This week I am at the Centre international de rencontres mathématiques (CIRM) in Luminy. I am attending and speaking at the “Higher Structures” conference. The whole event is wonderful! I hope I am not too out of place among the big names in the speakers’ list. I’ve learned a lot of new math during the talks, as well as to speak with people who I hadn’t had the chance to meet yet, or that I am not able to see very often. I’d like to thank Bruno Vallette and all the organizers for giving me this opportunity.
Many changes have happened in my life recently!
I defended my doctorate on November 17th. I guess I’m a doctor now! There are too many people to thank for that, so please see the “Thanks” section of my thesis. I am now entering the scary world of job applications. I am discovering the wonderful “GALAXIE” web application – fellow French job applicants know my pain.
Last week I was at the Max Planck Institute for the Conference for Young researchers in homotopy theory and categorical structures (which was, by the way, a great conference – thanks to the organizers), and I gave yet another talk about the Lambrechts–Stanley model for configuration spaces. So maybe it’s time I write a little bit about it on this blog. I’ll write a first post about the model itself, and later I will explain how the Fulton–MacPherson operad is involved in all this.
My paper Swiss-Cheese operad and Drinfeld center has finally been accepted! It is going to appear in the Israel Journal of Mathematics. I’ve made the few corrections suggested by the referee (the new version is available on the arXiv), and I’m now waiting for the final proofs before the paper can be published.
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, 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).
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.
I have uploaded a new preprint, The Lambrechts–Stanley Model of Configuration Spaces, which you can find on arXiv. Here is the abstract:
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…
Last week I was invited by Thomas Willwacher to ETH Zürich for a few days, during which I also had the opportunity to give a talk at the “Talks in Mathematical Physics” seminar. It was a very interesting few days, and I’m very grateful for this invitation!
The purpose of this post is to record the definition of \(\infty\)-operads and explain why it works like that. For this I’m using Lurie’s definition of \(\infty\)-operads; there is also a definition by Cisinski–Moerdijk–Weiss using dendroidal sets, about which I might talk later.
Indeed, the definition on an \(\infty\)-operad is a bit mysterious taken “as-is” – see [HA, §188.8.131.52]. My goal is to explain how to reach this definition, mostly for my own sake. Most of what follows is taken either from the book Higher Algebra, the \(n\)Lab, or the semester-long workshop about hgiher category theory in Lille in 2015.
This post is about the Milnor–Moore theorem, a powerful tool describing the structure of (co)commutative Hopf algebras. Like the Eckmann–Hilton argument, it shows that having multiple compatible operations on the same object can lead to unexpected results about the object. Briefly, the theorem says that as soon as the Hopf algebra is cocommutative and connected, then it is isomorphic to the universal enveloping algebra of a Lie algebra (and a similar dual statement is true for commutative Hopf algebras).
My first article has been accepted for publication with minor revisions! Entitled “Opérades et structures commutatives à homotopie près” (yes, it’s in French), it will appear in the Graduate Student Mathematical Diary, edited by the Mediterranean Institute For The Mathematical Sciences. The article is expository in nature, it contains a general introduction to the theory of operads, and then some applications of the theory, mostly in relation with the little disks operads. I’m psyched!
Now let’s hope that my preprint Swiss-Cheese operad and Drinfeld center meets the same fate… ☺
The theorem(s) of acyclic models are a rather powerful technique for proving that some functors defined on truncated chain complexes can be extended in higher dimensions, and that two such functors are homotopic, by proving it on a small class of “model” objects.
For some reason I only discovered this last year, and I always find myself forgetting the precise hypotheses and conclusion… Hopefully writing this blog post will fix them in my mind. My main reference will be:
This post is about something somewhat weird I noticed about infinitesimal bimodules over operads and their relationships with some \(E_n\) operads. I don’t know if it’s something significant, and I’d definitely be interested to hear more about it.
As promised, this post is about the famous Eckmann–Hilton argument. This argument, on the surface, looks like a simple algebraic trick; but it has deep consequences, which I will now try to explain. This post is an expanded version of a math.SE answer I wrote some time ago.