I will introduce Shimura varieties and discuss the role they play in the conjectural relashionship between Galois representations and automorphic forms. I will explain what is meant by a geometric realization of Langlands correspondences, and how the geometry of Shimura varieties and their local models conjecturally explains many aspects of these correspondences. This talk is intended as an introduction for non-number theorists to an approach to Langlands conjectures via arithmetic algebraic geometry.
This will be an introduction to special value formulas for L-functions and especially the uses of modular forms in establishing some of them -- beginning with the values of the Riemann zeta function at negative integers and hopefully arriving at some more recent work on the Birch-Swinnerton-Dyer formula.
The systolic inequality says that if we take any metric on an n-dimensional torus with volume 1, then we can find a non-contractible curve in the torus with length at most C(n). A remarkable feature of the inequality is how general it is: it holds for all metrics.
Given a Hamiltonian on $T^n\times R^n$, we shall explain how the sequence of suitably rescaled (i.e. homogenized) Hamiltonians, converges, for a suitably defined symplectic metric. We shall then explain some applications, in particular to symplectic topology and invariant measures of dynamical systems.
I will introduce l-adic representations and what it means for them to be automorphic, talk about potential automorphy as an alternative to automorphy, explain what can currently be proved (but not how) and discuss what seem to me the important open problems. This should serve as an introduction to half the special year for non-number theorists. The other major theme will likely be the `p-adic Langlands program', which I will not address (but perhaps someone else will).