On this page I'll provide some notes that I've written - mostly expository. They have mostly not been peer-reviewed in any way and were for the most part initially written for myself; so use them at your own risk.
If you spot any mistakes or typos, please let me know. I am also grateful for other types of feedback :) If some set of notes appears twice below, it is on purpose : a given set of notes can fit into several distinct categories.
Notes on homotopical algebra
Universality of multiplicative infinite loop space machines : these are notes I wrote based on Gepner, Groth and Nikolaus's paper with the same name for the course Topics in Topology at the university of Copenhagen (TopTop). They're essentially a summary of the paper, but might be helpful.
Picard spaces and Thom spectra: Also notes written for TopTop - I explain the basics of Picard spaces, and describe the \(\infty\)-categorical perspective on Thom spectra. As a bonus, I explain how Picard spaces relate to Hilbert 90.
The Eilenberg--Moore spectral sequence: This is more or less a translation into english of Toën's Appendix A in his paper on "schematization" which explains the Eilenberg--Moore theorem about derived tensor products of cochain algebras (from which one can derive the Eilenberg--Moore spectral sequence). As a bonus, here is a note explaining the dual version, for chain coalgebras, which works in slightly more generality (but the proof is more or less the same: the document itself is essentially copied from the previous one, and dualized).
Ramified extensions don't lift: A note to clarify that number rings that are ramified over \(\mathbb Z\) (or \(\mathbb Z_p\) ) do not lift to the sphere spectrum (mildly generalizing an obstruction found by Schwänzl, Vogt and Waldhausen for \(\mathbb Z[i]\), though the proof is based on the same idea).
Small idempotent algebras: A note proving that in any symmetric monoidal \(\infty\)-category, an idempotent algebra which is furthermore dualizable splits off of the unit.
\(f^2=0\) : A note to discuss the curious phenomenon that in homotopy theory, the map induced by a map \(f\) need not be null on \(X/f\), but its square always is.
Hochschild homology as an obstruction: Uses Hochschild homology to prove that while \(p=0\) in \(\mathrm{End}_{\mathbb Z}(\mathbb F_p)\), it is not a \(\mathbb Z/p\)-algebra.
Inverting elements : 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 elements 0: A short note on the notion of localization in homotopical algebra. This is in french.
On Waldhausen's obstruction to connectivity: Waldhausen has a beautiful result describing the failure of algebraic K-theory to preserve connectivity "more than it's supposed to" - for \(k\geq 1\), algebraic K-theory sends \(k\)-connective maps of connective ring spectra to \(k+1\)-connective maps of spectra, and Waldhausen explains what happens exactly in \(k+2\). This note explains that and relates it to the Dundas--McCarthy theorem (I think there is a minor indexing error somewhere in the note, be careful).
The Bass trace conjecture: Linnell, and then Berrick--Hesselholt prove a quantitative weak version of the Bass trace conjecture which concerns K-theory of integral group rings. The proof of Berrick--Hesselholt uses trace methods. This note gives a slightly different version of the same trace-methods-proof.
Semi-simplicity of numerical motives: This is a note to gather some of the proofs in Marcolli and Tabuada's works that add up to a proof that certain categories of "noncommutative numerical motives" are semi-simple.
Finiteness obstructions for G-spectra: I describe the K-theory of the category of compact genuine G-spectra for a finite group G (in terms of K-theory of spherical group rings) and explain how that can be used to differentiate between compact and finite G-sepctra. The calculation works more generally for localizing invariants, though it's not said explicitly in the note.
Examples of finiteness obstructions: In this note I spell out a few examples of finiteness obstructions, stemming from Lurie's generalization of Wall's finiteness obstruction in terms of K-theory (including Efimov's finiteness obstruction for stable \(\infty\)-categories).
K-theory and étale cohomology: Notes for a talk about the relation between \(K(1)\)-local K-theory and étale cohomology (the talk was in the context of a seminar on Blumberg and Mandell's K-theoretic Tate-Poitou duality, so that's what some of the implicit references in the note are about)
Hochschild homology as an obstruction: Uses Hochschild homology to prove that while \(p=0\) in \(\mathrm{End}_{\mathbb Z}(\mathbb F_p)\), it is not a \(\mathbb Z/p\)-algebra.
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. This is in french.
Notes on homotopy theory
Equivariant stable homotopy theory: An introduction to equivariant stable homotopy theory. This was written mostly for myself, to help me learn about the topic. It includes an introduction to classical induction theory, which is meant to prepare for Mathew-Naumann-Noel's derived induction theory.
p-adic stuff: Some notes on p-adic (stable) homotopy theory; they were left a bit uncompleted, I don't know if I'll ever get back to them. I found it helpful to work these things out on my own to integrate them.
An overview of chromatic homotopy theory: A summary of Lurie's lecture notes on chromatic homotopy theory. I wrote them to give a big picture overview (with relatively few proofs) of those lecture notes, to give an idea of what to look out for when reading said notes. They were also written for TopTop.
Large epimorphisms: A note on bounding the "size" of \(y\) in terms of that of \(x\) when there exists an epimorphism \(x\to y\) in a presentable (\(\infty\)-)category.
The Bousfield-Kan formula: An \(\infty\)-categorical interpretation of the Bousfield-Kan formula for homotopy colimits, from "first principles" in higher category theory.
Finite colimits and excisiveness: 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
Ordinals: A note about some problems with ordinals in "model-independent" homotopy theory
The effective Burnside category: 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. The one here seems simpler to me.
Filtered colimits are left exact: A somewhat roundabout, but "model-independent" proof that filtered colimits of anima are left exact (this is a bit of a joke, but not entirely)
Full dualizability: Something about fully dualizable presentable stable \(\infty\)-categories, and how sheaves typically aren't.
Notes on ordinary algebra and ordinary category theory
Adic spaces: An introduction to adic spaces, written for BabyTop at MIT, Spring 2024
Associative square zero extensions: A concrete working out of the classification of square zero extensions in the associative case, which happens to work un-derivedly.
Idempotent lifting : A proof of idempotent lifting for rings in the spirit of noncommutative geometry
Center of a monad: Discusses the notion of a commutative monad and introduces the "center" of a monad together with a proof that it is Morita-invariant. If you know how to do this \(\infty\)-categorically, let me know!
Polynomial equations: from local solutions to global solutions - 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)
A short introduction, written in french, to categories and universal properties.
A short introduction, also in french, to finite fields - about their classification.