On this page I'll provide some notes that I've written. They have mostly not been peer-reviewed in any way and were for the most part initially written for myself; so use them at your own risk. Click on the arrow to get a more precise description of each note.
If you spot any mistakes or typos, please let me know. I am also grateful for other types of feedback :)
Some of them are in french, as indicated by "(in french)".
Proofs of known facts that I wanted to re-write:
A note about how ramified extensions of the integers never lift to E_1-ring spectra over the sphere: Ramified extensions don't lift
A note on Goodwillie's theorem that periodic cyclic homology is nil-invariant and A^1-invariant over a rational base: The nil- and A^1-invariance of periodic cyclic homology
A note about Bass' trace conjecture, namely I reprove a result of Linnell's using trace methods, in a different way than the related paper of Berrick-Hesselholt : On Bass' trace conjecture
A note about a calculation of Waldhausen's, computing the lowest homotopy group of the fiber of the map induced on K-theory by a highly connective map of connective ring spectra: On Waldhausen's obstruction to connectivity
A note about the Eilenberg-Moore spectral sequence, based on an argument by Toën, but adapted slightly (the same proof works for a more general co-Eilenberg Moore type of statement for chain coalgebras instead; ask me about it if you are interested) : The Eilenberg-Moore spectral sequence
A proof of the fact that the free semi-additive category on BG is the effective Burnside category of finite free G-sets - this was a claim made by Barwick and proved in a paper of Glasman, but I didn't understand the proof so I came up with a different one, that feels simpler : The effective Burnside category of finite free G-sets as the free semi-additive category on BG
Some slides:
Slides for a talk at the workshop "Real THH in Venice" organized by Emanuele Dotto, about "The Goodwillie calculus of localizing invariants".
Expository material on higher algebra/homotopy theory/ (higher) category theory :
A note about some problems with ordinals in "model-independent" homotopy theory: Ordinals
A somewhat roundabout, but "model-independent" proof that filtered colimits of anima are left exact: Filtered colimits are left exact
A brief introduction to adic spaces, written for Babytop at MIT, Spring 2024 : Adic spaces
Notes about equivariant stable homotopy theory from the point of view of spectral Mackey functors (à la Barwick) - I have not proofread them as much as I would like to, so they will most likely change in the (hopefully near) future - any form of feedback is appreciated ! : Equivariant stable homotopy theory
A short incomplete note about full dualizability of presentable (stable) categories (I'd like to know a lot more than what's in there, if you do know more, let me know !): Full dualizability
A note about a proof that n-excisive functors "preserve" finite colimits - more precisely send finite colimits to other types of finite colimits, giving a sort of "formula" for f(colim), when f is only n-excisive : Finite colimits and excisiveness
A short (and incomplete in terms of references) note about a Hochschild homology obstruction to some endomorphism algebra being an F_p-algebra : HH as an obstruction to algebraic structure
A short note about finiteness obstructions in equivariant stable homotopy theory - a short explanation about the difference between finite and compact for G-spectra, based on a computation of the K-theory of compact G-spectra; also mentions quickly Wall's finiteness obstruction: Finiteness obstructions for G-spectra
Notes written for "Topics in Algebraic Topology" at Copenhagen University (some stuff on Picard spaces, on multiplicative infinite loop space machines, and on chromatic homotopy theory)
A note about square zero extensions in the associative setting - in this case, ordinary algebra behaves as well as homotopical algebra and we do have a classification in terms of derivations: Associative square-zero extensions
A note about the universal coefficient theorem with twisted coefficients - I wrote this after a discussion with David Roberts about whether such a thing existed, and a disappointment at the current state of affairs (it appears to only be in the literature in the form of an exercise of Spanier's), even though it requires no extra work with the appropriate point of view : A twisted universal coefficient theorem
Notes I wrote for myself, in an effort to understand some p-adic stuff in homotopy theory; the notes go back to the basics of p-adic stuff as I needed to understand some of the usual story first. The level varies along the document: the beginning is elementary (some knowledge of algebra and topology is required), then categories start being used, and from some point onwards, infinity-categories are used to make everything easier - the document is not fully finished, I intend to modify it (updated 18/08/21) : p-adic stuff
A note about the higher algebraic phenomenon that modding out by n need not make n null, but it does make it nilpotent (updated 21/05 - added a section on how one must be careful in formulating that statement) f^2 = 0
An introduction to algebraic K-theory, written for my "ENS diploma" validation. For simplicity, I focused on direct sum K-theory, but I tried to give a broad overview of applications (in french) : Introduction to algebraic K-theory
An infinity-categorical interpretation of the Bousfield-Kan formula for homotopy colimits, from "first principles" in higher category theory : The Bousfield-Kan formula
A definition of the "center" of a monad, together with a proof that the center of a monad is a Morita invariant, generalizing the classical result for rings (if you know how to do this with oo-categories, let me know !) : Center of a monad
A short and elementary proof that a dualizable idempotent algebra has to be a retract of the unit (I kept forgetting it, so this is just to record it) : Small idempotent algebras
A proof that if you invert an element in a commutative algebra, the result does not depend on whether you inverted it as a commutative algebra, or as an associative algebra: Inverting an element in a commutative algebra
Documents written for the quarantined tea of the maths department of the ENS - these are very short introductions to some mathematical object, and as such they're very far from complete; the format is Introduction-Definition-Examples-A nice theorem, they were supposed to last less than 5 minutes, and so unsurprisingly they lack a lot of detail and precision (in french): Fundamental infinity-groupoid of a space , The infinity-category of spectra
A short note about the notion of a symmetric monoidal adjunction - I prove there the well-known fact that if two adjoint functors have lax symmetric monoidal structures such that the unit and co-unit are symmetric monoidal, then the left adjoint is strong monoidal : Symmetric monoidal adjunctions
Logic, set theory,...:
An introduction to ordinals I had written for a friend; the end is not very precise because he didn't know anything about model theory (in french) : Introduction to ordinals
A document about the completeness theorem for propostional logic (it is meant to be self-contained) (in french): Completeness of propositional logic
The text of my first year memoir about topological dynamics and Ramsey theory, with Adrien Abgrall, under the supervision of N. de Rancourt and T. Tsankov (in french) : Topological dynamics and Ramsey theory
"Algebra":
A proof of idempotent lifting for rings in the spirit of noncommutative geometry : Idempotent lifting
An introduction to categories and universal properties, written for an algebra class (the examples were made to be compatible with the other classes we were having) (in french) : Introduction to universal properties and categories
A short note about finite fields, written for an algebra class - it's written without appealing to quotient rings, and is meant to be somewhat elementary, which explains some of the heaviness of the proofs (in french) : Finite fields
A short note about some polynomial equations in finite rings and the ring of integers; it was written for a facebook page (Mathematical Theorems you had no idea existed, because they're false) : Polynomial equations : from local solutions to global solutions (the false theorem is "A monic integer polynomial with roots in every finite ring has an integer root")
Documents and information about my master's thesis.