The seminar meets on Tuesdays, 14:30-15:30, in Math -101

This Week


Alex Lubotzky (Hebrew University)

First order rigidity of high-rank arithmetic groups

The family of high rank arithmetic groups is class of groups which is playing an important role in various areas of mathematics. It includes SL(n,Z), for n>2 , SL(n, Z[1/p] ) for n>1, their finite index subgroups and many more. A number of remarkable results about them have been proven including; Mostow rigidity, Margulis Super rigidity and the Quasi-isometric rigidity. We will talk about a new type of rigidity : “first order rigidity”. Namely if D is such a non-uniform characteristic zero arithmetic group and E a finitely generated group which is elementary equivalent to it ( i.e., the same first order theory in the sense of model theory) then E is isomorphic to D. This stands in contrast with Zlil Sela’s remarkable work which implies that the free groups, surface groups and hyperbolic groups ( many of which are low-rank arithmetic groups) have many non isomorphic finitely generated groups which are elementary equivalent to them. Joint work with Nir Avni and Chen Meiri.


Fall 2017 meetings

Upcoming Meetings

Date
Title
Speaker
Abstract
Nov 21 First order rigidity of high-rank arithmetic groups Alex Lubotzky (Hebrew University)

The family of high rank arithmetic groups is class of groups which is playing an important role in various areas of mathematics. It includes SL(n,Z), for n>2 , SL(n, Z[1/p] ) for n>1, their finite index subgroups and many more. A number of remarkable results about them have been proven including; Mostow rigidity, Margulis Super rigidity and the Quasi-isometric rigidity. We will talk about a new type of rigidity : “first order rigidity”. Namely if D is such a non-uniform characteristic zero arithmetic group and E a finitely generated group which is elementary equivalent to it ( i.e., the same first order theory in the sense of model theory) then E is isomorphic to D. This stands in contrast with Zlil Sela’s remarkable work which implies that the free groups, surface groups and hyperbolic groups ( many of which are low-rank arithmetic groups) have many non isomorphic finitely generated groups which are elementary equivalent to them. Joint work with Nir Avni and Chen Meiri.

Nov 28 TBA Itay Kaplan (Hebrew University)
Dec 5 TBA Gal Binyamini (Weizmann)
Dec 12, 14:00–17:00 TBA Math-Physics meeting
Dec 19 TBA Boaz Slomka (University of Michigan)
Jan 2 TBA Naomi Feldheim (Weizmann Institute)
Jan 9 An analogues of Borel’s Fixed Point Theorem for finite p-groups George Glaubermann (University of Chicago)
Jan 16 TBA David Evans (Cardiff University)

Past Meetings

Date
Title
Speaker
Abstract
Oct 24 Holography of traversing flows and its applications to the inverse scattering problems Gabriel Katz (MIT)

We study the non-vanishing gradient-like vector fields $v$ on smooth compact manifolds $X$ with boundary. We call such fields traversing.

With the help of a boundary generic field $v$, we divide the boundary $\d X$ of $X$ into two complementary compact manifolds, $\d^+X(v)$ and $\d^-X(v)$. Then we introduce the causality map $C_v: \d^+X(v) \to \d^-X(v)$, a distant relative of the Poincare return map.

Let $\mathcal F(v)$ denote the oriented 1-dimensional foliation on $X$, produced by a traversing $v$-flow.

Our main result, the Holography Theorem, claims that, for boundary generic traversing vector fields $v$, the knowledge of the causality map $C_v$ is allows for a reconstruction of the pair $(X, \mathcal F(v))$, up to a homeomorphism $\Phi: X \to X$ which is the identity on the boundary $\d X$. In other words, for a massive class of ODE’s, we show that the topology of their solutions, satisfying a given boundary value problem, is rigid. We call these results ``holographic” since the $(n+1)$-dimensional $X$ and the un-parameterized dynamics of the flow on it are captured by a single correspondence $C_v$ between two $n$-dimensional screens, $\d^+X(v)$ and $\d^-X(v)$.

This holography of traversing flows has numerous applications to the dynamics of general flows. Time permitting, we will discuss some applications of the Holography Theorem to the geodesic flows and the inverse scattering problems on Riemannian manifolds with boundary.

Oct 31 TBA Faculty Meeting (no colloquium)
Nov 7 Path connectedness of the space of hyperbolic ergodic measures Yakov Pesin (Penn State University)

In 1977 Sigmund proved that the space of ergodic measures supported on a basic set of an Axiom A diffeomorphism is path connected. In the talk I will describe a substantial generalization of this result to the space of hyperbolic ergodic measures supported on an isolated homoclinic class of a general diffeomorphism. Such homoclinic classes should be viewed as basic structural elements of any dynamics. Examples will be discussed. This is a joint work with A. Gorodetsky.

Nov 14 First steps of the symplectic function theory Michael Entov (Technion)

Symplectic function theory studies the space of smooth (compactly supported) functions on a symplectic manifold. This space is equipped with two structures: the Poisson bracket and the $C^0$ (or uniform) norm of the functions. Interestingly enough, while the Poisson bracket of two functions depends on their first derivatives, there are non-trivial restrictions on its behavior with respect to the $C^0$-norm. We will discuss various results of this kind and their applications to Hamiltonian dynamics.

Seminar run by Dr Michael Brandenbursky