Feb 02, 2015
Monday

09:00 AM  09:15 AM


Welcome

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
 Supplements



09:15 AM  10:30 AM


Homogeneous flows and the statistics of directions
Jens Marklof (University of Bristol)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Over the past three decades, the ergodic theory for flows on homogeneous spaces has produced many beautiful applications in number theory, geometry and mathematical physics. In these lectures I will provide an introduction to equidistribution theorems for large spheres and horospheres, and explain their very recent application to directional statistics for lattices, quasicrystals and other aperiodic point sets. These lectures are introductory and aimed at a nonspecialist mathematical audience. Much of the material is based on joint work with A. Strombergsson, D. ElBaz and I. Vinogradov.
 Supplements


10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Introduction to Ratner's Theorems on Unipotent Flows
Dave Morris (University of Lethbridge)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Let f be the obvious covering map from Euclidean nspace to the ntorus. It is well known that if L is any straight line in nspace, then the closure of f(L) is a very nice submanifold of the ntorus. In 1990, Marina Ratner proved a beautiful generalization of this observation that replaces Euclidean space with any Lie group G, and allows L to be any subgroup of G that is ``unipotent.'' We will discuss the statement of this theorem and related results, some of the ideas in the proofs, and a few of the important consequences.
 Supplements


12:00 PM  01:30 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



01:30 PM  02:30 PM


Masser’s conjecture on equivalence of integral quadratic forms
Han Li (Wesleyan University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
A classical problem in the theory of quadratic forms is to decide whether two given integral quadratic forms are equivalent. That is, whether their symmetric integral matrices A and B satisfy A=X’BX for some unimodular integral matrix X. For definite forms one can construct a simple decision procedure. Somewhat surprisingly, no such procedure was known for indefinite forms until the work of C. L. Siegel in the early 1970s. In the late 1990s, D. W. Masser made the following conjecture related to this problem: Let n be at least 3, and suppose A, B are equivalent. Then there exists a unimodular integral matrix X such that A=X’BX and X< C(A+B)^k, where the constants C, k depend only on the dimension n. In this talk we shall discuss our recent resolution of this conjecture based on a joint work with Professor Gregory A. Margulis.
 Supplements


02:40 PM  03:25 PM


Homogeneous flows and the statistics of directions
Jens Marklof (University of Bristol)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Over the past three decades, the ergodic theory for flows on homogeneous spaces has produced many beautiful applications in number theory, geometry and mathematical physics. In these lectures I will provide an introduction to equidistribution theorems for large spheres and horospheres, and explain their very recent application to directional statistics for lattices, quasicrystals and other aperiodic point sets. These lectures are introductory and aimed at a nonspecialist mathematical audience. Much of the material is based on joint work with A. Strombergsson, D. ElBaz and I. Vinogradov.
 Supplements


03:25 PM  03:55 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:55 PM  04:55 PM


Exponential decay of matrix coefficients.
Hee Oh (Yale University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
For a simple Lie group G and a discrete subgroup Gamma of G, we will discuss exponential decay of the matrix coefficients of G associated to functions on the quotient space Gamma\G. When G is of rank one, and Gamma is a lattice, this yields exponential mixing of the geodesic flow for the associated Riemannian manifold. We will also discuss a uniform exponential decay for congruence subgroups of Gamma when Gamma is contained in an arithmetic subgroup of G.
 Supplements



Feb 03, 2015
Tuesday

09:00 AM  10:00 AM


Homogeneous flows and the statistics of directions
Jens Marklof (University of Bristol)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Over the past three decades, the ergodic theory for flows on homogeneous spaces has produced many beautiful applications in number theory, geometry and mathematical physics. In these lectures I will provide an introduction to equidistribution theorems for large spheres and horospheres, and explain their very recent application to directional statistics for lattices, quasicrystals and other aperiodic point sets. These lectures are introductory and aimed at a nonspecialist mathematical audience. Much of the material is based on joint work with A. Strombergsson, D. ElBaz and I. Vinogradov.
 Supplements


