May 11, 2015
Monday

09:30 AM  09:45 AM


Welcome

 Location
 SLMath: Eisenbud Auditorium
 Video


 Abstract
 
 Supplements



09:50 AM  10:40 AM


Effective equidistribution of certain adelic periods
Amir Mohammadi (University of California, San Diego)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We will discuss a quantitative equidistribution statement for adelic homogeneous subsets whose stabilizer is maximal and semisimple. An application to certain equidistribution theorems will also be given. This is a joint work with Einsiedler, Margulis and Venkatesh.
 Supplements


10:40 AM  11:10 AM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:10 AM  12:00 PM


Quantum ergodicity on large graphs
Nalini Anantharaman (Université de Strasbourg)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We study eigenfunctions of the discrete laplacian on large regular graphs, and prove a ``quantum ergodicity'' result for these eigenfunctions : for most eigenfunctions $\psi$, the probability measure $\psi(x)^2$, defined on the set of vertices, is close to the uniform measure.
Although our proof is specific to regular graphs, we'll discuss possibilities of adaptation to more general models, like the Anderson model on regular graphs.
 Supplements


12:00 PM  01:30 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



01:30 PM  02:20 PM


Geodesic planes in hyperbolic 3manifolds
Hee Oh (Yale University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
The topic of my talk is what the possibilities are for the closure of a totally geodesic immersion of a hyperbolic plane into a complete hyperbolic 3 manifold. For finite volume hyperbolic 3 manifolds, work of Shah and Ratner shows that strong rigidity properties hold: any such immersed plane is either closed or dense. I will describe recent joint work with Curt McMullen and Amir Mohammadi which shows some of this rigidity persists in some cases of infinite volume hyperbolic 3manifolds
 Supplements


02:30 PM  02:55 PM


Equidistribution of expanding translates of curves in homogeneous spaces and its application to Diophantine approximation.
Lei Yang (University of Nevada)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
 
 Supplements


02:55 PM  03:20 PM


Entropy in the cusp and singular systems of linear forms
Shirali Kadyrov (Nazarbayev University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Singular systems of linear forms were introduced by Khintchine in the 1920s, and it was shown by Dani in the 1980s that they are in onetoone correspondence with certain divergent orbits of oneparameter diagonal groups on the space of lattices. We give a (conjecturally sharp) upper bound on the Hausdorff dimension of singular systems of linear forms (equivalently the set of points with divergent trajectories). This extends work by Cheung, as well as by Chevallier and Cheung, on the vector case. For a diagonal action on the space of lattices, we also consider the relation of the entropy of an invariant measure to its mass in a fixed compact set. Our technique is based on the method of integral inequalities developed by Eskin, Margulis, and Mozes. This is a joint work with D. Kleinbock, E. Lindenstrauss, and G. A. Margulis
 Supplements


03:20 PM  03:50 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:50 PM  04:40 PM


Local spectral gap
Alireza Golsefidy (University of California, San Diego)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
(Joint with R. Boutonnet, A. Ioana) The notion of local spectral gap for general measure preserving actions will be defined. We prove that the left translation action of a dense subgroup of a simple Lie group has local spectral gap if the subgroup has algebraic entries. This extends to the noncompact setting recent works of BourgainGamburd and Benoistde Saxce. We also extend BourgainYehudayoff’s result. The application of this result to BanachRuziewicz problem, delayed randomwalk, and monotone expanders will be explained.
 Supplements



May 12, 2015
Tuesday

09:30 AM  10:20 AM


Randomness in Diophantine approximation
Alexander Gorodnik (University of Bristol)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We discuss the problem of counting solutions of Diophantine inequalities. While a general asymptotic formula for the counting function has been established by W. Schmidt, finer statistical properties of this function are still not well understood. We investigate its limiting distribution and establish the central limit theorem in this context. This is a joint work with Anish Ghosh.
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  11:50 AM


