Generalised Kato classes and applications

This lecture series will be devoted to various arithmetic applications of the eponymous cohomology classes of the title, to such questions as the equivariant Birch and Swinnerton-Dyer conjecture, Perrin-Riou's conjecture relating Beilinson elements to rational points on elliptic curves, the theory of Stark-Heegner points, and the construction of explicit Selmer classes on elliptic curves in settings where the underlying Selmer group has rank two.

Nearby cycles over general bases and Thom-Sebastiani theorems

For germs of holomorphic functions $f : \mathbf{C}^{m+1} \to \mathbf{C}$, $g : \mathbf{C}^{n+1} \to \mathbf{C}$ having an isolated critical point at 0 with value 0, the classical Thom-Sebastiani theorem describes the vanishing cycles group $\Phi^{m+n+1}(f \oplus g)$ (and its monodromy) as a tensor product $\Phi^m(f) \otimes \Phi^n(g)$, where $(f \oplus g)(x,y) = f(x) + g(y), x = (x_0,...,x_m), y = (y_0,...,y_n)$. I will discuss algebraic variants and generalizations of this result over fields of any characteristic, where the tensor product is replaced by a certain local convolution product, as suggested by Deligne. The main theorem is a Künneth formula for $R\Psi$ in the framework of Deligne's theory of nearby cycles over general bases.

2015 Workshop on Inter-Universal Teichmuller Theory

The goal of this workshop is to understand some of the underlying ideas in absolute anabelian geometry and inter-universal Teichmuller theory, a subject that was developed by Shinichi Mochizuki over a period of of 15 - 20 years, with strong Diophantine consequences such as the ABC conjecture, the Szpiro conjecture for elliptic curves, and the Vojta conjecture for hyperbolic curves.

2015 Workshop on the Patched Eigenvariety

The goal of this workshop is to understand the recent work of Breuil, Hellmann and Schraen on the so-called patched eigenvariety and its application to modularity conjectures.

2015 Workshop on Number Theory

This workshop will be held from July 16th to July 31st, 2015. The subjects are analytic number theory and representation theory. The goal of this workshop is to introduce advanced theory and recent progress in number theory to researchers and graduate students in China.