10:00 AM  10:30 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



10:30 AM  11:30 AM


Introduction to Ratner's Theorems on Unipotent Flows
Dave Morris (University of Lethbridge)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Let f be the obvious covering map from Euclidean nspace to the ntorus. It is well known that if L is any straight line in nspace, then the closure of f(L) is a very nice submanifold of the ntorus. In 1990, Marina Ratner proved a beautiful generalization of this observation that replaces Euclidean space with any Lie group G, and allows L to be any subgroup of G that is ``unipotent.'' We will discuss the statement of this theorem and related results, some of the ideas in the proofs, and a few of the important consequences.
 Supplements


11:40 AM  12:40 PM


Quadratic Weyl Sums, Automorphic Functions, and Invariance Principles
Francesco Cellarosi (University of Illinois at UrbanaChampaign)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Hardy and Littlewood's approximate functional equation for quadratic Weyl sums (theta sums) provides, by iterative application, a powerful tool for the asymptotic analysis of such sums. The classical Jacobi theta function, on the other hand, satisfies an exact functional equation, and extends to an automorphic function on the Jacobi group. We construct a related, almost everywhere nondifferentiable automorphic function, which approximates quadratic Weyl sums up to an error of order one, uniformly in the summation range. This not only implies the approximate functional equation, but allows us to replace Hardy and Littlewood's renormalization approach by the dynamics of a certain homogeneous flow. The great advantage of this construction is that the approximation is global, i.e., there is no need to keep track of the error terms accumulating in an iterative procedure. Our main application is a new functional limit theorem, or invariance principle, for theta sums. The interesting observation here is that the paths of the limiting process share a number of key features with Brownian motion (scale invariance, invariance under time inversion, nondifferentiability), although time increments are not independent and the value distribution at each fixed time is distinctly different from a normal distribution. Joint work with Jens Marklof.
 Supplements


12:40 PM  02:10 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:10 PM  03:10 PM


Rigidity of higher rank diagonalisable actions
Manfred Einsiedler (ETH Zürich)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We will discuss some of the rigidity phenomena for higher rank diagonalisable actions on homogeneous spaces. In particular the measure rigidity results for higher rank actions under the positive entropy assumption.
 Supplements


03:10 PM  03:40 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:40 PM  04:40 PM


Exponential decay of matrix coefficients
Hee Oh (Yale University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
For a simple Lie group G and a discrete subgroup Gamma of G, we will discuss exponential decay of the matrix coefficients of G associated to functions on the quotient space Gamma\G. When G is of rank one, and Gamma is a lattice, this yields exponential mixing of the geodesic flow for the associated Riemannian manifold. We will also discuss a uniform exponential decay for congruence subgroups of Gamma when Gamma is contained in an arithmetic subgroup of G.
 Supplements


04:40 PM  06:20 PM


Reception

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements




Feb 04, 2015
Wednesday

08:40 AM  10:10 AM


Semigroups in semisimple groups
Yves Benoist (Centre National de la Recherche Scientifique (CNRS))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We will first explain three properties of a Zariski dense subsemigroups of a semisimple real Lie group:
 the existence of loxodromic elements,
 the convexity and non degeneracy of the limit cone,
 the density of the group spanned by the Jordan projection.
We will then explain how to deduce from them three limit theorems for a random walk on this semigroup:
 the law of large numbers,
 the central limit theorem,
 the local limit theorem.
 Supplements


10:10 AM  10:40 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



10:40 AM  11:40 AM


Introduction to Ratner's Theorems on Unipotent Flows
Dave Morris (University of Lethbridge)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Let f be the obvious covering map from Euclidean nspace to the ntorus. It is well known that if L is any straight line in nspace, then the closure of f(L) is a very nice submanifold of the ntorus. In 1990, Marina Ratner proved a beautiful generalization of this observation that replaces Euclidean space with any Lie group G, and allows L to be any subgroup of G that is ``unipotent.'' We will discuss the statement of this theorem and related results, some of the ideas in the proofs, and a few of the important consequences.
 Supplements


11:45 AM  12:45 PM


Unipotent flows on infinite volume manifolds
Amir Mohammadi (University of California, San Diego)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We will discuss the action of unipotent subgroups of G=SL(2,R), SL(2,C) on homogeneous spaces G/\Gamma where \Gamma is a discrete, Zariski dense subgroup of G. The focus will be on the case when \Gamma is not a lattice; some of the similarities and contrasts between this case and the lattice case will be described.
 Supplements



Feb 05, 2015
Thursday

08:40 AM  10:10 AM


Semigroups in semisimple groups
Yves Benoist (Centre National de la Recherche Scientifique (CNRS))

