Geometry working seminar

Wednesday, August 1, 2012 1:00 pm - 1:00 pm EDT (GMT -04:00)

Pure Mathematics Department, University of Waterloo

Matthew Beckett “Non-minimal Yang-Mills solutions”

Abstract:
Connections which satisfy the Yang-Mills equations are critical points of the Yang-Mills action. Minimal points on the 4-sphere are well understood via the ADHM construction, but it is a natural question to ask whether there exist non-minimal critical points. In this talk I will present some results from a 1991 paper by Thomas H. Parker which constructs some such examples of non-minimal Yang-Mills solutions.

Robert Garbary “Elliptic curves and the group law”

Abstract:
In number theory, an elliptic curve is the zero locus of a smooth irreducible cubic polynomial in P2. Such an object comes equipped with a group law, and much of number theory is concerned with studying this group law. However, the group law has a very strange definition — you add two points by doing two different intersections of lines with the curves. In this talk, I will show that the group law is a consequence of the Riemann–Roch theorem. In particular, for a compact Riemann surface X of genus 1, it implies a (non-canonical) bijection between X and Pic0(X) — the group of degree zero line bundles on X. Using this bijection, we can define a group law on X. I will show that, in the case that X is embedded into P2 as a smooth cubic (every X admits such an embedding), this group law is exactly the one that people in number theory care about.