Life in Paris

I’ve been in Paris for almost a month now. It’s been great! People at the math department and the math institute[^1] have all been welcoming and have helped me a lot in getting settled. There have been a lot of administrative procedures to complete – and I am unfortunately not done – and it’s great to have had people being able to guide me. And I finally found an apartment in Paris! It was unexpectedly hard: faculty salaries are not very high compared to the cost of living, and the first year is technically on “probation”, meaning I could theoretically get fired next August… Landlords in Paris have very rigid expectations and this made me fall outside them.

I’ve also started teaching a course called “Elementary Algebra and Analysis I”. It’s basically the course that gets taught everywhere around the world to first year students who just get into university. I’m really teaching the course (to a small group of 25 students) this time, and not just doing the exercise sessions, so it’s something new to get used to.

I have not been idle on the research side, either. At the end of July, joint with Ricardo Campos, Julien Ducoulombier, and Thomas Willwacher, we put a new preprint on the arXiv. We study the “framed” configuration spaces of a closed manifold. Recall that for a manifold $M$, the configuration space $\operatorname{Conf}_k(M)$ is the space of $k$ ordered pairwise distinct points in $M$. The framed configuration space $\operatorname{Conf}^{\mathrm{fr}}_k(M)$ is a bundle over $\operatorname{Conf}_k(M)$, with fiber the $k$-fold product of the frame bundle of $M$. Up to homotopy, there is a natural action of the framed little disks operad on the collection of all the framed configuration spaces of $M$. We give a model for the real homotopy type of these spaces and this action. As usual, this model is based on graph complexes and relies on an earlier model for the framed little disks operad due to Khoroshkin–Willwacher (arXiv:1705.08108). We hope to be able to use this new model for various things, for example in the Goodwille–Weiss calculus of embeddings.

I am also working on a solo project that I hope to release on the arXiv soon. (I am checking for the last typos ☺.) I am studying a higher-codimensional version of the classical Swiss-Cheese operad. While the Swiss-Cheese operad is naturally associated to manifolds with boundary, the operad I am studying would be naturally associated to a manifold with an embedded submanifold of codimension at least 2. I proved that this operad is formal over ℝ. I hope to be able to use this result to study configuration spaces of complements of embedded submanifolds.

PS: I’ve asked a question on MathOverflow about something that appears in my paper. I have a central derivation $B[n-m-1] \to A$ where $B$ is a Poisson $n$-algebra and $A$ is a Poisson $m$-algebra. So I was led to look at $A[\varepsilon]$, i.e. I add a new square-zero variable to $A$. I’m pretty much looking for a fancy interpretation of this new Poisson algebra in terms of the old one. If you think of something, let me know…

[^1]: For those who don't know, in France, there is a separation between the teaching department (UFR in French) and the research lab. The teaching department is part of the university, while the research lab is part of the university and the CNRS. The math research lab I'm at is called the "Institute of Mathematics of Jussieu-Paris Left Side" so I tend to call it "the institute" or "the IMJ". (In this case it's even part of CNRS and two university: Paris Diderot and Sorbonne Université.)