Mar 20

On pointwise periodicity and expansiveness

Tom Meyerovitch (BGU)

Following Kaul, a discrete (topological) group G of transformations of set
X is pointwise periodic if the stabilizer of every point is of finite index (cocompact) in G.
Equivalently, all Gorbits are finite (compact).
Generalizing a result of Montgomery, Kaul showed in the early 70’s that a
pointwise periodic transformation group is always compact when the group acts (faithfully) on a connected
manifold without boundary.
I will discuss implications of expansiveness and pointwise periodicity
of certain groups and semigroups of transformations.
In particular I’ll state implications for cellular automata and for planner tilings.
Based on joint work with Ville Salo.

Mar 27, 11:10–12:00

Derivative Algebras and Topological Conjugacies Between Cellular Automata

Jeremias Epperlein (BGU)

Topologial conjugacy is most probably the most natural notion of
isomorphism for topological dynamical systems. Classifying subshifts
of finite type up to topological conjugacy is a notoriously hard
problem with a long history of results. Much less is known about the
corresponding problem for
endomorphisms of subshifts of finite type (aka cellular automata).
I will discuss necessary and sufficient criteria under which periodic
cellular automata are topologically conjugate.
The main tool will be derivative algebras in the sense of Tarski and
McKinsey, an algebraic
structure based on the CantorBendixson derivative.

Apr 10

Discontinuity of the phase transition for the planar randomcluster and Potts models with q > 4

Matan Harel (Tel Aviv University)

The randomcluster model is a dependent percolation model where the weight of a configuration is proportional to q to the power of the number of connected components. It is highly related to the ferromagnetic qPotts model, where every vertex is assigned one of q colors, and monochromatic neighbors are encouraged. Through nonrigorous means, Baxter showed that the phase transition is firstorder whenever q > 4  i.e. there are multiple Gibbs measures at criticality. We provide a rigorous proof of this claim. Like Baxter, our proof uses the correspondence between the above models and the sixvertex model, which we analyze using the Bethe ansatz and transfer matrix techniques. We also prove Baxter’s formula for the correlation length of the models at criticality. This is joint work with Hugo DuminilCopin, Maxime Gangebin, Ioan Manolescu, and Vincent Tassion.

Apr 17

Irreducibility of random polynomials

Gady Kozma (Weizmann Institute)

Examine a polynomial with random, independent coefficients, uniform between 1 and 210. We show that it is irreducible over the integers with probability going to one as the degree goes to infinity. Joint work with Lior BarySoroker.

Apr 24

A powerlaw upper bound on the decay of correlations in the twodimensional randomfield Ising model

Ron Peled (Tel Aviv University)

The randomfield Ising model (RFIM) is a standard model for a disordered magnetic system, obtained by placing the standard ferromagnetic Ising model in a random external magnetic field. ImryMa (1975) predicted, and AizenmanWehr (1989) proved, that the twodimensional RFIM has a unique Gibbs state at any positive intensity of the random field and at all temperatures. Thus, the addition of an arbitrarily weak random field suffices to destroy the famed phase transition of the twodimensional Ising model. We study quantitative features of this phenomenon, bounding the decay rate of the effect of boundary conditions on the magnetization in finite systems. This is known to decay exponentially fast for a strong random field. The main new result is a powerlaw upper bound which is valid at all field strengths and at all temperatures, including zero. Our analysis proceeds through a streamlined and quantified version of the AizenmanWehr proof. Several open problems will be mentioned.
Joint work with Michael Aizenman.

Mon, Apr 30, In 101

Good lower bounds for multiple recurrence

Sebastián Donoso (Universidad de O’Higgins)

In 2005, Bergelson, Host and Kra showed that if $(X,\mu,T)$ is an ergodic measure preserving system and $A\subset X$, then for every $\epsilon>0$ there exists a syndetic set of $n\in\mathbb{N}$ such that
$\mu(A\cap T^{n}A\cap\dots\cap T^{kn}A)>\mu^{k+1}(A)\epsilon$ for
$k\leq3$, extending Khintchine’s theorem. This phenomenon is called multiple recurrence with good lower bounds.
Good lower bounds for certain polynomial expressions was studied by
Frantzikinakis but several questions remain open.
In this talk I will survey this topic, and present some progress regarding
polynomial expressions, commuting transformations,
and configurations involving the prime numbers.
This is work in progress with Joel Moreira, Ahn Le and Wenbo Sun.