Genericity along curves and applications
Corinna Ulcigrai (University of Bristol)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We will present two results in mathematical physics which can be obtained as applications of a result in homogeneous dynamics. The applications concern the dynamics in a class of pseudointegrable billiards in ellipses and the behaviour of light rays in arrays of Eaton lenses. They are based on an equidistribution result in the space of affine lattices, that guarantees that typical points on certain curves are Birkhoff generic for the geodesic flow. For the Eaton lenses application we also prove an Oseledets genericity result which generalizes in this set up a recent result by Eskin and Chaika. The talk is based on joint work with Krzysztof Fraczek and Ronggang Shi.
 Supplements


12:00 PM  01:30 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



01:30 PM  02:20 PM


Bernoulli convolutions for algebraic parameters
Peter Varju (University of Cambridge)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
The Bernoulli convolution with parameter lambda is the law of the random variable: Sum X_i lambda^i, where X_i are independent unbiased +1/1 valued random variables. If lambda <1/2, then the Bernoulli convolution is singular and is supported on a Cantor set. If 1> lambda >1/2, the question whether the Bernoulli convolution is singular or a.c. is a very interesting open problem. Recent papers of Hochman and Shmerkin prove that the set of lambda's such that the measure is singular is of Hausdorff dimension 0. I will discuss the problem for parameters lambda that are algebraic. Work in progress, joint with Emmanuel Breuillard.
 Supplements


02:30 PM  02:55 PM


Counting and dynamics in SL_2
Michael Magee (Institute for Advanced Study)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
In this talk I'll discuss a lattice point count for a thin semigroup inside SL_2(Z). It is important for applications I'll describe that one can perform this count uniformly throughout congruence classes.
The approach to counting is dynamical  with input from both the real place and finite primes. At the real place one brings ideas of Dolgopyat concerning oscillatory functions into play. At finite places the necessary expansion property
follows from work of Bourgain and Gamburd (at one prime) or Bourgain, Gamburd and Sarnak (at squarefree moduli).
These are underpinned by tripling estimates in SL_2(F_p) due to Helfgott. I'll try to explain in simple terms the key dynamical facts behind all of these methods.
This talk is based on joint work with Hee Oh and Dale Winter.
 Supplements


02:55 PM  03:20 PM


Hausdorff dimension of product sets in Lie groups
Nicolas de Saxce (Université de Paris XIII (ParisNord))

 Location
 SLMath: Eisenbud Auditorium
 Video


 Abstract
Given a Borel subset A in a Lie group G, we will study under what conditions the Hausdorff dimension of the product set AAA is strictly larger than that of A, especially in the case when the ambient group G is simple.
 Supplements



03:20 PM  03:50 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:50 PM  04:40 PM


Diophantine approximation for algebraic numbers
Hillel Furstenberg (The Hebrew University of Jerusalem)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
A number is well approximable (WA) if the error when approximatin by p/q can be made small compared to the sqare of 1/q. Almost all reals are WA. For good reasons quadratic irrationals are not. Nothing is known for cubic irrationals. We show how this relates to special orbits in homogeneous dynamics.
 Supplements


04:40 PM  06:20 PM


Reception

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements




May 13, 2015
Wednesday

09:00 AM  09:50 AM


Metric diophantine approximation on Lie groups
Emmanuel Breuillard (Université de Paris XI)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
A natural generalization of classical diophantine approximation consists in taking products of elements in a Lie group and ask how fast they can approach the identity. For example, classically, a number x is Diophantine if and only if sums of N terms each equal to 1, 1, x or x, cannot get closer to zero than an inverse power of N. Replacing x and 1 by two or more vectors in a vector space is the subject of simultaneous approximation, and diophantine approximation in matrices. In this talk I will consider the case when the elements are chosen from a more general Lie group and focus on nilpotent groups. It turns out that the computation of optimal exponents for nilpotent Lie groups can be reduced to that of certain algebraic subvarieties of matrices. We give a formula for the exponent of arbitrary submanifolds of matrices, showing it depends only on their algebraic closure, answering a question of Kleinbock and Margulis, and use it to express the exponent of nilpotent Lie groups in terms of various representation theoretic data. This is joint work with M. Aka, L. Rosenzweig and N. de Saxce
 Supplements


