Seminars Sorted by Series

An important theme in arithmetic combinatorics, which is closely related to the ergodic-theoretic project of understanding characteristic factors, is higher-order Fourier analysis. It has been well known for a long time that various norms defined in...

We shall discuss joint work with I Z Ruzsa in which it is shown that if A is a subset of {1,..,N} such that its difference set contains no number of the form $p-1$ for $p$ a prime, then $|A|=O(N\exp(-c\sqrt{4}{\log N}))$ for some absolute $c>0$.

Let A be subset of {1,...,n}. We say that A is square sum-free if the sum of any two different elements of A is not a square. Erdos and Sarkozy asked whether a square sum-free set can have more than n(1/3+epsilon) elements (motivated by the sequence...

This will be a survey of old and new problems and results in additive number theory.

In 1977 Szemeredi proved that any subset of the integers of positive density contains arbitrarily long arithmetic progression. A couple of years later Furstenberg gave an ergodic theoretic proof for Szemeredi's theorem. At around the same time...

Recent work of Alon-Shapira and Rodl-Schacht has demonstrated that every hereditary graph and hypergraph property is testable with one-sided error. This result appears definitive, but there are some subtleties to it that I will present here. For...

Let Q(n,p) denote the adjacency matrix of the Erdos-Renyi graph G(n,p), that is to say a symmetric matrix whose entries above the main diagonal are independently set to 1 with probability p and 0 with probability 1-p. We will examine the behavior of...

For a probability distribution X over a finite set, let D(X) denote the L_2-distance of X from the uniform distribution. Let X, Y be probability distributions over the finite group G and let Z be their G-convolution. Inspired by recent work of...

The aim is to present some of the more technical aspects of my joint project with Tim Gowers regarding the true complexity of a system of linear quations. Using so-called "quadratic Fourier analysis", we determined a necessary and sufficient...

Let $G$ be a finite Abelian group, say $Z/NZ$. A set $\Lambda = \{ lambda_1, \dots, \lambda_{m} \}$ is called {\it dissociated} if any equality $\sum_{i=1}^m \varepsilon_i \lambda_i = 0$, where $\varepsilon_i \in \{ 0,\pm 1 \}$ implies that all $...

We will study the following problem: Given $n$ subsets $A_1, A_2,\ldots, A_n\subset \mathbb{F}_q$ of a finite field $\mathbb{F}_q$ with $q$ elements. Let $|A_1|\cdot |A_2|\cdot\ldots dot|A_n|>q^{1+\varepsilon}$ for some $\varepsilon>0,$ one needs to...

The family of high-rank arithmetic groups is a class of groups playing an important role in various areas of mathematics.

It includes $\mathrm{SL}(n,\mathbb Z)$, for $n > 2$ , $\mathrm{SL}(n, \mathbb Z[1/p])$ for $n > 1$, their finite index...

Everyone is invited to the Simons 2021 workshop on High Dimensional Expanders in NY.

See https://www.simonsfoundation.org/event/2021-mps-conference-on-high-dimensional-expanders/

Somehow, despite the title, $SL(2,Z)$ is the poster child for arithmetic groups not satisfying the congruence subgroup property, which is to say that it has finite index subgroups which can not be defined by congruence conditions on their...

Let $f = \sum a_n x^n \in \mathbb Q[x]$ be a power series which is also a meromorphic function in some neighborhood of the origin. The subject of the talk will be how certain conditions on $f(x)$ as a meromorphic function actually guarantee that $f...

We explain our proof of the unbounded denominators conjecture. This talk will require the main theorem of the lecture on Nov. 17, 2021, as a “black box” but otherwise be logically independent of that talk.

We discuss a local to global profinite principle for being a commutator in some arithmetic groups. Specifically we show that $SL_2(Z)$ satisfies such a principle, while it can fail with infinitely many exceptions for $SL_2(Z[1/p])$. The source of...

We discuss a local to global profinite principle for being a commutator in some arithmetic groups. Specifically we show that $SL_2(Z)$ satisfies such a principle, while it can fail with infinitely many exceptions for $SL_2(Z[1/p])$. The source of...

If a monomorphism of abstract groups $H\hookrightarrow G$ induces an isomorphism of profinite completions, then $(G, H)$ is called a Grothendieck pair, recalling the fact that Grothendieck asked about the existence of such pairs with $G$ and $H$...

Taking up the terminology established in the first lecture, in 1970 Grothendieck showed that when two groups $(G,H)$ form a Grothendieck pair, there is an equivalence of their linear representations. For recent work showing that certain groups are...

In the first part of this talk, we take the ideas of the second talk and focus on the case of (arithmetic) lattices in $PSL(2,R)$ and $PSL(2,C)$. The required representation rigidity is achieved by what we call Galois rigidity. In particular if $...

This is joint work with Samuel Edwards and Hee Oh. Let $G$ be a connected semisimple real algebraic group, and $\Gamma G$ be a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We describe the asymptotic behavior of matrix...

A landmark result of Ratner states that if $G$ is a Lie group, $\Gamma$ a lattice in $G$ and if $u_t$ is a one-parameter $Ad$-unipotent subgroup of $G$, then for any $x \in G/\Gamma$ the orbit $u_t.x$ is equidistributed in a periodic orbit of some...

A landmark result of Ratner states that if $G$ is a Lie group, $\Gamma$ a lattice in $G$ and if $u_t$ is a one-parameter $Ad$-unipotent subgroup of $G$, then for any $x \in G/\Gamma$ the orbit $u_t.x$ is equidistributed in a periodic orbit of some...

This lecture serves as a background for the upcoming talk by Bharatram Rangarajan. I will review some aspects of bounded cohomology, including why it appears to have some relevance to stability questions. I will then explain vanishing results for...

In ongoing joint work with Glebsky, Lubotzky, and Monod, we construct an analog of bounded cohomology in an asymptotic setting in order to prove uniform stability of lattices in Lie groups (of rank at least two) with respect to unitary groups...

The Nielsen-Thurston theory of surface homeomorphism can be thought of as a surface analogue to the Jordan Canonical Form.  I will discuss my progress in developing a similar decomposition for free group automorphisms. (Un)Fortunately, free group...

In this talk, I will establish a sharp bound on the growth of cuspidal Bianchi modular forms. By the Eichler-Shimura isomorphism, we actually give a sharp bound of the second cohomology of a hyperbolic three manifold (Bianchi manifold) with local...

The Markoff equation $x^{2} + y^{2} + z^{2}=3xyz$, which arose in his spectacular thesis (1879), is ubiquitous in a tremendous variety of contexts.  After reviewing some of these, we will discuss joint work with Bourgain and Sarnak establishing...