Minimal log discrepancy of isolated singularities and Reeb orbits

Mark McLean
Stony Brook University
December 3, 2014
Let $A$ be an affine variety inside a complex $N$ dimensional vector space which either has an isolated singularity at the origin or is smooth at the origin. The intersection of $A$ with a very small sphere turns out to be a contact manifold called the link of $A$. If the first Chern class of our link is torsion (I.e. the singularity is numerically $\mathbb Q$ Gorenstein) then we can assign an invariant of our singularity called the minimal discrepancy. We relate the minimal discrepancy with indices of certain Reeb orbits on our link.

Level raising mod 2 and arbitrary 2-Selmer ranks

Chao Li
Harvard University
December 4, 2014
We prove a level raising mod $p = 2$ theorem for elliptic curves over $\mathbb Q$, generalizing theorems of Ribet and Diamond-Taylor. As an application, we show that the 2-Selmer rank can be arbitrary in level raising families. We will begin by explaining our motivation from W. Zhang's approach to the $p$-part of the BSD conjecture. Explicit examples will be given to illustrate different phenomena compared to odd $p$. This is joint work with Bao V. Le Hung.

Ball quotients

Bruno Klingler
Université Paris Diderot; Member, School of Mathematics
December 8, 2014
Ball quotients are complex manifolds appearing in many different settings: algebraic geometry, hyperbolic geometry, group theory and number theory. I will describe various results and conjectures on them.

More on sum-of-squares proofs for planted clique

Avi Wigderson
Herbert H. Maass Professor, School of Mathematics
December 9, 2014

While this talk is a continuation of the one I gave on Tue Nov 25, it will be planned as an independent one. I will not assume knowledge from that talk, and will reintroduce that is needed. (That first lecture gave plenty of background material, and anyone interested can watch it on https://video.ias.edu/csdm/2014/1125-AviWigderson).

A support theorem for the Hitchin fibration

Pierre-Henri Chaudouard
Université Paris 7; von Neumann Fellow, School of Mathematics
December 9, 2014
The main tool in Ngô's proof of the Langlands-Shelstad fundamental lemma, is a theorem on the support of the relative cohomology of the elliptic part of the Hitchin fibration. For $\mathrm{GL}(n)$ and a divisor of degree $> 2g-2$, the theorem says that the relative cohomology is completely determined by its restriction to any dense open subset of the base of the Hitchin fibration. In this talk, we will explain our extension of that theorem to the whole Hitchin fibration, including the global nilpotent cone (for $\mathrm{GL}(n)$ and a divisor of degree $> 2g-2$).

The Andre-Oort conjecture II

Bruno Klingler
Université Paris Diderot; Member, School of Mathematics
December 10, 2014
The Andre-Oort conjecture describes the expected distribution of special points on Shimura varieties (typically: the distribution in the moduli space of principally polarized Abelian varieties of points corresponding to Abelian varieties with complex multiplication). From the point of view of Hodge theory, it completely describes the geometric properties of the Hodge locus in some special instances. In these lectures I will try to introduce Shimura varieties, the Andre-Oort conjecture and recent work on it.