Timetable
| Time | Talk | Speaker | Location |
|---|---|---|---|
| 08.30 – 09.15 | Doors open | HG00.304 | |
| 09.15 – 09.30 | Welcome speech | Ben Moonen | HG00.304 |
| 09.30 – 10.15 | Heights and torsion points on elliptic curves | Robin de Jong | HG00.304 |
| 10.30 – 11.15 | Why do some polynomial equations have only finitely many solutions? | Ariyan Javanpeykar | HG00.307 |
| 11.30 – 12.15 | A landscape of curves | Emma Brakkee | HG00.307 |
| 12.30 – 13.45 | Lunch | HG00.00N | |
| 14.15 – 15.00 | Counting curves in the plane | David Holmes | CC2 |
| 15.45 – 16.30 | Surprising counts over finite fields | Victoria Hoskins | LIN1 |
| 16.45 – 17.00 | Closing speech | Ben Moonen | LIN1 |
| 17.00 – 18.00 | Drinks | HG00.00N | |
Robin de Jong: Heights and torsion points on elliptic curves
An elliptic curve is a plane curve given by a degree-three equation. Interestingly, the points on an elliptic curve admit a natural structure of abelian group. We are interested in two questions: (i) how to see if a point on an elliptic curve has a finite order? (ii) do there exist points of infinite order?
Ariyan Javanpeykar: Why do some polynomial equations have only finitely many solutions?
It was known more than a thousand years ago that the equation x^2 + y^2 = z^2 has infinitely many “really” different solutions in the integers. Fermat showed that this changes when we consider x^3+y^3=z^3 by showing that the finitely many “obvious” solutions are the only ones. It is a famous, maybe even infamous, theorem of Andrew Wiles that, for all n>2, the only integer solutions to the equation x^n+y^n = z^n are the “obvious” ones (commonly referred to as Fermat’s Last Theorem). It is natural to seek a geometric explanation for this change in behaviour. This will lead us to a striking theorem of Gerd Faltings from 1983 which gives a purely geometric explanation of why certain polynomial equations have only finitely many integer solutions, and to several analogous results in complex analysis.
Emma Brakkee: A landscape of curves
Elliptic curves, curves in the plane given by a degree three equation, are very interesting because they have both a geometric and a group structure. We will see that one cannot just study elliptic curves individually: one can describe them all simultaneously using a so-called moduli space.
David Holmes: Counting curves in the plane
Counting problems arising from geometry have been studied for thousands of years, yet new connections to theoretical physics and number theory continue to be discovered. A central example is counting algebraic curves through some points in the plane; there is one line through two points, one conic through 5 points, but what comes next?
Victoria Hoskins: Surprising counts over finite fields
One of the simplest techniques in number theory is to look for solutions of equations modulo a prime number, or more generally for solutions in a finite field. One obvious but surprisingly powerful fact is that one can count solutions in different finite fields, and these counts assemble into a powerful invariant, the so-called Zeta function, which miraculously even contains information about the topology of the set of complex solutions! There are other counting problems over finite fields coming from linear algebra and the representation theory of quivers, which unexpectedly behave in a similar way to point counts of algebraic varieties over finite fields, despite not coming from algebraic geometry. The goal of this talk is to explain the link between these two counting problems, and I will illustrate the ideas with many concrete examples.