| DATE |
SPEAKER
|
TOPIC
|
| February
11 |
|
No
Meeting
|
| February
18 |
Alex
Gorodnik
University of
Bristol/Princeton University |
Regularity
of conjugacy for actions of large
groups
It is well know that a small perturbation of an Anosov map is
topologically conjugate to the original map,
but the conjugacy is not smooth in general. We prove that for actions
of "large" (e.g., Zariski dense)
groups topological conjugacy is smooth. This is a joint work with
Hitchman and Spatzier. |
| February
25 |
|
No
Meeting
|
| March
3 |
Amir
Mohammadi
Yale
University |
Unipotent
flows in positive characteristic
Study of dynamics of action of unipotent subgroups on homogeneous
spaces and its
applications to Diophantine approximations has been attracting
considerable
attention over the past 40 years or so. Margulis's celebrated proof of
Oppenheim conjecture and Ratner's seminal work on Raghunathan's
conjecture are
two inspiring works in the subject. Although Raghunathan's
conjectures in characteristic zero have been settled affirmatively,
very little
is known in positive characteristic case. In this talk we will address
this
issue. In particular the main focus will be on a recent joint work with
M.
Einsiedler on classification of joinings for the action of certain
unipotent
subgroups. As it turns out this has some applications to
quasi-isometries of
lattices. This connection is drawn explicit by K. Wortman in an
appendix to our
work. |
| March
10 |
Alireza
Salehi Golsefidy
Princeton
University |
Translates
of horospherical measures and
counting problems.
(Joint with A. Mohammadi) In this talk, I will briefly explain the
relation between some of the counting problems, mixing, and ergodic
theory. The counting problems might be of geometric or number theoretic
nature.
For instance consider V=G/H a homogeneous variety, and one would like
to study the integer or rational points on V. Eskin, Mozes, and Shah
attacked this problem via unipotents flows. However they had to assume
that H is maximal and reductive (in particular not inside any parabolic
subgroup of G.) I will explain an ergodic theoretic approach toward
such problem for a flag variety.
For a geometric example, consider SL(n,Z)-translates of a horosphere in
the symmetric space of SL(n,R). Question is how many of them intersect
a ball of radius R. In fact, Eskin and McMullen answered this question
for n=2, using mixing. I will explain why mixing is not enough and how
one can get such a result for any n.
I will show that the main ingredient for both of the mentioned
questions is understanding the limits of translates of horospherical
measures, i.e. the probability measure supported on U SL(n,Z)/SL(n,Z),
where U is the set of upper triangular unipotent matrices. |
| March
24 |
|
No
meeting
|
| March
31 |
Danijela
Damjanovic
Harvard
University |
Local
rigidity for some rank two algebraic
abelian actions
I will discuss local rigidity for some rank-two actions: partially
hyperbolic on SL(n, R)/L, for n>3; and parabolic on SL(2,
R)XSL(2,
R)/L. |
| April
7 |
Kariane
Calta
Vassar College
|
Translation
Surfaces and Notions of Periodicity
I will provide a brief introduction to translation surfaces, which are
surfaces that can be constructed by gluing finitely many polygons in
R^2 along parallel edges to form a closed surface with cone points. I
will discuss the geometric notion of complete periodicity as it relates
to the classification of lattice translation surfaces in genus two
given by Calta and McMullen. Then I will introduce the notion of
algebraic periodicity which generalizes that of complete periodicity
and discuss recent results of Calta and Smillie related to algebraic
periodicity. |
| April
14 |
Speaker
University
|
No
meeting
|
| April
21 |
Francois
Maucourant
Rennes 1
University |
Two
examples of nonhomogeneous orbit closures.
We will explain how to construct orbits of non-homogeneous closure for
some subgroup of the diagonal group
acting on the space of lattices of dimension at least 6. Also, a
similar example for the action of *2,*3 on the four dimensionnal torus
is discussed. |
| April
28 |
Alex
Furman
University of
Illinois at Chicago |
L^1-Measure
Equivalence, Simplicial volume,
and rigidity of hyperbolic lattices.
I will report on a recent work with Uri Bader and Roman Sauer on Orbit
Equivalence (or Measure Equivalence) of lattices in SO(n,1),
n>2.
Measure Equivalence (ME) is a rather weak equivalence relation between
groups, which may be viewed as a measurable analogue of Quasi-Isometry.
This concept is closely related to questions of orbit structures of
group actions in Ergodic Theory, and has connections to von Neumann
algebras and Descriptive set theory. Most existing rigidity results are
related to higher rank phenomena, while most ME-invariants of groups
(such as amenability, property (T), L^2-Betti numbers) involve unitary
representations. In this work we consider a restricted notion of ME,
which seems to be prevalent among actions of non-amenable groups. In
this context we prove rigidity results for all lattices in SO(n,1), and
show that in the wider class of fundamental groups of negatively curved
manifolds, the dimension and the simplicial volume (a topological
invariant) are also preserved by L^1-OE. |