Homotopy Theory

Speaker: Dan Christensen (Western)

"Models of (homotopy) type theory"

Time: 14:00

Room: MC 107

I will describe what it means to give a model of type theory, and give some examples of models, including some that arise from Quillen model categories.

Comprehensive Exam Presentation

Speaker: Somnath Chakraborty (Western)

"Etale cohomology of Classifying Simplicial Scheme of Affine Group Schemes, with torsion coefficients"

Time: 14:00

Room: MC 107

We consider the category of etale sheaves on a simplicial scheme, define etale cohomology in terms of derived functors, and then mention some spectral sequences connecting Cech and etale cohomology. We consider the model structure on the category of simplicial sheaves on an arbitrary Grothendieck site C , and for a fibrant simplicial sheaf X, and a sheaf of abelian groups F, we consider morphisms to the Eilenberg-MacLane constructions K(F; n), in Ho(C ). For a linear algebraic group scheme G, we show that this is the sheaf cohomology arising out of the model structure coincides with etale cohomology. Finally, we mention about the etale cohomology of BGLn(k), and carry out the calculation when n = 1 for any finite extension of Q.

Noncommutative Geometry

Speaker: Branimir Cacic (Texas A&M)

"An introduction to strict deformation quantisation"

Time: 14:30

Room: MC 106

In deformation quantisation, one passes from a classical system to a quantum system by deforming the commutative algebra of classical observables into a noncommutative algebra of quantum observables in a manner consistent with the correspondence principle. In the presence of a smooth action of a compact abelian Lie group $G$ (e.g., a torus), one can rigorously effect deformation quantisation through a method, due to Rieffel, called strict deformation quantisation. In this talk, I'll give an introduction to strict deformation quantisation of Frechet $G$-pre-$C^\ast$-algebras, together with the corresponding deformation of $G$-equivariant finitely generated projective modules and $\ast$-representations. If time permits, I'll also introduce Connes--Landi deformation, which, as was observed by Sitarz and Varilly, is precisely strict deformation quantisation for $G$-equivariant spectral triples.

Noncommutative Geometry

Speaker: Branimir Cacic (Texas A&M)

"A reconstruction theorem for noncommutative G-manifolds"

Time: 13:00

Room: MC 106

Just as one can construct the noncommutative $2$-torus as the strict deformation quantisation of the commutative $2$-torus along the translation action, so too can one more generally construct noncommutative $G$-manifolds, namely, strict deformation quantisations of commutative spectral triples along the action of a compact abelian Lie group $G$. I will propose an abstract definition of noncommutative $G$-manifold, analogous to the definition of commutative spectral triple, and show that the deformation of an abstract noncommutative $G$-manifold with deformation parameter $\theta \in H^2(\hat{G},\mathbb{T})$ by a class $\theta^\prime \in H^2(\hat{G},\mathbb{T})$ yields an abstract noncommutative $G$-manifold with deformation parameter $\theta+\theta^\prime$; combined with Connes's reconstruction theorem for commutative spectral triples, this yields the analogue of Connes's reconstruction theorem for noncommutative $G$-manifolds. If time permits, I will also discuss a Pontrjagin-dual version of the Connes--Dubois-Violette splitting homomorphism and use it to show that sufficiently well-behaved rational noncommutative $\mathbb{T}^N$-manifolds are, in fact, almost-commutative.