29 votes
Accepted

Gabber's original proof of his purity theorem

It's different, but it also uses Weil II. See Purity for intersection cohomology after Deligne-Gabber for my translation of the original.
user avatar
  • 31.8k
24 votes

Applications of microlocal analysis?

Microlocal analysis is used in computed tomography and other tomographic imaging techniques e.g. in medicine . Specifically, it is used to describe which wavefront sets (here: boundaries of objects, e....
user avatar
  • 11.6k
16 votes
Accepted

When does a perverse sheaf occur in the decomposition theorem?

In general it is a difficult problem. For example, the core of Ngô's proof of the fundamental lemma is his support theorem which implies that in the context of the Hitchin fibration, all simple ...
user avatar
16 votes

Applications of microlocal analysis?

There are striking applications in dynamical systems, due to Dyatlov and Zworski, where dynamical zeta functions were analysed using techniques from microlocal analysis. These zeta functions have a ...
user avatar
  • 21.5k
16 votes

So what exactly are perverse sheaves anyway?

If you're looking for a more geometric interpretation of perverse sheaves, you might be interested in MacPherson's 1990 lecture notes "Intersection Homology and Perverse Sheaves." As far as I know, I ...
user avatar
15 votes

Applications of microlocal analysis?

Although microlocal analysis was developed originally exclusively for linear problems, it has played an increasingly important role in nonlinear PDE via what's known as paradifferential calculus. ...
user avatar
  • 24.9k
12 votes

Applications of microlocal analysis?

(1) The older and more widely known applications are to regularity and solvability of PDEs of any order that are not necessarily elliptic. (2) The phenomenon of the propagation of singularities ...
user avatar
10 votes

So what exactly are perverse sheaves anyway?

After some searching I've found the notes An illustrated guide to perverse sheaves, by Geordie Williamson, which is a beautifully illustrated (and sort of topologically oriented) introduction to ...
user avatar
  • 1,241
10 votes

About an application of BBD decomposition theorem

One approach: Because $\pi$ is smooth, the $Y_i$s in the decomposition theorem must all be the entire space $G/Q$. Because $G/Q$ is simply-connected, the local systems are all constant. To calculate ...
user avatar
  • 116k
9 votes
Accepted

Perverse sheaves and tensor product

This is extremely false. Consider the skyscraper sheaf on a smooth point of a positive dimensional variety; this is always perverse (since it is Verdier self-dual). The tensor product of this with ...
user avatar
  • 41.5k
8 votes
Accepted

$\ell$-adic monodromy theorems (over $\mathbb{C}$)

The sages (for whom I am but an unworthy mouthpiece) say: Tell the OP to look at 6.2.9 in Deligne’s Weil II paper, it’s quite close to answering what he is asking in his Q2.
user avatar
  • 93.8k
8 votes

Examples of calculating perverse sheaves on algebraic varieties with easy stratification

I would suggest you two papers Gudiel Rodríguez, Félix; Narváez Macarro, Luis: Explicit models for perverse sheaves. Rev. Mat. Iberoamericana 19 (2003), no. 2, 425–454. This deals with the two-...
user avatar
  • 8,201
8 votes

Counterexamples to gluing complexes of sheaves

This doesn't quite answer your question, but it might be useful. First of all, note that it is extremely unbelievable (please, correct me if I am wrong) that you can glue objects of (bounded) ...
user avatar
  • 2,716
8 votes
Accepted

A recommendation for a book on perverse sheaves

Pramod Achar is working on a book on perverse sheaves and applications in representation theory. It's a great book! EDIT (2021): The book has now been published by the AMS: Perverse Sheaves and ...
8 votes
Accepted

Are equivariant perverse sheaves constructible with respect to the orbit stratification?

