Activities This Week

I will briefly discuss the general things that Magdalena Georgescu, Joav Orovitz, and I determined one needs to take into consideration for constructing inverse sequences of groupoids with Haar systems such that the pullback morphism induce a directed sequence of groupoid C*-algebras (to be clear, the groupoid C*-algebra of the inverse limit groupoid is the direct limit of the induced directed system of groupoid C*-algebras). Then I will proceed to discuss a variety of examples of how to create, in a simple way, groupoids whose groupoid C*-algebras are matrix algebras, UHF-algebras, infinite tensor powers of direct sums of such things, and dimension drop algebras $Z_{m,n}$ where $m$ and $n$ are natural or even supernatural numbers. I will briefly discuss my work with Atish Mitra on our current project for making the Jiang-Su algebra as a groupoid C*-algebra of an inverse limit groupoid (which, I believe is much more understandable and geometric than other groupoids which have Jiang-Su algebra as groupoid C*-algebra that show up in the literature). I will also discuss my project with Magdalena Georgescu on taking inverse limits of sigma-compact groupoids by second countable groupoids as a way to bootstrap known results about second countable groupoids to sigma-compact groupoids.

In the first part of the talk I briefly outline Hindman’s finite sums theorem, a famous Ramsey-theoretic result in algebra, its precursors and some generalizations, and the algebra of ultrafilters, a powerful technique providing a tool for getting similar results in combinatorics, algebra, and dynamics, see [1]. In the second (main) part I apply a multidimensional generalization of Hindman’s theorem (proved by Hindman and Bergelson) to show topologizability of certain algebras.

The topologization problem for groups and rings was first posed by Markov Jr. and then studied by various authors. I consider universal algebras consisting of an Abelian group and a family of additional operation (of arbitrary arity) distributive w.r.t. the group addition. Such algebras are called here polyrings; their instances include rings, modules, algebras over a field, differential rings, etc. Given a polyring $K$, a closed subbasis of the Zariski topology on the Cartesian product $K^n$ consists of finite unions of sets of roots of equations $t(x_1,\ldots,x_n)=0$ for all terms $t$ in $n$ variables.

The main theorem (a proof of which I plan to sketch) states that, for every infinite polyring $K$ and every $n>0$, sets definable by terms in $<n$ variables are nowhere dense in the space $K^n$. In particular, $K^n$ is nowhere dense in $K^{n+1}$. A fortiori, all the spaces $K^n$ are non-discrete (this fact was earlier stated by Arnautov for $K$ a ring and $n=1$). For more details, see [2].

References [1] N. Hindman, D. Strauss, Algebra in the Stone-Cech compactification: Theory and applications, W. de Gruyter, 2nd ed., 2012. [2] D. I. Saveliev, On Zariski topologies on polyrings, Russian Math. Surveys, 72:4 (2017), in press.

14:15-14:30-coffee break

14:30-14:40 Inna Entova

Title: Superalgebras and tensor categories

Abstract: I will briefly describe what are Lie superalgebras, and present some questions on their representations which have been studied in the last few years.

14:45-14:55 Daniel Berend

Title: Applied Probability.

Abstract: We will present an example of a problem in this area.

15:00-15:10 Shelomo Ben Abraham

Aperiodic tilings – an overflight

15:15-15:25 Yair Glasner

Title: A probabilistic Kesten theorem and counting periodic orbits in finite graphs.

Abstract: I will describe the notion of Invariant ransom subgroups and how we used it to give precise estimates on the asymptotic number of closed (non-backtracking) circuits in finite graphs.

15:30-15:40 Tom Meyerovitch

Title: Gibbs measures and Markov Random Fields

Abstract: From an abstract mathematical point of view, a Markov Random Field is a random function on the vertices of some (finite or countable) graph, with a certain conditional independence property. Every Gibbs measure (for a local interaction) is a Markov random Field. An old theorem due to Hammersley and Clifford establishes the converse, under some extra assumptions. I will present these notions and state some (slightly more) recent results and questions.

Coffee break

16:00-16:10 Doron Cohen

Title: Stochastic Processes and Quantum Chaos

Abstract: Our recent study considers the dynamics of stochastic and quantum models, in particular ring geometry: (a) with classical particles that perform random walk in disordered environment; (b) with quantum Bose particles whose dynamics is coherent. One theme that arises in both cases is the Anderson-type localization of the eigenstates.

16:15-16:25 Ilan Hirshberg

Title: C*-dynamical systems and crossed products. Abstract: I’ll briefly say a few words on what the words above mean, and loosely what kinds of problems I tend to look at.

16:30-16:40 Victor Vinnikov Noncommutative Function Theory

One of my main interests in recent years have been in developing a theory of functions of several noncommuting variables. It turns out, following the pioneering ideas of Joseph L. Taylor in the early 1970s, that such functions can be naturally viewed as functions on tuples of square matrices of all sizes that satisfy certain compatibility conditions as we vary the size of matrices. Noncommutative functions are related, among other things, to the theory of operator spaces (including such topics as complete positivity and matrix convexity) and to free probability.

Some other topics that I am interested in, and that I can discuss in case of interest, are function theory on the unit ball and on the polydisc in C^n and related operator theory, line and vector bundles on compact Riemann surface, especially theta functions and Cauchy kernels, determinantal representations of algebraic varieties, and various topics on convexity in real algebraic geometry related to hyperbolic polynomials.

16:45-16:55 David Eichler

Vortex-based, zero conflict routing in networks

Abstract: A novel approach is suggested for reducing traffic conflict in 2D spatial networks. Intersections without primary conflicts are defined as zero traffic conflict (ZTC) designs. A provably complete classification of maximal ZTC designs is presented. It is shown that there are 9 four-way and 3 three-way maximal ZTC intersection designs, to within mirror, rotation, and arrow reversal symmetry. Vortices are used to design networks where all or most intersections are ZTC. Increases in average travel distance, relative to unrestricted intersecting flow, are modest, and represent a worthwhile cost of reducing traffic conflict.

Singular spaces appear everywhere. And the singularity is often non-isolated, i.e. the singular locus is of positive dimension. These non-isolated singularities are more complicated and less studied.

Let X be a variety with singular locus Z, the simplest example being the surface {y^2=x^2z}. Generically along Z the singularity “factorizes”, i.e. X is locally at each point the product: (the germ of Z)\times (the germ of space with an isolated singularity).

But at some special points of Z the picture degenerates and the family of sections of X, transversal to Z, becomes not equi-singular (in whichever sense). These points form the discriminant of transversal singularity type. We study this discriminant, assuming X,Z are locally complete intersections and X is of “ordinary type” generically along Z.

First I will define the discriminant, as a subscheme of Z, and formulate its properties. This discriminant is a (effective) Cartier divisor in Z, nef but not necessarily ample, with nice pullback/pushforward properties under some maps. The discriminant deforms flatly under some deformations of X.

Then I will give the formula for the class of this discriminant in the cohomology/Chow group/Picard group of Z. This class “counts the number of points” where the transversal type degenerates as one travels along the singular locus. In most cases this class is not zero (when Z is complete or projective). This places a “topological” obstruction to the naive expectation (from differential geometry): “In the generic case the transversal type does not degenerate”.

The talk is based on arXiv:1705.11013 and arXiv:1308.6045.