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I'd like to mention the resolution of the redshift conjecture for $E_{\infty}$-rings. The latter are a homotopical refinement of usual commutative rings; at least if we restrict to connective ones, we …

I am not sure whether it is in the spirit of the original question, but let me add a wordy version of Emily's extensive and excellent bibliography -- a bit more of a road map. Let me divide to this pu …

I want to apologize in advance if my question is too elementary as I am not an expert in Lie theory. I have posted it before on stackexchange without receiving an answer. Let $G$ be a compact Lie grou …

For any $n\geq 1$, one can define a functor $\mathcal{M}_1(n)\colon \mathrm{Schemes}/\mathbb{Z}[\frac1n] \to \mathrm{Groupoids}$, sending a scheme to the groupoid of elliptic curves over it with a cho …

I would like to know if there is some online source where Boardman's 1964 thesis is available or his Warwick mimeographed notes. This is because by what I've heard Boardman's construction has a more m …

Much has already been said in the other answers and comments, but let me summarize a few points. One way to obtain from a category a 'homotopy theory' (aka an $\infty$-category) is to specify a notion …

The theorem can be found in more general form in Land, Tamme On the K-theory of pullbacks, Lemma 2.4.

If you want to learn about stacks, I can recommend 'Fundamental Algebraic Geometry: Grothendieck's FGA Explained'. Vistoli's exposition of the basic theory of stacks is hard to beat, I think. Moreover …

For the sake of the readers who are not fluent in German, I provide a translation of the German Wikipedia page (link to the revision at the time of posting this answer): Hermann Künneth (1892-1975) w …

Let $R$ be a commutative ring. If I am not mistaken, there is the following fact: For a finitely generated abelian group $A$, the $R$-module $A\otimes R$ is free if and only if we can write the to …

How about the following problem: Given an integer $n$, how many ways are there to write it as the sum of $k$ squares? Or, equivalently, in $\mathbb{R}^k$, how many lattice points (in the standard int …

As you probably know, the existence of Greek letter elements relies on the existence of (generalized) Smith-Toda complexes -- the best introduction to those is probably still Section 1.3 of Ravenel's …

The following strategy should work although I do not claim that it is the most elegant. Claim 1: The coarse moduli stack of elliptic curves $\mathcal{M}_{1,R}$ is affine. Proof: It is enough to sho …

For a sequence of positive integers $a_0, \ldots, a_n$ and a ring $R$, there is a graded ring $R[x_0,\ldots, x_n]$ where $x_i$ is in degree $a_i$. There is a corresponding $\mathbb{G}_m$-action on $Sp …

I always found Algebraic Stacks by Tomas Gomez to be a very readable quick introduction. It is virtually without proofs but explains on 34 pages the most relevant definitions and constructions and dis …