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Research Fellows Directory

Trevor Wooley

Professor Trevor Wooley

Research Fellow

Organisation

University of Bristol

Research summary

Equations governing phenomena of interest provide the universal language of

much of modern science. The basic equations of number theory are polynomial

equations to be solved in integers (whole numbers). Studied since Antiquity, these

Diophantine equations influence the development of codes and cryptosystems

applied, for example, in data storage systems such as DVDs, in mobile phones,

and in banking security. Two problems occupy the number theorist, the first being

to identify quasi-random families of objects of use in applications, and the second

to provide assurance that hidden patterns underlying these families do not unravel

their usefulness.

Investigation of Diophantine equations therefore has significance in real and

potential applications. One might guess that integer solutions are scattered

randomly throughout the real set of solutions. A polynomial might be biased to

take odd values more often than not, however, and this would decrease the

probability of its vanishing. A precise modification of this heuristic accounting for

divisibility by 2, 3, and so on, is embodied in the local-global principle. Although

the circle method, which makes use of Fourier analysis to count solutions, often establishes such a local-global principle in a precise manner, exotic equations

exist that fail to satisfy the local-global principle.

My research centres on a modified version of the circle method that incorporates

ingredients from recent work in additive combinatorics and arithmetic geometry.

Large classes of the problematic equations failing the local-global principle can be

successfully studied using this method, and it leads in turn to new conclusions in

the latter subdisciplines. What is compelling about this line of enquiry is that the

circle method, refined already by a century of technical development, is emerging

now as a flexible, dynamic method that provides a fundamental technique in the

emerging field of arithmetic harmonic analysis.

Grants awarded

Diophantine geometry via modern variants of the circle method

Scheme: Wolfson Research Merit Awards

Dates: Jul 2007 - Jun 2012

Value: £75,000

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