I’ve been neglecting this blog a lot. Juggling research, teaching, organizing a seminar, and a personal life leaves little time for writing articles! (Wait, isn’t that the same complaint as last time?)

Most prominently I’ve been spending a lot of time working on my paper about the Lambrechts–Stanley model for configuration spaces (see my previous post). The good news is, I’m done (or as done as one can be with a paper). I’ve just uploaded the third version of the paper on the arXiv (also available on my lab webpage), and I’ve submitted it. I’ve finally managed to remove this bothersome hypothesis about the Euler characteristic of the manifold, and I’ve fixed an issue about my use of the propagator (PA forms are hard). The new abstract is:

We prove the validity over $\mathbb{R}$ of a CDGA model of configuration spaces for simply connected manifolds of dimension at least 4, answering a conjecture of Lambrechts–Stanley. We get as a result that the real homotopy type of such configuration spaces only depends on a Poincaré duality model of the manifold. We moreover prove that our model is compatible with the action of the Fulton–MacPherson operad when the manifold is framed, by relying on Kontsevich’s proof of the formality of the little disks operads. We use this more precise result to get a complex computing factorization homology of framed manifolds.

I’ve also had the opportunity to give three talks about the paper: at the annual meeting of the GDR Topologie Algébrique in Amiens, at the conference in the honor of Said Zarati in Tunis, and at the seminar of the LAGA at Paris 13. I’d like to thank the organizers of these events, and especially Sadok Kallel and Bruno Vallette for inviting me at the latter two.

Hopefully I will now have more time to fill this blog! (Or more realistically, do more research.)

My first real post in a while! It turns out that writing an actual paper (cf. previous blog post) takes a lot of time and effort. Who knew?

The Voronov product of operads is an operation introduced by Voronov in his paper The Swiss-cheese operad (he just called it “the product”). It combines an operad and a multiplicative operad to yield a new colored operad; the main example I know is the homology of the Swiss-cheese operad. This is a construction that I use in my preprint Swiss-Cheese operad and Drinfeld center, where as far as I know I coined the name “Voronov product” – I haven’t seen this operation at all outside of Voronov’s paper. I wanted to advertise it a bit because I find it quite interesting and I’m eager to see what people can do with it.

### Voronov products

The setting is as follows. Consider two symmetric one-colored operads, $\mathtt{P}$ and $\mathtt{Q}$, in some monoidal category. Suppose that you’re also given a morphism of operads $\mathtt{Com} \to \mathtt{P}$, where $\mathtt{Com}$ is the operad of commutative algebras. Then Voronov builds a new, bicolored operad $\mathtt{P} \otimes \mathtt{Q}$.

I have uploaded a new preprint, The Lambrechts–Stanley Model of Configuration Spaces, which you can find at arXiv:1608.08054 . Here is the abstract:

We prove the validity over $\mathbb{R}$ of a CDGA model of configuration spaces for simply connected manifolds with vanishing Euler characteristic, answering a conjecture of Lambrechts–Stanley. We get as a result that the real homotopy type of such configuration spaces only depends on a Poincaré duality model of the manifold. We moreover prove that our model is compatible with the action of the Fulton–MacPherson operad, by relying on Kontsevich’s proof of the formality of the little disks operads. We use this more precise result to get a complex computing factorization homology of manifolds.

This week I was at the Young Topologists Meeting! It’s gotten even bigger than two years ago, as there were more than 180 participants this year. The conference was quite interesting, and Copenhagen is a really nice city! The main theme was homological stability, about which I have learned a lot. The organizers should be applauded, because I can’t imagine how hard it must have been to plan a conference this big.

Now that I’ve done all my (math-related) travelling for the summer, I hope I’ll be able to post actual content here…