Activities This Week

Given a Z^d shift of finite type and a nearest-neighbour interaction, we present sufficient conditions for efficient approximation of pressure and, in particular, topological entropy. Among these conditions, we introduce a combinatorial analog of the measure-theoretic property of Gibbs measures known as strong spatial mixing and we show that it implies many desirable properties in the context of symbolic dynamics. Next, we apply our results to some classical 2-dimensional statistical mechanics models such as the (ferromagnetic) Potts, (multi-type) Widom-Rowlinson, and hard-core lattice gas models for certain subsets of both the subcritical and supercritical regimes. The approximation techniques make use of a special representation theorem for pressure that may be of independent interest.

Part of this talk is joint work with Stefan Adams, Brian Marcus, and Ronnie Pavlov.

For every finite ordered graph $H$ there is a natural number $k(H)>1$ such that whenever all copies of $H$ in the ordered inverse limit of all finite ordered graphs are partitions to finitely many Borel parts, then there is a (closed) copy of the inverse limit graph in itself whose copies of $H$ meet at most $k(H)$ many parts.

The probability that a random ordered graph on $n$ vertices satisfies $k(H)=1$ tends to 1 as $n$ grows.

Joint work with S. Geschke and S. Huber.

While the topic of geometric incidences has existed for several decades, in recent years it has been experiencing a renaissance due to the introduction of new polynomial methods. This progress involves a variety of new results and techniques, and also interactions with fields such as algebraic geometry and harmonic analysis.

A simple example of an incidences problem: Given a set of n points and set of n lines, both in R^2, what is the maximum number of point-line pairs such that the point is on the line. Studying incidence problems often involves the uncovering of hidden structure and symmetries.

In this talk we introduce and survey the topic of geometric incidences, focusing on the recent polynomial techniques and results (some by the speaker). We will see how various algebraic and analysis tools can be used to solve such combinatorial problems.

Perhaps the simplest dilation result in operator theory is the dilation of an isometry to a unitary. However, when one generalizes an isometry to a Toeplitz-Cuntz-Krieger family of a directed graph, things become much more complicated.

The analogue of a unitary operator in this case is a (full) Cuntz-Krieger family, and a result of Skalski and Zacharias on C-correspondences supplies us with a such a dilation when the graph is row-finite and sourceless. We apply Arveson’s non-commutative Choquet theory to answer this question for arbitrary graphs. We compute the non-commutative Choquet boundary of graph tensor algebras and are able to recover a result of Katsoulis and Kribs on the computation of the C-envelope of these algebras.

However, as the non-commutative Choquet boundary of the operator algebra is a more delicate information than the C*-envelope, we are able to dilate any TCK family to a (full) CK family. In fact, we are able to make progress on a decade old problem of Skalski and Zacharias, that asks for the multivariable analogue, generalizing a result of Ito’s dilation theorem. More precisely, we are able to show that TCK families of graphs $G_1,…,G_d$ that commute according to a higher rank row-finite sourceless directed graph have CK dilations that still commute according to the higher rank graph structure.

פונקצית זטא של רימן היא אחת הפונקציות המרתקות במתמטיקה. למשל, היא יודעת את כל מה שאפשר לדעת על התפלגות הראשוניים, למרות שבינתיים היא לא מגלה לנו. בהרצאה נספר קצת על פונקצית זטא, תכונותיה, ועל הכללות שלה בכיוונים שונים.