### Categorification and Topological Invariants in Luminy

#### Posted by Alexander Hoffnung

Yesterday began the *petit groupe de travail* with the title “Categorification and Topological Invariants” at the Centre International des Rencontres Mathématiques (CIRM) in Luminy. This week-long meeting is meant to be both introductory as well as an opportunity to explore interesting questions in the fields of interest of the various participants. So far this seems to be going quite well.

I will give a little diary below the fold, and I will try to fill in some of what I hear or say in the talks as comments.

I arrived in Marseille on Sunday with my girlfriend Vera Dao and we fumbled our way through the airport, bus station, metro station, the house and car of a kind (although odd) stranger and the campus at Luminy using a mix of English, broken Spanish, and mutilated French until we happily found ourselves at the door to the institute. (We even saw a small brawl on the metro!)

The grounds here are very nice from what I have seen so far, and the weather is quite pleasant. I am told that I have been inside during the “hot” hours though. The sun seems as though it will never set, although around 9:30 or so it finally dips out of site. Vera is doing all of the site-seeing, along with her new friend Carola, and I am site-seeing vicariously though them, or rather her camera. With some luck I will be able to figure out how to get those picture up here. In the meantime, here is a picture courtesy taken by George Gollin.

There is another meeting here this week on non-linear dispersive evolution equations, so we are quite outnumbered by the analysts at mealtime. However, it was nice to see a friendly face from Riverside among this crowd. I had never talked to Ben Dodson while in Riverside, but he recognized my shirt (possibly a hint that I should change more often). We each leave Riverside for new cities this summer, so this was probably a first and last chance to speak with him a bit, at least for some time. It is proving difficult, however, to get the PDE people to explain their work to me beyond a few sentences!

Now a little about our meeting. This is actually a brand new experience for me, as well as the others I believe. The first day of the meeting consisted of four people:Yael Fregier (Luxembourg), Raphael Zentner (Muenster), Andrew Lobb (Stony Brook), and myself. There was not a lot of organization prior to the meeting as we were not sure exactly who would be in attendance. Nonetheless, it seems to be working out quite nicely.

At this first meeting, Yael began to speak about $A_\infty$-modules, although most of the day was spent on various aspects of $A_\infty$-algebras, in particular the correspondence with Maurer-Cartan elements in a graded Lie algebra. I will try to recap what he explained soon, mostly to see if I have understood him correctly.

Today, Adrian Lim (Luxembourg) joined us and we heard from Raphael in the morning on introductory knot theory. This was very nice since I very often find myself in knot theory talks (I am not completely sure why) without the requisite definitions on hand. Even better someone suggested yesterday that the audience be obliged to step up to the board and recap some parts of the lecture at various points throughout the day. Raphael had us work through some nice exercises in calculating the Alexander polynomial of various knots.

After lunch, Andrew took over. He spoke about a conjecture of Milnor on the slice genus of a knot, and more briefly about categorifications of various famous knot polynomials. This got us into some low-dimensional topology, which I just do not know. If I can manage, I will try to recap what Andrew has discussed after his next talk.

Tomorrow we will quit early and I will go to the beach and undoubtedly return looking like a lobster. Then Thursday, I will speak on categorification of various sorts, but will likely focus on some of the more basic category theory which tends to show up.

Friday, Adrian will speak on Chern-Simons theory, and then Vera and I will head off to Paris.

## Re: Categorification and Topological Invariants in Luminy

Alex wrote:

Please do - Being deeply engrossed with the infty version of Mau

Rer-Cartan elements, I can hardly wait.Also delving into Yael’s and Zambon’s pre-pre-print.