Oct

27

Fri

Speaker: Les Wilson (UHM)

Title: Singularity Theory

Abstract: Singularity Theory studies singular phenomena in various fields: singular points of sets (where the set is not locally a manifold), singular points of functions (where the partial derivatives are all 0) or of mappings (where the Jacobian matrix is not maximal rank), singular points of vector fields or differential forms (where it is zero), singular points of geometric structures, etc. I will give examples, discuss common techniques (e.g. stability, genericity, finite determinacy, bifurcation), and some areas I’m particularly interested in.

Oct

31

Tue

Title : Harmonic functions on Sierpinski carpets

Abstract : I will discuss a notion of Sobolev spaces and harmonic functions on Sierpinski carpets, which differs from the classical approach of potential theory in metric measure spaces. The goal is to define a notion that takes into account also the ambient space, where the carpet lives. As an application of carpet-harmonic functions we obtain a quasisymmetric uniformization result for Sierpinski carpets.

Nov

14

Tue

Title : A dichotomy for groupoid C*-algebras.

Abstract : Notions of paradoxical decompositions appear in the work of Hausdorff, Banach, and Tarski who showed that a discrete group satisfies the amenable/paradoxical divide. In this talk we study paradoxical phenomena in the field of operator algebras; directing our focus to C*-algebras arising from dynamical systems, graphs, and groupoids. Like Tarski, we use the type semigroup construction to move from non-paradoxicality to the existence of means or traces. These semi-groups witness the stably finite/purely infinite nature of the corresponding C*-algebras.

Nov

21

Tue

Title : Poincaré type inequalities on Hamming cube via martingale inequalities.

Abstract : Harmonic analysis is intimately related with martingale estimates.

But there is another type of discrete analysis, namely, harmonic analysis on Hamming cube (the math. foundation of Big Data science) that seemed to be disjoint from this relationship. We show how many classical (and some new) estimates on Hamming cube follow from martingale estimates. We also show why this is related to solving certain non-linear PDE of Monge–Ampère type and with classical inequalities in Gaussian spaces.

## University of Hawaiʻi at Mānoa