I will show that the universal cover of the commutative classifying space has the homotopy type of a wedge of spheres when $G$ is an extraspecial $p$-group. Commutative classifying space is a certain subspace of the usual classifying space whose set of $n$-simplices is given by the set of commuting $n$-tuples in $G$. Its universal cover can be described as a partially ordered set obtained from the collection of abelian subgroups of $G$. Such objects are closely related to Tits buildings associated to algebraic groups. I will also mention about possible applications of such objects in quantum computation.

The notion of irreducibility in the conventional Zariski topology is too coarse for real algebraic sets. For example, the hyperbola xy=1 is an irreducible algebraic set which is not even connected in the Euclidean topology. Cartan umbrella is another irreducible algebraic set, which is connected, but decomposes into the union of a 'sheet' and a 'stick.' Inspired by Nash and his notion of 'sheets,' one might require then to distinguish, as called by Kurdyka, the 'rigid pieces' of a real algebraic set. We will review different approaches to defining irreducibility and irreducible components in real algebraic geometry, in which more 'good' functions than just polynomials should be considered. We may work with the ring of Nash functions, continuous rational functions, or, most notably, arc-analytic functions. Nash functions are able to detect two components for the hyperbola xy=1. Continuous rational functions are able to detect a sheet and a stick component for Cartan umbrella. But we find examples in which only the phenomenon called 'arc-symmetricity' can among the others realize a decomposition.

Speaker's web page: http://www.math.uwo.ca/index.php/profile/59/

Let $X$ be a quasicategory. Then it is an $\infty$-topos, if it is an accessible left exact localization of the presheaf category of a small quasicategory. We will introduce two sets of intrinsic conditions which are equivalent to being an $\infty$-topos:

1) the $\infty$-categorical Giraud axioms, and

2) colimits in $X$ are universal, and it has small object classifiers for large enough regular cardinals,

and we discuss the equivalences.

Let H be the C*-algebra of a non-trivial compact quantum group acting freely on a unital C*-algebra A. Baum, Dabrowski and Hajac conjectured that there does not exist an equivariant *-homomorphism from A to the equivariant noncommutative join C*-algebra A*H. When A is the C*-algebra of functions on a sphere, and H is the C*-algebra of functions on Z/2Z acting antipodally on the sphere, then the conjecture becomes the celebrated Borsuk-Ulam theorem. Recently, Chirvasitu and Passer proved the conjecture when H is commutative. The main goal of this talk is to show how to extend the Chirvasitu-Passer result to a far more general setting assuming only that H admits a character different from the counit. Also, assuming that our compact quantum group is a q-deformation of a compact connected semisimple Lie group, we prove that there exists a finite-dimensional representation of the compact quantum group such that, for any C*-algebra A admitting a character, the finitely generated projective module associated with A*H via this representation is not stably free. (Based on joint work with L. Dabrowski and S. Neshveyev.)

We explain the structure of the reduced group C∗-algebras of complex semisimple quantum groups, and discuss a connection to the Baum-Connes assembly map for classical complex groups.

This talk is based on my joint paper with Masoud Khalkhali and Ali Moatadelro (arXiv:1610.04740). First I will recall Gilkey’s theorem on asymptotic expansion of heat kernels for the special case of Laplacians. I will also introduce the noncommutatvie 3-torus (NCT3) and then I will conformally perturb the standard volume form on it. Then the corresponding perturbed Laplacian will be discussed, and using Connes’ pseudodifferential calculus, I will define the scalar curvature of NCT3. Finally, introducing a rearrangement lemma I will compute an explicit formula for the scalar curvature of the curved noncommutative 3-torus.