## Chow Rings, Decomposition of the Diagonal and the Topology of Families

Lecture 3: Decomposition of the Small Diagonal and the Topology of Families

Claire Voisin

CNRS, Institut de Mathematiques de Jussieu, Paris

November 3, 2011

Lecture 3: Decomposition of the Small Diagonal and the Topology of Families

Michael Khanevsky

Institute for Advanced Study

November 2, 2011

John Mather

Princeton University; Institute for Advanced Study

November 2, 2011

Claire Voisin

CNRS, Institut de Mathematiques de Jussieu, Paris

November 2, 2011

Lecture 2: On the Generalized Bloch and Hodge Conjectures for Complete Intersections

Tatyana Shcherbina

Institute for Low Temperature Physics, Kharkov

November 1, 2011

We consider asymptotics of the correlation functions of characteristic polynomials of the hermitian Wigner matrices $H_n=n^{-1/2}W_n$ and the hermitian sample covariance matrices $X_n=n^{-1}A_{m,n}^*A_{m,n}$. We use the integration over the Grassmann variables to obtain a convenient integral representation.

Claire Voisin

CNRS, Institut de Mathematiques de Jussieu, Paris

November 1, 2011

Summary: These lectures are devoted to the interplay between cohomology and Chow groups of a complex algebraic variety, and also to the consequences, on the topology of a family of smooth projective varieties, of statements concerning Chow groups of the general fiber. A crucial notion is that of coniveau of the cohomology and its conjectural relation with the shape of Chow groups of small dimension. A common theme will be that of decomposition of the diagonal, which will appear in various contexts.

Jeff Kahn

Institute for Advanced Study

October 31, 2011

Penka Georgieva

Princeton University

November 11, 2011

I will first discuss the orientability of the moduli spaces of J-holomorphic maps with Lagrangian boundary conditions. It is known that these spaces are not always orientable and I will explain what the obstruction depends on. Then, in the presence of an anti-symplectic involution on the target, I will give a construction of open Gromov-Witten disk invariants. This is a generalization to higher dimensions of the works of Cho and Solomon, and is related to the invariants defined by Welschinger

Frederic Bourgeois

Universite Libre Bruxelles

November 11, 2011

Enrico Bombieri

Institute for Advanced Study

November 10, 2011

The Davenport-Heilbronn function (introduced by Titchmarsh) is a linear combination of the two L-functions with a complex character mod 5, with a functional equation of L-function type but for which the analogue of the Riemann hypothesis fails. In this lecture, we study the Moebius inversion for functions of this type and show how its behavior is related to the distribution of zeros in the half-plane of absolute convergence. Work in collaboration with Amit Ghosh.