I am currently a third-year Maths PhD student at the University of Bristol under the supervision of Prof. Trevor Wooley.
My research interests lie in the area of analytic number theory, especially applications of the circle method. This is a Fourier-analytic method for estimating the number of solutions of diophantine equations which has been first developed by Hardy and Littlewood in the 1920s. But even though the basic method is quite classical, its versatility makes it a strong tool that continues to yield new and exciting results. Furthermore, since the problem of solving polynomial equations occupies a very central position in not only number theory, but also many of its neighbouring fields such as geometry, combinatorics or ergodic theory, results obtained by the circle method or modifications thereof find applications in a wide range of problems.
A field in which the results obtained by the circle method are generally not as strong as one would like is the application to the very general setting of forms in many variables. Since Davenport and Birch first succeeded in applying the method to problems of this type around 1060, relatively little progress has made. This is partially due to the extreme generality of the question that makes it hard to find a method capable of capturing the various shapes and shades in which such general forms can occur. In particular the behaviour of highly singular forms as well as the conditions under which one can guarantee local solubility prove elusive. One of my main interests consists therefore in finding more efficient methods to deal with difficulties of this type.
The flexibility of the circle method allows for applications far beyond finding simple points on hypersurfaces, and indeed it can be adapted to count lines or higher-dimensional linear spaces as well. On the one hand, these objects have a lot of structure which can be exploited to obtain stronger results than one would naively expect, but understanding them and capturing the greater generality of linear spaces compared to points is also challenging in a way that has not been experienced in earlier applications. In my research, I try to obtain a better understanding of the behaviour of linear spaces contained in hypersurfaces.
Over the last couple of years I have been involved in undergraduate teaching, covering a range of courses.
My contact details are:
| Address: | University of Bristol |
| Howard House | |
| Queen's Avenue | |
| Bristol, BS8 1SN | |
| Email: | mazjb@bristol.ac.uk |