Colloquium Ical Atom Mailing List

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.

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.

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.

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.

Joint work with Pierre Simon. I will present a criterion that ensures that Aut(M) has a 2-generated dense subgroup when M is a countable structure (which holds in many examples), and discuss related subjects.

I will describe a link between tame geometry and diophantine geometry that has been unfolding in the past decade following the fundamental theorem of Pila-Wilkie in the theory of o-minimal structures. In particular I will describe how this theorem has been used in proofs of the Manin-Mumford conjecture (by Pila-Zannier), the Andre-Oort conjecture for modular curves (by Pila) and many other questions of “unlikely intersections” in diophantine geometry. I will then discuss questions related to effectivity of the Pila-Wilkie theorem and its implications for the diophantine applications. In particular I will discuss our recent proof (joint with Novikov) of the restricted form of Wilkie’s conjecture, and more recent results on effectivity for the larger class of semi-Noetherian sets.

14:15-14:30-coffee break

14:30-14:40 Inna Entova

Title: Superalgebras and tensor categories

Abstract: I will briefly describe what are Lie superalgebras, and present some questions on their representations which have been studied in the last few years.

14:45-14:55 Daniel Berend

Title: Applied Probability.

Abstract: We will present an example of a problem in this area.

15:00-15:10 Shelomo Ben Abraham

Aperiodic tilings – an overflight

15:15-15:25 Yair Glasner

Title: A probabilistic Kesten theorem and counting periodic orbits in finite graphs.

Abstract: I will describe the notion of Invariant ransom subgroups and how we used it to give precise estimates on the asymptotic number of closed (non-backtracking) circuits in finite graphs.

15:30-15:40 Tom Meyerovitch

Title: Gibbs measures and Markov Random Fields

Abstract: From an abstract mathematical point of view, a Markov Random Field is a random function on the vertices of some (finite or countable) graph, with a certain conditional independence property. Every Gibbs measure (for a local interaction) is a Markov random Field. An old theorem due to Hammersley and Clifford establishes the converse, under some extra assumptions. I will present these notions and state some (slightly more) recent results and questions.

Coffee break

16:00-16:10 Doron Cohen

Title: Stochastic Processes and Quantum Chaos

Abstract: Our recent study considers the dynamics of stochastic and quantum models, in particular ring geometry: (a) with classical particles that perform random walk in disordered environment; (b) with quantum Bose particles whose dynamics is coherent. One theme that arises in both cases is the Anderson-type localization of the eigenstates.

16:15-16:25 Ilan Hirshberg

Title: C*-dynamical systems and crossed products. Abstract: I’ll briefly say a few words on what the words above mean, and loosely what kinds of problems I tend to look at.

16:30-16:40 Victor Vinnikov Noncommutative Function Theory

One of my main interests in recent years have been in developing a theory of functions of several noncommuting variables. It turns out, following the pioneering ideas of Joseph L. Taylor in the early 1970s, that such functions can be naturally viewed as functions on tuples of square matrices of all sizes that satisfy certain compatibility conditions as we vary the size of matrices. Noncommutative functions are related, among other things, to the theory of operator spaces (including such topics as complete positivity and matrix convexity) and to free probability.

Some other topics that I am interested in, and that I can discuss in case of interest, are function theory on the unit ball and on the polydisc in C^n and related operator theory, line and vector bundles on compact Riemann surface, especially theta functions and Cauchy kernels, determinantal representations of algebraic varieties, and various topics on convexity in real algebraic geometry related to hyperbolic polynomials.

16:45-16:55 David Eichler

Vortex-based, zero conflict routing in networks

Abstract: A novel approach is suggested for reducing traffic conflict in 2D spatial networks. Intersections without primary conflicts are defined as zero traffic conflict (ZTC) designs. A provably complete classification of maximal ZTC designs is presented. It is shown that there are 9 four-way and 3 three-way maximal ZTC intersection designs, to within mirror, rotation, and arrow reversal symmetry. Vortices are used to design networks where all or most intersections are ZTC. Increases in average travel distance, relative to unrestricted intersecting flow, are modest, and represent a worthwhile cost of reducing traffic conflict.

Problems pertaining to approximation and their applications have been extensively studied in the theory of convex bodies. In this talk we discuss several such problems, and focus on their extension to the realm of measures. In particular, we discuss variations of problems concerning the approximation of convex bodies by polytopes with a given number of vertices. This is done by introducing a natural construction of convex sets from Borel measures. We provide several estimates concerning these problems, and describe an application to bounding certain average norms.

Based on joint work with Han Huang

A family of lines through the origin in Euclidean space is called equiangular if any pair of lines defines the same angle. The problem of estimating the maximum cardinality of such a family in $R^n$ was extensively studied for the last 70 years. Answering a question of Lemmens and Seidel from 1973, in this talk we show that for every fixed angle$\theta$ and sufficiently large $n$ there are at most $2n-2$ lines in$R^n$ with common angle $\theta$. Moreover, this is achievable only when $\theta =\arccos \frac{1}{3}$. Various extensions of this result to the more general settings of lines with $k$ fixed angles and of spherical codes will be discussed as well. Joint work with I. Balla, F. Drexler and P. Keevash.

A Gaussian stationary process is a random function f:R–>R or f:C–>C, whose distribution is invariant under real shifts, and whose evaluation at any finite number of points is a centered Gaussian random vector. The mathematical study of these random functions goes back at least 75 years, with pioneering works by Kac, Rice and Wiener. Nonethelss, many basic questions about them, such as the fluctuations of their number of zeroes, or the probability of having no zeroes in a large region, remained unanswered for many years.

In this talk, we will give an introduction to Gaussian stationary processes, and describe how a spectral perspective combined with tools from harmonic, real and complex analysis, yields new results about such long-lasting questions.

Borel’s Fixed Point Theorem states that a solvable connected algebraic group G acting on a non-empty complete variety V must have a fixed point. Thus, if V consists of subgroups of G, and G acts on V by conjugation, then some subgroup in V is normal in G.

Although G is infinite or trivial here, we can use the method of proof to obtain applications to finite p-groups. We plan to discuss some applications and some open problems. No previous knowledge of algebraic groups is needed.

I will discuss the programme to understand conformal field theory via subfactors and twisted equivariant K-theory. This has also resulted in a better understanding of the double of the Haagerup subfactor, which was previously thought to be exotic and un-related to known models.