The goal of this student seminar is to understand some situations in which Ratner theory applies. In many cases, there is no obvious homogeneous space or even a group action. Yet, topological and measure rigidity of homogeneous actions play a key role. We will try to focus on concrete applications.

Time: Wednesdays 3-4:30pm
Location: E 203

Provisional schedule

Date Speaker Topic
1 Oct 2014 Simion Filip Overview of topics. Ratner's theorem in different guises.
8 Oct 2014 David Aulicino Dynamics of SL(2,R) in genus 2 (McMullen).
15 Oct 2014 Ian Frankel Gaps in \(\sqrt {n} \) mod 1 (Elkies-McMullen).
22 Oct 2014 Simion Filip Intro to K3 surfaces, hyperkahler manifolds.
29 Oct 2014 Bena Tshishiku Teichmuller space of HK manifolds and ergodic theory.
5 Nov 2014 Dan Le Hermitian symmetric domains, Shimura varieties.
12 Nov 2014 Sean Howe Equidistribution of Hecke orbits and special subvarieties.
19 Nov 2014 Ronen Mukamel Periods of abelian differentials (Kapovich).
26 Nov 2014 Thanksgiving
McMullen's work in genus 2
This work is concerned with classifying the orbit closures of the SL(2,R)-action on the moduli space of genus 2 surfaces with a holomorphic 1-form. It reduces the question to the homogeneous setting, to which Ratner theory applies.
Follow "Dynamics of SL(2,R) over moduli space in genus 2" available here [1 lecture]
Kapovich on periods of abelian differentials
Let \(X\) be a (topological) surface and pick a cohomology class \([c]\in H^1(X;\mathbb{C})\). For which \([c]\) is there a complex structure on \(X\) and a holomorphic 1-form \(\omega\) such that \(\omega\) represents the cohomology class \([c]\)? The paper of Kapovich answers this question. It would be interesting to generalize this to relative cohomology classes (with some number of punctures) and more generally, strata or orbit closures of flat surfaces.
Follow "Periods of abelian differentials and dynamics" available here [1 lecture]
Elkies-McMullen on Gaps in \(\sqrt{n}\) mod 1
Project the numbers \(\sqrt{1},\sqrt{2},\ldots,\sqrt{N}\) to the unit circle and cut the circle at these points. Rescaling the segments by \(N\) (so that their length is roughly \(1\)), what is the distribution of lengths? The paper of Elkies-McMullen answers this question.
Follow "Gaps in \(\sqrt{n}\) mod 1 and ergodic theory" available here [1 lecture]
Equidistribution, Andre-Oort and related topics
This topic is about equidistribution results on hermitian symmetric domains and their interpretation in algebraic geometry. Often, ergodic theory provides softer proofs which do not appeal to spectral theory or L-function estimates.
Intro to Shimura varieties and Hermitian Symmetric Domains (and spectral theory?) [1 lecture]
Equidistribution of Hecke orbits (Eskin-Oh) and special subvarieties (Clozel-Ullmo) [1 lecture]
Verbitsky's work
The Hodge structures of K3 surfaces (and more generally, hyperkahler manifolds) have a nice associated period domain. It is useful in studying their geometry, and recently Verbitsky found some applications of Ratner's orbit closure theorems to some questions in algebraic geoemtry.
Intro to K3 surfaces and hyperkahler manifolds [1 lecture]
Teichmuller spaces, ergodic theory, applications [1 lecture]
Verbitsky's ICM report and the paper Ergodic complex structures
Vatsal on special values of L-functions (time permitting)
This topic is concerned with non-vanishing (or divisibility) properties of special values of L-functions. For example, it was known to Kummer that for odd \(n,n'\):
If \((p-1)\not \mid n+1 \) and \(n\equiv n' \mod p-1 \) then we have $$ (1-p^n)\zeta(-n) \equiv (1-p^{n'})\zeta (-n') \mod p $$ Orbit closure theorems play a role in some recent proofs, and it would be nice to have an introduction to this topic.
An introduction to this work is in Vatsal's ICM report [1 lecture]