The Markoff equation x2+y2+z2=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 forms of strong...

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Arithmetic Groups

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 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 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...

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...

A landmark result of Ratner states that if G is a Lie group, Γ a lattice in G and if ut is a one-parameter Ad-unipotent subgroup of G, then for any x∈G/Γ the orbit ut.x is equidistributed in a periodic orbit of some subgroup L less than G, and...

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

A key motivation behind Ratner's equidistribution theorem for one-parameter unipotent flows has been to establish Raghunathan's conjecture regarding the possible orbit closures of groups generated by one-parameter unipotent groups; using the equidistribution theorem Ratner proved that if G and Γ are as above, and if H

These results have had many beautiful and unexpected applications in number theory, geometry and other areas. A key challenge has been to quantify and effectify these results. Beyond the case of actions of horospheric groups where there are several fully quantitative and effective results available, results in this direction have been few and far between. In particular, if G is semisimple and U is not horospheric no quantitative form of Ratner's equidistribution theorem was known with any error rate, though there has been some progress on understanding quantitatively density properties of such flows with iterative logarithm error rates.

In these two talks, we report on a fully quantitative and effective equidistribution result for orbits of one-parameter unipotent groups in quotients of SL2(C) and SL2(R)×SL(2,R).

This is joint work with Samuel Edwards and Hee Oh. Let G be a connected semisimple real algebraic group, and Γ

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 Γ1 is a...

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...