Seminars

A product system is a measurable field of Hilbert spaces (indexed over a semigroup), along with a compatible product rule. Product systems were introduced by William Arveson in his study of E0- semigroups over [0, ∞). He showed that the product system associated with an E0- semigroup is a complete invariant, and proceeded to prove seminal results concerning one parameter E0- semigroups using their product systems. Later, product systems (and E0- semigroups) over semigroups other than [0, ∞) were studied, especially those over convex cones. In this talk, will discuss some results on multiparameter product systems (product systems over closed, convex cones in R d ). This talk is divided into two parts. In the first part, we discuss multiparameter decomposable product systems. Roughly, a product system is said to be decomposable, if it possesses enough left coherent sections to generate the product system. We will first describe the structure of a multiparameter decomposable product system. Then, we will illustrate these structural results in a particular case. CAR flows associated to isometric representations of cones form basic exam- ples of E0- semigroups. In the second part of the talk, we characterise isometric representations of a cone, , with commuting range projections, whose asso- ciated CAR flow is type I. Moreover, we compute the index and gauge group (certain invariants associated to a product system/ E0- semigroups) for this class of CAR flows. This talk is based on the work with S. Sundar, and is the thesis defense talk. The Zoom link is given below. https://zoom.us/j/98206472628 Meeting ID: 982 0647 2628 Passcode: 560304

I will discuss a problem in uniform multiplicative diophantine approximation, motivated by the celebrated Littlewood Conjecture on the simultaneous rational approximation of two real numbers. I will explain how this uniform-multiplicative problem connects to homogeneous dynamics and ergodic theory, and then present some recent results that estimate the size (in terms of the Lebesgue measure) of the corresponding Diophantine sets.

Discrete homotopy theory is a homotopy theory designed for studying graphs, detecting combinatorial (rather than topological) holes. Central to this theory are the discrete homotopy groups, defined using maps out of grids of suitable dimensions. Of these, the discrete fundamental group in particular has found applications in various areas of mathematics, including matroid theory, subspace arrangements, and topological data analysis. In this talk, based on joint work with Chris Kapulkin, we introduce the discrete fundamental groupoid, a multi-object generalization of the discrete fundamental group, and use it as a starting point to develop some robust computational techniques. A new notion of covering graphs allows us to extend the existing theory of universal covers to all graphs, and to prove a classification theorem for coverings. We also prove a discrete version of the Seifert?van Kampen theorem, generalizing a previous result of H. Barcelo et al. We then use it to solve the realization problem for the discrete fundamental group through a purely combinatorial construction. Currently, a central open problem in the field is to determine whether the cubical nerve functor, which associates a cubical Kan complex to a graph is a DK-equivalence of relative categories. If true, this would allow the import of important results like the Blakers-Massey theorem from classical homotopy theory to the discrete realm. We propose a new line of attack, by breaking it into more tractable problems comparing the homotopy theories of the respective $n$-types, for each integer $n ≥ 0$. We also solve this problem for the first nontrivial case, $n = 1$.