathematics home page
     
     

    Revised 14 April 2004