פעילויות השבוע

The Hardy space $H^2(\mathbb{D})$ is the Hilbert space of analytic functions on the unit disc with square summable Taylor coefficients is a fundamental object both in function theory and in operator algebras. The operator of multiplication by the coordinate function turns $H^2(\mathbb{D})$ into a module over the polynomial ring $\mathbb{C}[z]$. Moreover, this space is universal, in the sense that whenever we have a Hilbert module $\mathcal{H}$ over $\mathbb{C}[z]$, such that $z$ acts by a pure row contraction, we have that $\cH$ is a quotient of several copies of $H^2(\mathbb{D})$ by a submodule.

There are two multivariable generalizations of this property, one commutative and one free. I will show why the free generalization is in several ways the correct one. We will then discuss quotients of the noncommutative Hardy space and their associated universal operator algebras. Each such quotient naturally gives rise to a noncommutative analytic variety and it is a natural question to what extent does the geometric data determine the operator algebraic one. I will provide several answers to this question.

Only basic familiarity with operators on Hilbert spaces and complex analysis is assumed.

In this talk we will discuss a type of visibility problem (in Euclidean spaces), with an infinite, discrete, set of obstacles. A dense forest refers to a discrete point set Y that satisfies dist(L,Y)=0 for every ray L in $R^d$, and moreover, the distance between Y and every line segment decays uniformly, as the length of the segments tend to infinity. The constructions of dense forests that are known today were given using tools from Diophantine approximations (Bishop+Peres), homogeneous dynamics (Solomon-Weiss), Fourier analysis (Adiceam), the Lovász local lemma (Alon), and more tools form number theory and dynamics (Adiceam-Solomon-Weiss). We will discuss some of these constructions of dense forests, as well as the speed of which the visibility decays in them. Some of the results that I will discuss come from a joint work with Faustin Adiceam and with Barak Weiss.

Tropicalizations and tropical reductions provide a convenient tool to control degenerations of algebraic objects. Roughly speaking, a tropicalization is a piecewise linear object, associated to an algebraic object over a non-Archimedean field, that contains essential information about one of its integral models. The tropical reduction is then the reduction of the model over the residue field. For applications, it is often important not only to describe the tropicalization process, but also to be able to decide whether something that looks like the tropicalization and the tropical reduction comes from an algebraic object. Such statements are called lifting theorems. Tropical techniques have been applied successfully to a number of problems in algebraic geometry, such as enumerative questions, dimension estimates, descriptions of compactifications etc. In particular, in a recent work of Bainbridge, Chen, Gendron, Grushevsky, and Moeller, a tropical approach was used to describe a new compactification of the space of smooth curves with differentials (although the authors don’t use this terminology). The proofs of BCGGM rely on transcendental techniques. In my talk, I will present a modified version of BCGGM tropicalization, and will indicate an algebraic proof of the main result. The talk is based on a joint work with M.Temkin.