09:50 AM  10:20 AM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



10:20 AM  11:10 AM


Escape of mass for measures invariant under the diagonal group
Uri Shapira (TechnionIsrael Institute of Technology)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Let A denote the diagonal group in SL(n,R) acting on the space of unimodular lattices in R^n. In this talk I will explain a construction of a sequence of Ainvariant ergodic probability measures (supported on compact Aorbits) which converge to the zero measure. In fact the "geometry" of these orbits will be described in some detail.
 Supplements


11:20 AM  12:10 PM


Oppenheim conjecture and related problems
Gregory Margulis (Yale University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
The Oppenheim conjecture proved in mideighties states that the set of values at integral points of an irrational indefinite quadratic form in n>2 variables is dense in R. I will talk about the history and proof of this conjecture and will also talk about various problems related to the conjecure
 Supplements



May 14, 2015
Thursday

09:00 AM  09:50 AM


Effective density of unipotent orbits
Elon Lindenstrauss (The Hebrew University of Jerusalem)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Raghunathan conjectured that If G is a Lie group, Gamma a lattice, p in G/Gamma, and U an (ad)unipotent group then the closure of U.p is homogeneous (a periodic orbit of a subgroup of G). This conjecture was proved by Ratner in the early 90's via the classification of invariant measures; significant special cases were proved earlier by Dani and Margulis using a different, topological dynamics approach. Neither of these proofs is effective, nor do they provide rates  e.g. if p is generic in the sense that it does not lie on a
periodic orbit of any proper subgroup L<G with U<=L, an estimate (possibly depending on diophantinetype properties of the pair (p,U))) how large a piece of an orbit is needed so that it comes within distance epsilon of any point in a given compact subset of G/Gamma
 Supplements


10:00 AM  10:50 AM


On integer points in the Hitchin moduli space
Marc Burger (ETH Zürich)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Hitchin representations form a special component in the space of SL(n, R)representations of a compact
surface group; in the talk we will discuss two notions of integrality for such representations and show some nonfiniteness phenomena for n large. We will also show how finiteness can be restored by the
use of an appropriate height function. This is joint work with F.Labourie and A.Wienhard
 Supplements


10:50 AM  11:20 AM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



11:20 AM  12:00 PM


Concert

 Location
 SLMath: Eisenbud Auditorium
 Video


 Abstract
 
 Supplements



12:00 PM  01:30 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



01:30 PM  02:20 PM


Random Dynamics and a formula for Furstenberg, KullbackLedrappier Entropy
Federico Rodriguez Hertz (Pennsylvania State University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Joint with Aaron Brown, we found the following property for stationary measures of random composition of surface diffeomorphisms. Either the stable space is nonrandom, or the stationary measure is atomic or it is and SRB measure. This result generalize the work of Y. Benoist and JF. Quint as well as the ones by A. Eskin and M. Mirzakhani to nonhomogeneous, non affine setting. In the meantime we found a formula for the Furstenberg or KullbackLedrappier entropy involving Lyapunov exponents and dimensions. In this talk I will describe the results and some consequences of it.
 Supplements


02:30 PM  02:55 PM


Word metric asymptotics for actions of hyperbolic groups on Gromov hyperbolic spaces
Ilya Gekhtman (Rheinische FriedrichWilhelmsUniversität Bonn)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
I will discuss work in progress with Sam Taylor and Giulio Tiozzo in which we study some asymptotics with respect to the word metric for actions of hyperbolic groups on Gromov hyperbolic spaces. For example for a large class of such actions we prove that the proportion of elements in a Cayley ball of radius R which do not act loxodromically decays to zero.
 Supplements


