About the Project
The School of Mathematical Sciences of Queen Mary University of London invite applications for a PhD project commencing either in January 2017 or September 2017 for students seeking funding, or at any point in the academic year for self-funded students.
This project will be supervised by Dr Ivan Tomasic.
This project is aimed at a student with interests not only in algebraic geometry and number theory, but also in logic and model theory.
The motivating problem of Diophantine geometry is determining the integral solutions of systems of multi-variate polynomial equations.
This global problem is too hard to attack directly, so we typically aim to replace it with a local problem, for example by reducing the equation modulo a prime p. This can be helpful, because if an integer polynomial equation does not have a solution modulo p, it cannot have an integer solution.
In other words, studying the number N_n of solutions of a polynomial congruence modulo the n-th power of p for varying n, can reveal a great deal of information about the global problem. It is convenient to pack the information about the numbers N_n, into a generating function called the Poincare series. This series has been studied in much detail by Igusa and Denef, who proved that it is in fact a rational function. As a consequence, there is a strong regularity (a linear recurrence relation) in the sequence N_n, and the situation is completely clarified.
A beautiful idea used in their proof was to replace the problem of counting solutions of congruences by a calculation of certain p-adic integrals, where one can use standard integration techniques such as Fubini’s Theorem and Change of Variables (recall that the field of p-adic numbers is the completion of the rationals with respect to the p-adic norm).
Logic enters the picture through the techniques of quantifier elimination for first-order formulae over the p-adics, which allow us to simplify the expressions we are integrating.
The goal of this project is to extend these considerations to the study of difference polynomial equations (a difference polynomial in variable x is a polynomial expression involving terms x, s(x), s(s(x)) etc, where s is the so-called difference operator). The solutions will be sought in the maximal unramified extension of the p-adic integers, where the difference operator is a lift of Frobenius.
Full details can be found in the project abstract: http://www.maths.qmul.ac.uk/sites/default/files/PhD%20Projects%202016/Geometry%20and%20Analysis/Tomasic%20Project%202017.pdf
The application procedure is described on the School website. For further enquiries please contact Dr Ivan Tomasic ([Email Address Removed]). This project is eligible for full funding, including support for 3.5 years’ study, additional funds for conference and research visits and funding for relevant IT needs. Applicants interested in the full funding will have to participate in a highly competitive selection process. The best candidates will be eligible to receive a prestigious Ian Macdonald Postgraduate Award of £1000, for which you will be considered alongside your application. The application deadline for full funding is November 30th 2016 to start in January 2017, and January 31st to start in September 2017.
Funding Notes
There is also 50% funding scheme available for students who are able to find the matching 50 % of the cost of their studies. Competition for these half-funded slots will be less intensive, and eligible students should mention their willingness to be considered for them in their application. The application deadline for 50 % funding is January 31st 2017.
This project can be also undertaken as a self-funded project, either through your own funds or through a body external to Queen Mary University of London. Self-funded applications are accepted year-round.
Continue with Facebook
