School of Mathematics

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.

One of the great challenges posed by the laws of quantum mechanics is that the number of parameters required to describe a quantum state in general grows exponentially in the number of particles. This complexity is directly related to the phenomenon...

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...

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...

Problems analogous to Bloch's conjecture, and some tentative methods of attacking them, will be discussed.

For a finitely presented group, the Word Problem asks for an algorithm which declares whether or not words on the generators represent the identity. The Dehn function is the time-complexity of a direct attack on the Word Problem by applying the...

The study of Cayley graphs of finite groups is related to the investigation of pseudo-random graphs and to problems in Combinatorial Number Theory, Geometry and Information Theory. I will discuss this topic, describing the motivation and focusing on...

The Parallel Repetition Theorem upper-bounds the value of a repeated (tensored) two prover game in terms of the value of the base game and the number of repetitions. In this work we give a simple transformation on games -- "fortification" -- and...

Finding large cliques in random graphs and the closely related "planted" clique variant, where a clique of size \(k\) is planted in a random \(G(n,1/2)\) graph, have been the focus of substantial study in algorithm design. Despite much effort, the...

Welcome Remarks
Robbert Dijkgraaf, Director and Leon Levy Professor Institute for Advanced Study