Seminars Sorted by Series

Goldston & Montgomery and Montgomery & Soundararajan have established formulae for the variance of sums of the von Magoldt function over short intervals (i.e. for the variance of the number of primes in these intervals) assuming, respectively, the...

The large-time behavior of the return probabilities of a random walk is controlled by the behavior of the Green's function $G_r (x,y)$ at the radius $r=R$ of convergence. For nearest neighbor random walks on virtually free groups it is known that...

I'll describe a physical implementation of zero knowledge proofs whose goal is to verify that two physical objects are identical, without revealing any information about them. Our motivation is the task of verifying that an about-to-be-dismantled...

The talk will have 2 parts (between the parts we will have a break). In the first part, we will discuss two options for using groups to construct expander graphs (Cayley graphs and Schreier diagrams). Specifically, we will see how to construct...

I will talk about two natural notions of convergence for sequences of graphs of bounded degree and their connection to groups and group actions. The first is Benjamini-Schramm convergence, which is strongly related to parameter testing. The second...

Consider the problem when we want to construct some structure on a bounded degree graph, e.g. an almost maximum matching, and we want to decide about each edge depending only on its constant radius neighborhood. We show that the information about...

We study the limits of efficient computation in the context of constructing random-looking graph distributions that can be used to emulate huge random graphs over N=2^n vertices. In particular we study k-wise independent graphs where (as in the...

The curvelet transform is a directional wavelet transform over R^n, originally due to Candes and Donoho (2002). It is used to analyze functions that have singularities along smooth surfaces. I demonstrate how this can lead to new quantum algorithms...

The cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite Markov chains to equilibrium. Many natural families of chains are believed to exhibit cutoff, and yet establishing this fact is often extremely...

A nxn matrix M over GF(2) is said to be (r,\delta)-rigid if every matrix M' within \delta n^2 Hamming distance from M has rank at least r. A long standing open problem is to construct explicit rigid matrices. In a recent remarkable result, Alman and...

Using ideas from computational learning theory, I'll show that "for most practical purposes," one can learn a quantum state using a number of measurements that grows only linearly with the number of qubits n. By contrast, traditional quantum state...

A randomness extractor is an algorithm which extracts randomness from a low-quality random source, using some additional truly random bits. Extractors have proved useful in a variety of seemingly unrelated areas. We construct new extractors which...

We consider a locally finite (Radon) measure on SO(d,1)/Gamma invariant under a horospherical subgroup of SO(d,1) where Gamma is a discrete, but not necessarily geometrically finite, subgroup. We show that whenever the measure does not observe any...

Let Q be a non-degenerate indefinite quadratic form in d variables. In the mid 80's, Margulis proved the Oppenheim conjecture, which states that if d ≥ 3 and Q is not proportional to a rational form then Q takes values arbitrarily close to zero at...

In this talk I will discuss some definitions of exponential mixing and other rates of mixing and discuss some of its consequences.

In classical or quantum systems with time scales widely separated by a small slowness parameter, the fast system is driven by the slow and reacts back on it. The asymptotics of the driven fast motion has been much studied, but understanding the...

I shall discuss two related local regularity results for asymptotically hyperbolic (or complex hyperbolic) Einstein metrics, near a point at infinity: local polyhomogeneity and unique continuation.

A three dimensional strongly pseudoconvex Cauchy Riemann manifold can be considered as a contact manifold with a conformal structure on the hyperplane distribution. In this talk, we will discuss CR structures near the standard one on S^3 , the...

Simple cell decompositions of surfaces are in one to one correspondence with certain conjugacy classes of subgroups of the modular group. Polycrystalline structures in 2 dimensional materials give rise to statistical simple cell decompositions of...

In 2007, Dvir proved the Kakeya conjecture over finite fields. His proof uses the polynomial method: we take a hypothetical strange set, find a polynomial that vanishes on the set, and use properties of polynomials to show that the set does not...

The k-dilation of a mapping F measures how much the mapping stretches k-dimensional areas. If a map F has k-dilation L , then it maps each k-dimensional surface of k-volume V to an image of k-volume at most LV. If the k-dilation of a map is very...

Let $H$ be any smooth function on $R^4$ and let $Y$ be any compact and regular level set. I'll explain a proof that $Y$ admits an infinite family of proper compact subsets that are invariant under the Hamiltonian flow, which moreover have dense...

In joint work with Barry Mazur, we obtain lower bounds for Selmer ranks of elliptic curves over dihedral extensions of number fields, giving evidence for a generalized Parity Conjecture. Heegner points can account for large Selmer groups in dihedral...

Since the proof of Iwasawa's main conjecture in the classical case (theorem of Mazur and Wiles), automorphic forms on GL(2) have played an essential role to understand Iwasawa theory for GL(1). In this lecture, Leopoldt 's conjecture for totally...

For the Rankin-Selberg L-function L(s, , ), there are central value formulas or derivative formulas depending on its root number. Using theta-lifting, Waldspurger showed an explicit formula relating the central value to certain period integral. We...

Let F be a local field of characteristic 0. We consider distributions on GL(n+1,F) which are invariant under the adjoint action of GL(n,F). We prove that such distributions are invariant under transposition. This implies that an irreducible...

The US political system is built on representatives chosen by geographically localized regions. This presents the government with the problem of designing these districts. Every ten years, the US census counts the population and new political...

In these two lectures, I will come back and develop the material outlined in the Monday Seminar. More precisely, I will explain with more details the notion of microsupport of sheaves and its behaviour with respect to the operations, in particular...