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First, some background. I was trying to read the article Loop Spaces and Langlands Parameters but I get immediately stuck at Theorem 2.1 in the introduction. This was actually forward-referring to C …

The following answers might be useful: Fourier-Mukai transform - a first example Intuition for Integral Transforms Fourier transform for dummies The last one has my sketch of an answer which I'll …

First of the questions about derived algebraic geometry from a noobie. The way I understand it, every discrete ring $R$ corresponds to some ring spectrum whose $\pi_0$ is $R$. Now consider $p$-adic nu …

Ah, great question! I'm not a big expert, but one thing it obviously does is constructing the sheaf M from the locally free bundles (locally free = projective). For example, consider a skyscraper she …

There's an important piece of geometric knowledge usually quoted as Beilinson-Bernstein-Deligne. Here's a refresher: by $IC$ one means the intersection complex, which is just $\mathbb Q$ for a smooth …

This question comes (heavily edited) from my notes, thus slightly unusual structure. We know that algebraic maps have very strict structure, and in many settings the operations f_*, f_!, their adjo …

Yes, it's defined at least sometimes. See Constructible Sheaves and the Fukaya Category by Nadler and Zaslow, math/0604379

This is a completely different answer. If my intuition is correct, in the case of noncompact Calabi-Yau, the Fukaya category is (1) expected to be defined (2) for a good physical reason (3) but not …

There's a very interesting text by Cumrun Vafa called Geometric Physics. Here I'm particularly interested in Chapter 4, where we take a Calabi-Yau manifold presented as a degenerating fibration: …

I've read about model categories from an Appendix to one of Lurie's papers. What are the examples of model categories? What should be my intuition about them? E.g. I understand the typical examples …