For the academic year 2026-27, the seminar is organised by Víťa Kala and myself, and runs on Wednesdays 14:00 in room K2.
The seminar is intended for students and faculty members, so the level of the talks should be accessible to these people.
We have a mailing list for announcing upcoming talks. If you wish to subscribe to the mailing list, please contact any of the organisers.
Bhargava and Shankar proved that the average of the ranks of elliptic curves is bounded, and they accomplished that by bounding the average size of their 2-Selmer groups using a parametrisation by binary quartic forms. Since then, similar parametrisations have yielded similar results for other families of curves. Recently, Koymans and Smith obtained some other related results using very different methods. We will give a friendly overview of these results and the main ideas that go into them, and towards the end we will use these techniques to obtain new statistical results for the sizes of isogeny Selmer groups associated to certain hyperelliptic curves.
More than a decade ago, Blomer and Kala showed that, for every positive integer n, there exist real quadratic fields admitting no universal quadratic form in n variables. It has since been shown that the same phenomenon occurs for totally real fields of every degree divisible by 2 or 3. Yet, even for quintic fields, this remained unknown. In this talk, we discuss recent progress on this problem.