We present natural families of coordinate algebras of noncommutative Euclidean spaces and noncommutative products of Euclidean spaces. These coordinate algebras are quadratic ones associated with an R-matrix which is involutive and satisfies the quantum Yang–Baxter equation. As a consequence they enjoy a list of nice properties, being regular of finite global dimension. Notably, we have spherical manifolds, and noncommutative quaternionic planes as well as noncommutative quaternionic tori. On these there is an action of the classical quaternionic torus SU(2)×SU(2) in parallel with the action of the torus U(1)×U(1) on a complex noncommutative torus.
Abstract

The irrational rotation algebra is known to be self-dual in a KK-theoretic sense. The required K-homology fundamental class was constructed by Connes out of the Dolbeault operator on the 2-torus, but there has not been an explicit description of the dual element. In this talk, I will geometrically construct that K-theory class by using a pair of transverse Kronecker flows on the 2-torus.This is based on joint work with Heath Emerson (University of Victoria).
Abstract

In recent years, Gromov proposed studying the geometry of positive scalar curvature (abbreviated by "psc") via various metric inequalities. In particular, he proposed the following conjecture: Let $M$ be a closed manifold which does not admit a metric of psc. Then for any Riemannian metric on $V = M \times [-1,1]$ of scalar curvature $\geq n(n-1)$ the estimate $d(\partial_- V, \partial_+ V) \leq 2\pi/n$ holds, where $\partial_\pm V = M \times \{\pm 1\}$ and $n = \dim V$. Previously, Rosenberg and Stolz conjectured similarly that if $M$ does not admit psc, then $M \times \mathbb{R}$ does not admit a complete metric of psc and $M \times \mathbb{R}^2$ does not admit a complete metric of uniformly psc. In this talk, we will discuss a new geometric phenomenon consisting of a precise quantitative interplay between distance estimates and scalar curvature bounds which underlies these three conjectures. We will explain that this phenomenon arises if $M$ admits an obstruction to psc using the index theory of Dirac operators.
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