May 1

Limit theorems for a counting process with extendable dead time (Type II counter)

Chen Dubi (BGU)

Measuring occurrence times of random events, aimed to determine the statistical properties of the governing stochastic process, is a basic topic in science and engineering, and has been the topic of numerous mathematical modeling techniques. Often, the true statistical properties of the random process deviate from the measured properties due to the so called “dead time” phenomenon, defined as a time period after a reaction in which the detection system is not operational. From a mathematical point of view, the dead time can be interpreted as a rarefied series of the original time series, obtained by removing all events which are within the dead time period inflicted by previous events.
When the waiting times between consecutive events form a series of
independent identically distributed random variables, a natural setting for analyzing the distribution of the number of event or the event counter is a renewal process. In particular, for high rate measurements (or, equivalently, large measurement time), the limit distribution of the counter is well understood, and can be described directly through the first two moments of the waiting time between consecutive events.
In the talk we will discuss limit theorems for counters with paralyzing dead time (type II counter), expressed directly through the probability density function of the waiting time between consecutive events. This is done by writing explicit formulas for the for the first and second moments of a waiting time distribution between consecutive events in the rarefied process, in terms of the probability density function of the waiting of the original process.

May 8

The tadic Littlewood conjecture is false

Erez Nesharim (University of York)

The Littlewood and the padic Littlewood conjectures are famous open problems on the border between number theory and dynamics. In a joint work with Faustin Adiceam and Fred Lunnon we show that the analogue of the padic Littlewood conjecture over $F_3((1/t))$ is false. The counterexample is given by the Laurent series whose coefficients are the regular paper folding sequence, and the method of proof is by reduction to the non vanishing of certain Hankel determinants. The proof is computer assisted and it uses substitution tilings of $Z^2$ and a generalisation of Dodson’s condensation algorithm for computing the determinant of any Hankel matrix.

May 22

Harmonic functions on locally compact groups

Idan Perl (BGU)

Spaces of harmonic functions on a given group have a strong relationship with its largescale geometry. Classically, mostly bounded harmonic functions have been studied. We review some results about bounded harmonic functions and present the recent research on spaces of unbounded harmonic functions.

Jun 5

Equidistribution of Iwasawa components of lattices and asymptotic properties of primitive points

Tal Horesh (IHES)

I will discuss the equidistribution of certain parameters of primitive integral points in Euclidean space, as their norms tend to infinity.
These parameters include directions of integral points on the unit sphere, the integral grids in their orthogonal hyperplanes, and the shortest solutions to their associated gcd equations.
These equidistribution statements follow from counting lattice points in the real Special Linear group.

Jun 12

Kakutani’s splitting procedure for multiscale substitution schemes

Yotam Smilansky

In 1975, S. Kakutani introduced a splitting procedure which generates a sequence of partitions of the unit interval [0,1], and showed that this sequence is uniformly distributed in [0,1]. We present generalizations of this procedure in higher dimensions, which correspond to constructions used when defining substitution and multiscale substitution tilings of Euclidean space. We prove uniform distribution of these sequences of partitions using new path counting results on graphs and establish Kakutani’s result as a special case.

Jun 19

The MultiLane Totally Asymmetric Simple Exclusion Process

Ofer Busani (Bar Ilan)

The Totally Asymmetric Simple Exclusion Process (TASEP) is a wellstudied model where one assumes every site on Z to be either occupied by a particle or vacant. Each site has a Poisson clock attached to it, if the clock rings for site x, where there happens to be a particle, the particle makes a jump to site x+1 if it is vacant. The TASEP is often used to model traffic on a one lane road.
In this work we generalize this model to a finite number of lanes where cars can move from one lane to another at different rates, and having different speed on each lane. We consider the problem of finding the stationary measures for this model as well as its hydrodynamics (what would the traffic look like from the point of view of a helicopter).
The talk will be as selfcontained as possible. Joint work with Gidi Amir, Christoph Bahadoran and Ellen Saada.
