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:

Second part of the talk from the previous week.

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).

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. I will begin by describing why Jean-Paul formulated this prob-

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.

I will discuss the inverse Monge-Ampere flow and its applications to the existence, and non-existence, of Kahler-Einstein metrics. To motivate this discussion I will first describe the classical theory of the Donaldson heat flow on a Riemann surface, and its relationship with the Harder-Narasimhan filtration of an unstable vector bundle.

Abstract: The classical theory of metric Diophantine approximation is very well developed and has, in recent years, seen significant advances, partly due to connections with homogeneous dynamics. Several problems in this subject can be viewed as particular examples of a very general setup, that of lattice actions on homogeneous varieties of semisimple groups. The latter setup presents significant challenges, including but not limited to, the non-abelian nature of the objects under study.

The recent work of Abe--Henniart--Herzig--Vigneras gives a classification of irreducible admissible mod-$p$ representations of a $p$-adic reductive group in terms of supersingular representations. However, supersingular representations remain mysterious largely, and in general we know them very little. Up to date, there are only a classification of them for the group $GL_2 (Q_p)$ and a few other closely related cases.

I will review the Kostant-Souriau geometric quantization procedure for passing from functions on a symplectic manifold (classical observables) to operators on a Hilbert space (quantum observables). With the "half-form correction" that is required in this procedure, one cannot quantize a complex projective space of even complex dimension, and one cannot equivariantly quantize the two-sphere nor any symplectic toric manifold. I will present a geometric quantization procedure that uses metaplectic-c structures to incorporate the half-form correction into the earlier

We will give a short review of various topics discussed in the first semester and last Monday and then we'll pick the fruits: namely, we will show how to get groups which are no approximated.

A random linear (binary) code is a dimension lamba*n (0