Seminars Sorted by Series

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...

We use the recent developments of Jacquet in order to obtain an explicit expression for a compact unitary period of certain automorphic form on GL(n) in terms of special values of L-functions. Jacquet obtains a characterization of the image of...

A celebrated result of Waldspurger provides a relation between Fourier coefficients of half-integral weight modular forms and the central value of L-functions of the associated Shimura lift (which is a modular form of integral weight). This has a...

I will discuss the following theme: starting with an a priori Diophantine result (typical flavour: integer solutions to such-and-such equation are well-spaced) and turning it into an equidistribution-type statement on a homogeneous space. This (in...

I will explain a conjectural generalisation of Ihara's lemma in the theory of modular forms to higher dimensional automorphic forms and sketch how this conjecture implies the Sato-Tate conjecture for rational elliptic curves with somewhere...

We will describe some recent ergodic theorems for general families of averages on semisimple Lie groups, and explain how they can be used to 1) Solve the lattice point counting problem for general domains in the group, with explicit estimate of the...

Ideal classes in (totally real) number fields give naturally rise to compact orbits inside SL(n,Z)\SL(n,R) for the diagonal subgroup. We will discuss their (equi-)distribution properties as the field varies, and the two main ideas in our approach...

Given a higher rank arithmetic group (E.g. SL(3,Z)) it has r(n) complex irreducible representations of degree n. We will study the the rate of growth of r(n), the associated zeta function SUM(r(n)n^(-s)), its Euler factorisation etc. Some...

I will describe recent joint work with Janos Pintz and Cem Yildirim on small gaps between primes and primes in tuples. Perhaps the most surprising result is that if the primes have level of distributed in arithmetic progressions greater than 1/2...

We obtain asymptotic lower bounds for the spectral function of the Laplacian and for the remainder in local Weyl's law on compact manifolds. In the negatively curved case, thermodynamic formalism is applied to improve the estimates. Our results can...

The remarkable phenomenon of Superrigidity, discovered by Margulis in the context of linear representations of lattices in higher rank semi-simple groups, has motivated and inspired a lot of research on other "higher rank" groups and representations...

We consider maps between smooth projective curves and some arithmetic and geometric properties of such maps. In particular, we will discuss the case of maps from the generic Riemann surface of genus g -- a problem first seriously looked at by...