Consider a simple random walk on $\mathbb{Z}$ with a random coloring of $\mathbb{Z}$.
Look at the sequence of the first $N$ steps taken and colors of the visited locations.
From it, you can deduce the coloring of approximately $\sqrt{N}$ integers.
Suppose an adversary may change $\delta N$ entries in that sequence. What can be deduced now?
We show that for any $\theta<0.5,p>0$, there are $N_{0},\delta_{0}$
such that if $N>N_{0}$ and $\delta<\delta_{0}$ then with probability $>1-p$ we can reconstruct
the coloring of $>N^{\theta}$ integers.
In recent years, topological dynamics have become an important tool in model theory. I will talk about some topological dynamical results from my PhD thesis about the so-called group-like equivalence relations. I plan to give a glimpse of the motivations in model theory (mostly related to the model-theoretic Galois groups and connected components of definable groups) and to show some ideas of the proofs.
I will briefly recall the required notions from topological dynamics. Some knowledge of model theory will help to understand the motivations, but otherwise, it will not be necessary.
Expander graphs have been a topic of great interest in the last 50 years for mathematicians and computer scientists. In recent years a high dimensional theory is emerging. We will describe some of its main directions and questions.
The Mass Transport Principle is a useful technique that was introduced to the study of automorphism-invariant percolations by Häggström in 1997. The technique is a sort of mass conservation principle, that allows us to relate random properties (such as the random degree of a vertex) to geometric properties of the graph.
I will introduce the principle and the class of unimodular graphs on which it holds, as well as a few of its applications.
Abstract:
In 1934, Loewner proved a remarkable and deep theorem about matrix monotone functions. Recently, the young Finnish mathematician, Otte Heinävarra settled a 10 year old conjecture and found a 2 page proof of a theorem in Loewner theory whose only prior proof was 35 pages. I will describe his proof and use that as an excuse to discuss matrix monotone and matrix convex functions including, if time allows, my own recent proof of Loewner’s original theorem.
Symbolic dynamics is a tool that simplifies the study of dynamical systems in various aspects. It is known for almost fifty years that uniformly hyperbolic systems have ``good'' codings. For non-uniformly hyperbolic systems, Sarig constructed in 2013 ``good'' codings for surface diffeomorphisms. In this talk we will discuss some recent developments on Sarig's theory, when the map has discountinuities and/or critical points, such as multimodal maps of the interval and Bunimovich billiards.
Let X be a stationary Z^d-process. We say that X is a factor of an i.i.d. process if there is a (deterministic and translation-invariant) way to construct a realization of X from i.i.d. variables associated to the sites of Z^d. That is, if there is an i.i.d. process Y and a measurable map F from the underlying space of Y to that of X, which commutes with translations of Z^d and satisfies that F(Y)=X in distribution. Such a factor is called finitary if, in order to determine the value of X at a given site, one only needs to look at a finite (but random) region of Y.
Abstract: We will discuss the main steps in the proof of the theorem stating that if (G,+, ...) is a strongly minimal expansion of a group interpretable in an o-minimal expansion of a field, and \dim(G)=2 then G is a pure algebraic group.
Joint work with P. Eleftheriou and Y. Peterzil.
