In the past decades There has been considerable interest in the probability that two random elements of (finite or certain infinite)
groups commute.
I will describe new works (by myself and by others) on probabilistically nilpotent groups, namely groups in which the probability that [x_1,...,x_k]=1 is positive/bounded away from zero.
It turns out that, under some natural conditions,
these are exactly the groups which have a finite/bounded index
subgroup which is nilpotent of class < k.
The proofs have some combinatorial flavor.

In this talk we recall Conlon's random construction of sparse 2-dim simplicial complexes arising from Cayley graphs of F_2^t . We check what expansion properties this construction has (and doesn't have): Mixing of random walks, Spectral gap of the 1-skeleton, Spectral gap of the links, Co-systolic expansion and the geometric overlap property.

Let X be the spherical building associated to the group G=GL(n,F) ,
where F is a finite field. We will survey some results on the homology of X with constant and twisted coefficients, and on the corresponding expansion properties.

In this talk we shall review a paper by Gromov and Guth, in which they introduced several ways to measure the geometric complexity of an embedding of simplicial complexes to Euclidean spaces.
One such measurement is strongly related to the notion of high dimensional expanders introduced by Gromov, and in fact, it is based on a paper of Kolmogorov and Barzadin from 1967, in which the notion of an expander graph appeared implicitly.
We shall show one application of bounded degree high dimensional expanders, and present many more open questions arising from the above mentioned paper.

Locally testable codes are error-correcting codes that admit
super-efficient checking procedures. In the first part of the talk, we will
see why expander based codes are NOT locally testable. This is in contrast
to typical "good" error correcting properties which follow from expansion.
We will then see that despite this disconnect between expansion and
testability, all known construction of locally testable codes follow from
the high-dimensional expansion property of a related complex leaving open

Computing R=P.Q ,the product of two mXm Boolean matrices [BMM] is an ingredient
of many combinatorial algorithms.
Many efforts were made to speed it beyond the standard m^3 steps, without using
the algebraic multiplication.
To divide the computation task, encoding of the rows and column indices were
used (1.1) j by (j1,j2) k by (k1,k2)
e.g. using integer p j2=j mod p ,j1=ceiling of j/p.
Clearly, the product of the ranges of the digits= m1.m2 - is approximately m.

The set of real numbers is often identified with
Cantor Space 2^omega, with which it shares many important
properties: not only the cardinality, but also other
"cardinal characteristics" such as cov(null), the smallest
number of measure zero sets needed to cover the whole space,
and similarly cov(meager), where meager="first category";
or their "dual" versions non(meager) (the smallest
cardinality of a nonmeager set) and non(null).

Many ZFC results and consistency results (such as

Suppose that a countable group $G$ acts on a compact metric space $X$ and that $S \subset G$ is a semigroup not containing the identity of $G$. If every continuous function $f$ on $X$ is contained in the closed algebra generated by $\{sf : s \in S\}$, the action is said to be $S$-predictable.
In this talk, we consider the following question due to Hochman:
When $G$ is amenable, does $S$-predictability imply zero topological entropy?
To provide an affirmative answer, we introduce the notion of a random invariant order.

09:00 - 10:50 Noga Ron-Zewi, "Locally testable codes"

14:00 - 14:50 Dorit Aharonov, "Quantum error correcting codes"
​15:00 - 15:50 Dorit Aharonov, " Quantum Locally Testable codes and High dimensional expansion"



Abstract for Noga Ron-Zewi's talk:

In 1970, Furstenberg made a number of conjectures about the expansions of real numbers in non-comensurable bases, e.g. bases 2 and 3. The most difficult remains wide open, but several related problems, which can be stated in terms of the dimension theory of certain fractal sets, were recently settled. In the first talk I will try to describe the conjectures and some of what was known. In the second talk I will present Meng Wu's proof of the "slice conjecture" (it was also proved independently by Pablo Shmerkin, and I will try to also say a little about that proof too).

Second part of the talk from the previous week.

Both talks will be given by Oren Becker.
9:00 - 10:50
Title: Proving stability via hyperfiniteness, graph limits and invariant random subgroups

Abstract: We will discuss stability in permutations, mostly in the context of amenable groups. We will characterize stable groups among amenable groups in terms of their invariant random subgroups. Then, we will introduce graph limits and hyperfinite graphings (and some theorems about them), and show how the aforementioned characterization of stability follows.

14:00 - 16:00

Second part of the talk from last week:
An ergodic system (X;B; μ; T) is said to have the weak Pinsker
property if for any ε > 0 one can express the system as the direct
product of two systems with the first having entropy less than ε and
the second one being isomorphic to a Bernoulli system. The problem
as to whether or not this property holds for all systems was open for
more than forty years and has been recently settled in the affirmative
in a remarkable work by Tim Austin.

Let $\Sigma$ be a Riemann surface of genus $g \geq 2$, and p be a point on $\Sigma$. We define a space $S_g(t)$ consisting of certain irreducible representations of the fundamental group of $\Sigma \setminus p$, modulo conjugation by SU(n). This space has interpretations in algebraic geometry, gauge theory and topological quantum field theory; in particular if Σ has a Kahler structure then $S_g(t)$ is the moduli space of parabolic vector bundles of rank n over Σ.