Number Theory & Geometry
This group is interested in various aspects of number theory and geometry and their interconnections. More specifically, we have members interested in analytic number theory and algebraic number theory, especially over function fields; and members interested in differential and complex geometry, algebraic geometry and arithmetic and Diophantine geometry.
There is a weekly number theory seminar in which topics in some of these areas are studied. Recent topics include:
- Using modular symbols to compute spaces of modular forms (complex geometry and number theory)
- The circle method (analytic number theory)
- Elliptic curves (number theory and algebraic geometry)
- Tate’s thesis (harmonic analysis, algebraic number theory)
Here is a more detailed description of our staff members and their interest:
Dr Gareth Boxall works in the area of o-minimality and Diophantine geometry. Central to this is the Pila-Zannier strategy for combining counting results for rational points on transcendental sets with lower bounds on Galois orbits of points of interest on algebraic varieties.
Dr Sophie Marques is interested in extending ramification theory to Algebraic geometry. Moreover, she wants to understand low degree extensions of the projective line (and more generally extensions of function fields) but also low degree extension of the rational field (and more generally extensions of number fields). For instance, she would like to classify such extensions, describe precisely their ramification, obtain explicit genus formulae, obtain explicit integral basis, describe moduli spaces…
Dr Arnold Keet is working on towers of rank two Drinfeld modular curves.
Dr Bruce Bartlett is interested in how number theory relates to topology. He is a very active member of the number theory seminar.
Dr Naina Ralaivaosaona works in analytic number theory. More precisely, he is interested in the asymptotic enumeration of integer partitions and its applications. For instance, he has studied the distribution of the number of summands in prime partitions of large positive integers. The techniques used in this research area include the circle method, the saddle-point method and the Mellin transform method.
Dr Dirk Basson works in the field of function field arithmetic, and specifically extending the theory Drinfeld modular forms to higher rank. This also involves studying the moduli space of rank r Drinfeld modules both as an algebraic variety and as a rigid analytic variety. Recently he has also started collaborating in the area of computer algebra.