Activities This Week

A topological structure on a set enables one to give an exact meaning to the phrase “whenever $x$ is sufficiently near $a$, $x$ has the property $P(x)$”; we introduce a notion of a generalised topological space which enables one to give a similar exact meaning to the phase “every $n$-tuple of sufficiently similar points $x_1, x_2, \ldots, x_n$ has property $P (x_1,\dots, x_n)$” for $n > 1$. (Uniform spaces were introduced to do this for $n = 2$ and “similar” meaning “at small distance”, as explained in the introduction to (Bourbaki, General Topology).)

These spaces generalise uniform and topological spaces, filters, and simplicial sets, and the concept is designed to be flexible enough to formulate category-theoretically a number of standard basic elementary definitions in various fields, e.g. in analysis, limit, (uniform) continuity and convergence, equicontinuity of sequences of functions; in algebraic topology, being locally trivial and geometric realisation; in geometry, quasi-isomorphism; in model theory, stability, simplicity and several Shelah’s dividing lines in classification theory.

In the talk, I shall explain that convergence and homotopical triviality are defined by the same simplicial formula using the decalage (shift) endomorphism of the category of generalised topological spaces; that is, in some precise sense, a sequence $(a_i)_i$ converges to a point $a$ iff the associated map $i\mapsto a_i$ is homotopically equivalent to the constant map $i\mapsto a$, in the category of generalised topological spaces.

If time permits, I shall try to explain how a construction of geometric realisation by Amnon Besser (1998) is interpreted as defining an endomorphism of the category of generalised topological spaces. In short, the geometric realisation of a simplex is the space of upper semi-continuous maps $[0,1]\rightarrow [0<1<\cdots < N]$ with Levy-Prokhorov metric. Equivalently, the set of such maps is the $Hom$ of the simplicial sets $X.$ and $Y.$ represented by these linear orders $[0,1]$ and $[0<1<\cdots < N]$. Now, we define the endofunctor to be the inner Hom $Hom(X.,Y.)$ equipped with a certain extra structure depending functorially on $Y.$

The precise definition of the category of generalised topological space is simple enough to fit in the abstract: it is the category of simplicial objects of the category of filters on sets, or, equivalently, the category of finitely additive measures taking values $0$ and $1$ only. Thus a generalised topological space is a simplicial set equipped, for each $n ≥ 0$, with a filter on the set of $n$-simplices such that under any face or degeneration map the preimage of a large set is large.

The exact meaning we assign to the phase “every $n$-tuple of (sufficiently similar) points $x_1, x_2, \ldots, x_n$ has property $P(x_1,\ldots,x_n)$” for $n > 1$, is very much the same as done in topology: In a topological space, this exact meaning of “a property $P(x)$ holds for all points sufficiently near $a$” is that the set $\{x : P(x)\}$ belongs to the neighbourhood filter of a point $a$. Similarly, in a generalised topological space, it is that the set $\{(x_1, ..., x_n) : P (x_1, ..., x_n)\}$ belongs to the “neighbourhood” filter defined on $n$-simplices.

ההרצאה תתקיים לכבוד פרישתו לגמלאות של פרופ’ גרגורי משביצקי.

Stable processes is an important class of stochastic processes, including Gaussian processes, Cauchy processes and Levy processes. In an analogy to that the ergodicity of a Gaussian process is determined by the spectral measure, it was shown by Rosinski and Samorodnitsky that the ergodicity of a stationary symmetric stable process is characterized by its spectral representation. While this result was known when the process is indexed by $\mathbb{Z}^d$ or $\mathbb{R}^d$, the classical techniques fail when it comes to non-Abelian groups and it was an open question whether the ergodicity of such processes admits a similar characterization.

In this talk I will introduce the fundamentals of stable processes, the ergodic theory behind their spectral representation, and the key ideas of the characterization of the ergodicity for processes indexed by amenable groups. If time permits, I will mention recent results in non-singular ergodic theory that allow the constructions of weakly-mixing but not strongly-mixing stable processes indexed by many groups (Abelian groups, Heisenberg group).

In this talk, I will discuss a noncommutative (nc) analogue of the Gleason problem and its application in the “NC Cowen-Douglas” class. The Gleason problem was first studied by Andrew Gleason in studying the maximal ideals of a commutative Banach algebra. In particular, he showed that if the maximal ideal consisting of functions in the Banach algebra $\mathcal{A} ( \mathbb{B} ( 0, 1 ) )$ vanishing at the origin is finitely generated then it has to be generated by the coordinate functions where $\mathcal{A} ( \mathbb{B} ( 0, 1 ) )$ is the Banach algebra of holomorphic functions on the open unit ball $\mathbb{B} ( 0, 1 )$ at $0$ in $\mathbb{C}^n$ which can be continuously extended up to the boundary. The question – whether the maximal ideals in algebras of holomorphic functions are generated by the coordinate functions – has been named the Gleason problem. It turns out that the existence of a local solution of the Gleason problem in a reproducing kernel Hilbert space provides a sufficient condition for the membership of the tuple of adjoint of multiplication operators by coordinate functions in the Cowen-Douglas class.

After briefly discussing these classical aspects of the Gleason problem, I will introduce its nc counterpart for uniformly analytic nc functions and show that such a problem in the nc category is always locally uniquely solvable unlike the classical case. As an application one obtains a characterization of nc reproducing Hilbert spaces of uniformly analytic nc functions on a nc domain in $\mathbb{C}^d_{ \text{nc} }$ so that the adjoint of the $d$ - tuple of left multiplication operators by the nc coordinate functions are in the nc Cowen-Douglas class. Along the way, I will recall necessary materials from nc function theory.

This is a part of my ongoing work jointly with Professor Vinnikov on the nc Cowen-Douglas class.