Which positive integers are the area of a right angled triangle with rational sides? In this talk I will discuss this classical problem, its reformulation in terms of rational points on elliptic curves and Tunnell's theorem which gives a complete solution to this problem assuming the Birch and Swinnerton-Dyer conjecture.
This talk will attempt to say something about the p-adic zeta function, a p-adic analytic object which encodes information about Galois cohomology of Tate twists in its special values. We first explain the construction of the p-adic zeta function, via p-adic Fourier theory. Then, after saying something about Coleman integration, we will explain the interpretation of special values of the p-adic zeta function as limiting values of p-adic polylogarithms, in analogy with the Archimedean case. Finally, we will explore the consequences for the de Rham and etale fundamental groupoids of the projective line minus three points.
Vinogradov's three prime theorem resolves the weak Goldbach conjecture for sufficiently large integers. We discuss some of the ideas behind the proof, and discuss some of the obstacles to completing a proof of the odd goldbach conjecture.
Complex structures on a closed surface of genus at least 2 are in
one-to-one correspondence with hyperbolic metrics, so that there is a
single space, Teichmüller space, parametrising all possible complex
and hyperbolic structures on a given surface (up to isotopy). We will
explore how complex and hyperbolic geometry interact in Teichmüller
space.
We show how the reduction procedure for generalized Kahler
structures can be used to recover Hitchin's results about the
existence of a generalized Kahler structure on the moduli space of
instantons on bundle over a generalized Kahler manifold. In this setup
the proof follows closely the proof of the same claim for the Kahler
case and clarifies some of the stranger considerations from Hitchin's
proof.