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We will first explain three properties of a Zariski dense subsemigroups of a semisimple real Lie group:
 the existence of loxodromic elements,
 the convexity and non degeneracy of the limit cone,
 the density of the group spanned by the Jordan projection.
We will then explain how to deduce from them three limit theorems for a random walk on this semigroup:
 the law of large numbers,
 the central limit theorem,
 the local limit theorem.
 Supplements


10:10 AM  10:40 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



10:40 AM  11:40 AM


Unipotent flows on infinite volume manifolds
Amir Mohammadi (University of California, San Diego)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
 
 Supplements


11:45 AM  12:45 PM


Unipotent flows and Lfunctions
Akshay Venkatesh (Institute for Advanced Study)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
 Supplements



12:45 PM  02:15 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:15 PM  03:15 PM


Geometric and arithmetic aspects of bounded orbits on homogeneous spaces
Dmitry Kleinbock (Brandeis University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
This will be a case study of a particular theme in homogeneous dynamics which will highlight connections with Diophantine approximation. I will start with a historical survey and then will mention some new work, joint with Jinpeng An and Lifan Guan
 Supplements


03:15 PM  03:45 PM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:45 PM  04:45 PM


Unipotent flows and quadratic forms (after Linnik)
Akshay Venkatesh (Institute for Advanced Study)

 Location
 MSRI: Simons Auditorium
 Video


 Abstract
 
 Supplements



Feb 06, 2015
Friday

08:40 AM  09:40 AM


Grids with dense values
Uri Shapira (TechnionIsrael Institute of Technology)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The talk will be concerned with a property of lattices in the geometry of numbers which I refer to as "almost sure gridDV with respect to a function F". The goal of the talk will be threefold:
(1) Discuss a theorem relating this property to the dynamics of the lattice under the invariance group of the function.
(2) Apply the theorem to various classical discussions and see what it gives (it relates nicely to Minkowski's conjecture and to diophantine exponents).
(3) Discuss a mistake I have in the published paper and some open problems
 Supplements


09:40 AM  10:10 AM


Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



10:10 AM  11:10 AM


Thin groups: arithmetic and geometric viewpoints
Elena Fuchs (University of California, Davis)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
In the past decade, substantial progress has been made in several beautiful arithmetic problems connected to thin groups: Apollonian packings, Zaremba's conjecture, and so on. This progress was made possible by a series of developments, in particular the Affine Sieve of BourgainGamburdSarnak from a decade ago. In this talk we will focus on the arithmetic of Apollonian packings, and then move on to consider thin groups in their own right, apart from arithmetic applications. Specifically, we will discuss how hyperbolic geometric methods have shed light on thinness of certain monodromy groups of hypergeometric equations, and how one might go about distinguishing thin groups from their nonthin counterparts in general.
 Supplements


11:20 AM  12:20 PM


Martin boundary and local limit theorem of Brownian motion on negativelycurved manifolds
Seonhee Lim (Seoul National University)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Let $p(t,x,y)$ be the heat kernel on the universal cover $\widetilde{M}$ of a compact Riemannian manifold of negative curvature. We show that $$C(x,y) = \lim_{t \to \infty} e^{\lambda_0 t} t^{3/2} p(t,x,y) $$ is a positive function depending only on $x,y \in \widetilde{M}$, where $\lambda_0$ is the bottom of the spectrum. The function $C(x,y)$ can be described in terms of a PattersonSullivan density on $\partial \widetilde{M}$. We also show that $\lambda_0$Martin boundary of $\widetilde{M}$ coincides with its topological boundary. We will explain how Margulis argument for counting geodesics as well as a uniform version of Dolgopyat's rapidmixing of the geodesic flow are used to prove the results. This is a joint work with François Ledrappier.
 Supplements


12:20 PM  01:50 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



01:50 PM  02:50 PM


Diophantine approximation on group varieties
Anish Ghosh (Tata Institute of Fundamental Research)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will discuss the problem of Diophantine approximation on group varieties and outline a strategy to establish analogues of classical results in Diophantine approximation in this setting. Specifically, I will talk about the problem of counting solutions to Diophantine inequalities on group varieties and its connection to spectral gap. This is joint work with Alexander Gorodnik and Amos Nevo.
 Supplements