02:55 PM  03:20 PM


Some Effective Estimates in Reduction Theory of Quadratic Forms
Han Li (Wesleyan University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
The aim of this talk is to present a recent joint work with Professor Gregory A. Margulis. Our main result is that any indefinite ternary integral quadratic form is integrally equivalent to many forms whose coefficients are small. This is deduced from a concentration property of the periodic SO(1, 2)orbits in SL(3, R)/SL(3,Z).
 Supplements


03:20 PM  03:55 PM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



03:50 PM  04:40 PM


Bottom of spectrum and equivariant family of measures at the boundary in negative curvature.
François Ledrappier (University of Notre Dame)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
The universal cover of a compact negatively curved manifold has strong homogeneity properties at infinity. We present such properties related to the bottom of the spectrum of the Laplacian, equivariant family of measures at the boundary and large time asymptotic of the heat kernel. This is joint work with Seonhee Lim.
 Supplements



May 15, 2015
Friday

09:00 AM  09:50 AM


Random walks on semisimple groups
JeanFrançois Quint (CNRS  Université de Montpellier)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
I will present recent results on the asymptotic behaviour of random walks on semisimple groups and ask open questions. This is joint work with Yves Benoist
 Supplements


09:50 AM  10:20 AM


Tea

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



10:20 AM  11:10 AM


Global rigidity of Anosov actions by higher rank lattices
Zhiren Wang (Pennsylvania State University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We will discuss a recent work with Aaron Brown and Federico Rodriguez Hertz on smooth classification of Anosov actions by higher rank lattices on nilmanifolds. In particular, we will explain how the existence of an Anosov diffeomorphism from the group action leads to Anosov property of generic elements in the acting group, allowing to make use of large abelian subgroups.
 Supplements


11:20 AM  12:10 PM


Dynamics on the moduli space of flat structures
Maryam Mirzakhani (Stanford University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
The moduli space of holomorphic oneforms on a closed surface of genus $g$ has a natural piecewise linear structure and carries an action of GL(2,R). One can investigate the geometry and dynamics of an individual flat surface by studying its orbit under this linear group action. I will discuss several open problems regarding the properties of these orbit closures. This talk is based on joint work with Alex Eskin, Amir Mohammadi and Alex Wright.
 Supplements


12:10 PM  01:30 PM


Lunch

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



01:30 PM  02:20 PM


Kinetic transport in quasicrystals
Andreas Strombergsson (Uppsala University)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
We study the dynamics of a point particle in an array of spherical scatterers centered at the points of a quasicrystal (of cutandproject type). It turns out that, as in the case of a periodic scatterer configuration, the stochastic process that governs the time evolution for random initial data in the limit of low scatterer density (BoltzmannGrad limit) is Markovian after an appropriate extension of the phase space. The definition and properties of the transition kernel for this limiting Markov process are more complicated in the quasicrystal case, and we will discuss this both in general and in specific examples. Homogeneous dynamics enters as a key tool in the proofs. This is joint work with Jens Marklof
 Supplements


02:30 PM  03:20 PM


Stiffness for Random Walks on Locally symmetric spaces
Alex Eskin (University of Chicago)

 Location
 SLMath: Eisenbud Auditorium
 Video

 Abstract
Let G be a semisimple Lie group, and \Gamma a lattice. in G. Let \mu be a measure on G. We show that under a certain condition on \mu, any \mustationary measure on G/\Gamma is in fact invariant under the group generated by the support of \mu. We give an alternative argument (which bypasses the Local Limit Theorem) for some of the breakthrough results of Benoist and Quint in this area. This is joint work with Elon Lindenstrauss
 Supplements


03:20 PM  03:50 PM


Tea & Goodbye

 Location
 SLMath: Atrium
 Video


 Abstract
 
 Supplements



