### Alan Carey’s 60th Birthday Conference

#### Posted by Urs Schreiber

This week at MPI Bonn is (or has been) taking place a conference in honor of Alan Carey’s 60th birthday. on “noncommutative geometry and index theory, statistical models, geometric issues in quantum field theory. Hamiltonian anomalies and bundle n-gerbes”.

I missed most of it, but am on my way now for the last day at least.

This abstract here sounds interesting:

Speaker: **Christian BÃ¤r (Potsdam)**

Title: *Renormalized integrals and a path integral representation of the heat kernel*

Abstract: We propose a concept of renormalized integrals which generalizes integrals on measure spaces. This is a suitable framework for defining path integrals without having to construct an actual measure on path space. We show how this works for the heat kernel of a Laplace type operator on a compact Riemannian manifold. The hope is that this approach can circumvent problems with the standard approach using Wiener measure when one wants to deal with the SchrÃ¶dinger equation instead of the heat equation.

On the following we once had a little bit of discussion here on the blog. I am hoping to learn more from the talk:

Speaker: **Dennis Borisov (Yale)**

Title: *Higher dimensional infinitesimal groupoids of manifolds*

Abstract: The construction (by Kapranov) of the space of infinitesimal paths on a manifold is extended to include higher dimensional infinitesimal objects, encoding contractions of infinitesimal loops. This full infinitesimal groupoid is shown to have the algebra of polyvector fields as its non-linear cohomology.

Am also very glad to see former colleague Danny Stevenson. Naturally, he speaks about nonabelian cohomology represented by gerbes. In – let’s seee – 15 minutes! I better run now, just arrived at the station in Bonn only…

## Nonabelian sheaf cohomology

In the first part of his talk Danny Stevenson gave a nice introduction to how the abelian sheaf cohomology and the nonabelian cohomology of a topological space $X$ are all unified into the intrinsic cohomology of the $(\infty,1)$-topos $\mathbf{H} = Sh_{(\infty,1)}(X)$ (or its hypercompletion) over that space, as described at $n$Lab: cohomology.

He has been working for some time now on results that show how this intrinsic cohomology of $\mathbf{H}$ relates to that back in $\infty Grpd \simeq Top$. Not sure if he will make it to that in his talk, but roughly the statement is that for $\mathbf{B}A \in \mathbf{H}$ the $(\infty,1)$-sheaf induced by a suitably well behaved topological $\infty$-group $A$, and for $\mathcal{B}|A| \in Top$ its geometric realization, the corresponding cohomologies computed in $\mathbf{H}$ coincide with those computed in $Top$ in that $H(X,A) := \pi_0 \mathbf{H}(X= *, \mathbf{B}A) \simeq \pi_0 Top(X,\mathcal{B}|A|)$.

(This is not in the talk, though, I see now, hopefully I am allowed to say it here anyways.)

This generalizes the corresponding reesult by Baez-Stevenson and Baas, Kro et al. which we discussed here at some point, and which is the restriction of the above to the case that $A$ is a topological 2-group, in which case the cohomology in question classifies $A$-principal 2-bundles on $X$, or equivalently generalized $A$-gerbes on $X$.

Danny approaches this by representing $A$ as a simplicial topological group and looking at its Moore complex.