The Infinite Seminar

The blog is in quite a summer lull, and I’m not doing near my share of keeping it up. I’m blaming my vacation and my marriage ceremony tomorrow.

Anyway, here is a question I was thinking about on my way home. About appropriate terminology for my own research. I’m putting quite a bit of thought into calculation of $A_{\mathrm{\infty }}$-structures on Ext-algebras in cases where the usual calculation methods - Homotopy perturbation theory, Merkulov’s method, et.c. - do not really work well since the required data about the endomorphism ring of the appropriate chain complex ends up being much too large. So far I have been calling this local computation, but it struck me that it might end up confusing those more used to local being used for .. say .. localization in various contexts.

On the way I thought about blind computation, to indicate the lack of information compared to the more global methods, but this doesn’t seem to be quite it either.

Thus a question to the readers (and writers?) of this blog: what would be a good word to describe my particular brand of computation of $A_{\mathrm{\infty }}$-structures on $H_{*}(A)$, for $A$ a DG-algebra which gets viewed as a black box, capable of performing calculations, but not of displaying its internals in any good way?

Some of the recent preprints seen in the arXiv mailings include:

Mikael Johansson: A partial $A_{\mathrm{\infty }}$-structure on the cohomology of $C_n\times C_m$

Suppose k is a field of characteristic 2, and n,m≥2 powers of 2. Then the $A_{\mathrm{\infty }}$-structure of the group cohomology algebras $H^{*}(C_n,k)$ and $H^{*}(C_m,k)$ are well known. We give results characterizing an $A_{\mathrm{\infty }}$-structure on $H^{*}(C_n\times C_m,k)$ including limits on non-vanishing low-arity operations and an infinite family of non-vanishing higher operations.

Alastair Hamilton & Andrey Lazarev: Cohomology theories for homotopy algebras and noncommutative geometry

This paper builds a general framework in which to study cohomology theories of strongly homotopy algebras, namely $A_{\mathrm{\infty }},C_{\mathrm{\infty }}$ and $L_{\mathrm{\infty }}$-algebras. This framework is based on noncommutative geometry as expounded by Connes and Kontsevich. The developed machinery is then used to establish a general form of Hodge decomposition of Hochschild and cyclic cohomology of $C_{\mathrm{\infty }}$-algebras. This generalizes and puts in a conceptual framework previous work by Loday and Gerstenhaber-Schack.

Alastair Hamilton & Andrey Lazarev: Symplectic $A_{\mathrm{\infty }}$-algebras and string topology operations

In this paper we establish the existence of certain structures on the ordinary and equivariant homology of the free loop space on a manifold or, more generally, a formal Poincar\’e duality space. These structures; namely the loop product, the loop bracket and the string bracket, were introduced and studied by Chas and Sullivan under the general heading `string topology’. Our method is based on obstruction theory for $C_{\mathrm{\infty }}$-algebras and rational homotopy theory. The resulting string topology operations are manifestly homotopy invariant.

Yes, this post includes a bit of shameless self-promotion.

There are more and more mathematical research group blogs entering the scene lately. I would like to try and start one more, this one with a focus on various closely related subjects - the ${*}_{\mathrm{\infty }}$ algebras.

We know of the $A_{\mathrm{\infty }}$ algebras. We know of the $L_{\mathrm{\infty }}$ algebras. And we know of $E_{\mathrm{\infty }}$ ring spectras. I most probably forget about several highly interesting similar constructions, which I however expect will end up being more than welcome here.

The tagline I expect the prevalent theme here to follow, though, will be “…up to homotopy”. This seems, to my mind, to be a unifying factor of most ${*}_{\mathrm{\infty }}$-theories I have seen so far. Thus, as the technical host of discussions, I wish you all welcome.

If you feel you can contribute more than just as a regular reader and commenter, and might even wish to write posts, then please create a user and drop me a note on mik@math.uni-jena.de. As soon as I see the user created, I will be able to grant further privileges. Hopefully, this way, we can get enough of the conversation going here to make this as vibrant and interesting a group blog as all the other examples out there.