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Update: The lecture notes of the CAGA lecture series on perfectoid spaces at the IHES can now be found online, cf. http://www.ihes.fr/~abbes/CAGA/scholze.html. It seems that it's my job to answer thi …

No such space exists. Even better, let's generalize your proof by converting information about path components into homology groups. For an open inclusion of spaces $X \setminus \{x\} \subset X$ and …

The concept of definable real number, although seemingly easy to reason with at first, is actually laden with subtle metamathematical dangers to which both your question and the Wikipedia article to w …

To read Higher Topos Theory, you'll need familiarity with ordinary category theory and with the homotopy theory of simplicial sets (Peter May's book "Simplicial Objects in Algebraic Topology" is a goo …

Let $K$ be one of the following fields: the complex numbers, a finite field, a number field (and we could amalgamate the last two into the more general case of a field finitely generated overs its pri …

It's sort of like the inverse function theorem, and that is why it is so strong. If you have $n$ functions vanishing at the origin of $k^n$ and want to know if they give a local coordinate system, yo …

I know nothing about work of ``idelic nature'' by Von Neumann or Pruefer. Already in the 1930's Weil understood that Chevalley was wrong to ignore the connected component, because Weil understood alr …

When MacPherson and I first started thinking about intersection homology, we realized that there was a number that measured the "badness" of a cycle with respect to a stratum. This number had the pro …

One relevant thing here is that you are referring to differentiating and integrating within the class of so-called elementary functions, which are built recursively from polynomials and complex expone …

Here is a brief answer: Lie groups provide a way to express the concept of a continuous family of symmetries for geometric objects. Most, if not all, of differential geometry centers around this. By d …

The surface area $|\partial S|$ of a (bounded, smooth) body $S$ is the derivative of the volume $|S_r|$ of the $r$-neighbourhoods $S_r$ of $S$ at $r=0$: $$ |\partial S| = \frac{d}{dr} |S_r| |_{r=0}.$ …

I took a pane of clear glass and touched two balls at once I put my light, perhaps, by chance, above the pane. Alas, the shining pile on the same side in its arrangement lay, and no matter what …

Mnemonic: $\quad M=IM \Rightarrow m=im$ The version of Nakayama described: If $I$ is an arbitrary ideal of an arbitrary commutative ring $A$ and if a finitely generated module $M$ satisfies $M=IM$, …

Note that in linear algebra matrices describe at least two different things: linear maps between vector spaces (we consider only finite-dimensional vector spaces here) and bilinear forms. When thinkin …

The 2-adic rationals $\mathbb{Q}_2$ and the 3-adic rationals $\mathbb{Q}_3$ are homeomorphic, because each one is a countable disjoint union of Cantor sets. They are also isomorphic as groups if you …