Every complex of sheaves has a unique maximal open subset on which it is locally constant, because if it is locally constant on two open sets, it is locally constant on their union. Let $U$ be then ...
user avatar
  • 116k
7 votes
Accepted

Relation between Milnor fiber and its restriction via vanishing cycles

There is no general relationship between the cohomology of the Milnor fibers of $\tilde f$ and $f$, in this setting. At a point $p\in X_0$, the Milnor fiber of $f$ is the intersection of $X$ and the ...
user avatar
7 votes
Accepted

What's known about the stalks of Lusztig's perverse sheaves on quiver varieties?

The dimensions of the stalks of Lusztig's sheaves give the coefficients when a canonical basis element is expanded in a PBW basis. These stalks satisfy a parity vanishing condition. For these ...
user avatar
7 votes
Accepted

Examples of calculating perverse sheaves on algebraic varieties with easy stratification

Geordie Williamson has a very nice set of notes on perverse sheaves: http://people.mpim-bonn.mpg.de/geordie/perverse_course/lectures.pdf it deals with some examples on curves (section 10). You can ...
user avatar
  • 9,482
7 votes

"Correct" definition of stratified spaces and reference for constructible sheaves?

Consult the book "Sheaves on manifolds" by Kashiwara and Schapira. It's a hard nut to crack, but it is the most efficient presentation I've seen. In Chapter 8, they work with stratifications ...
user avatar
7 votes
Accepted

Counterexamples to gluing complexes of sheaves

One can ask whether the derived category forms a stack (of triangulated categories). The answer is no: Let $S^2$ denote the two-sphere (or $\mathbb{P}^1$ if you prefer) and let $k$ denote a ring of ...
user avatar
7 votes
Accepted

So what exactly are perverse sheaves anyway?

An $\infty$-categorical perspective is given here https://arxiv.org/abs/1507.03913 and a triangulated expansion of those ideas is here https://arxiv.org/abs/1806.00883 More or less, perverse sheaves ...
user avatar
  • 11.5k
7 votes

Understanding an involution of the category of perverse sheaves on $\mathbb{C}$

It is the Fourier–Sato transform. You can find a detailed discussion in e.g. section 4D of this article: Bezrukavnikov and Kapranov - Microlocal sheaves and quiver varieties.
user avatar
  • 8,468
6 votes

"Correct" definition of stratified spaces and reference for constructible sheaves?

A bit more reader friendly than Kashiwara-Schapira is Borel's Intersection Cohomology. It doesn't treat perverse sheaves, but it has a good overview of stratified spaces in the first few chapter and ...
user avatar
6 votes
Accepted

Riemann Hilbert Correspondence with fixed stractification

One should phrase the constructibility in terms of six-functors and then, since the RH correspondence respects those, one will see what is the corresponding notion. First, to be a local system for a ...
user avatar
  • 5,322
6 votes

A recommendation for a book on perverse sheaves

Here are two possibilities: Topological Invariants of Stratified Spaces by Markus Banagl Intersection Homology & Perverse Sheaves: with Applications to Singularities by Laurenţiu G. Maxim
6 votes

A recommendation for a book on perverse sheaves

I'm currently taking a course on perverse sheaves and we are using Kashiwara & Schapira's Sheaves on Manifolds (published by Springer). It has all the things you mention and I've found it very ...
6 votes

Purity of perverse cohomology sheaves

By Deligne's theorem, the complex $f_*(K)$ is pure. The fact that a pure complex has pure perverse cohomology sheaves follows from Theorem 5.4.5 of Faisceaux Pervers by Beilinson, Bernstein, Deligne ...
user avatar
  • 116k
5 votes

Applications of microlocal analysis?

Search for papers by Maarten de Hoop (Colorado School of Mines, then Purdue, now a Simons chair at Rice University), who has used microlocal analysis very extensively to study problems in global and ...
user avatar
  • 1,067
5 votes

Applications of microlocal analysis?

Broadly speaking, microlocal analysis helps one in 'geometrization' of certain results on the singularities of distributions. In this direction, one striking application of microlocal analysis is the ...
user avatar
  • 2,089

Only top scored, non community-wiki answers of a minimum length are eligible