#Math

Defense, postdoc, new paper
#math #job #conf-spaces #paper

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.

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A Model for Configuration Spaces of Closed Manifolds
#math #conf-spaces

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.

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Configuration Spaces Woes
#math #conf-spaces #paper

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).

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The Voronov Product of Operads
#math #operads #swiss-cheese

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.

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Young Topologists Meeting 2016
#math #trip

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…

Infinity-Operads Demystified
#math #algtop #operads #higher-cat

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, §2.1.1.10]. 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.

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List of Facts
#math

I just started a list of facts, mainly rather classical facts that I don’t want to forget. Before, that list lived on sheets of papers strewn across my desk, which was clearly not optimal. Now it’s in a more permanent form.

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The Milnor–Moore Theorem
#math #algtop

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).

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Paper Accepted!
#math #paper

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… ☺

Acyclic Models
#math #algtop

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:

  • Samuel Eilenberg and Saunders MacLane. “Acyclic models”. In: Amer. J. Math. 75 (1953), pp. 189–199. ISSN: 0002-9327. JSTOR: 2372628. MR0052766.
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About Abelian Bimodules
#math #algtop #operads

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.

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