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25 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....
Dirk's user avatar
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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 ...
Anthony Quas's user avatar
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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 ...
Greg Friedman's 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. ...
Deane Yang's user avatar
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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 ...
T. Amdeberhan's user avatar
11 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 ...
Dat Minh Ha's user avatar
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11 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 ...
Will Sawin's user avatar
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11 votes
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Intersection cohomology and Poincaré duality

Let $E$ be an elliptic curve and let $Y$ be the union of two $\mathbb P^1$s joined by a node (say, the solution set of $xy = 0$ in $\mathbb Z/2$). Let $\sigma$ be an involution of $E \times Y$ that ...
Will Sawin's user avatar
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10 votes
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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 ...
Ben Webster's user avatar
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10 votes
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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 ...
9 votes
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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 ...
Will Sawin's user avatar
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8 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 ...
Liviu Nicolaescu's user avatar
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) ...
gdb's user avatar
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8 votes

Equivariant perverse sheaves and orbit stratification

There is a monodromy action of $\pi_1(G)$ on every $G$-constructible perverse sheaf. It must be trivial for equivariant sheaves. Simplest example is $\mathbb{C}^*$ acting on $\mathbb{C}$. In the ...
Justin Hilburn's user avatar
7 votes
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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 ...
Geordie Williamson's user avatar
7 votes
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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 ...
fosco's user avatar
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7 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 ...
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.
Vivek Shende's user avatar
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7 votes
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Extending IC sheaves across smooth divisors with normal crossings

The local monodromy around $D_i$ can be obtained by taking a $\eta$ a geometric generic point of $D_i$, $R$ the etale local ring of $X$ at $\eta$ with uniformizer $\pi$, then pulling $\mathcal L$ back ...
Will Sawin's user avatar
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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 ...
Greg Friedman's user avatar
6 votes
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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 ...
Sasha's user avatar
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6 votes

Operations on perverse sheaves on disk

I'm not sure how the notation you're using matches up to different presentations of this equivalence. I'm going to assume $E$ is nearby cycles, $F$ is vanishing cycles, $c$ is the obvious map from ...
Will Sawin's user avatar
  • 141k
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 ...
Will Sawin's user avatar
  • 141k
6 votes

Tensor product and semisimplicity of perverse sheaves

No. For example if $X = \mathbb A^1$, $j \colon \mathbb G_m \to \mathbb A^1$ the open immersion, $\mathcal L$ a rank one lisse sheaf on $\mathbb G_m$ with monodromy of order $2$, and $M = N = j_! \...
Will Sawin's user avatar
  • 141k
5 votes

Applications of microlocal analysis?

A pretty recent and impressive application of microlocal analysis is to analytic number theory or automorphic forms, via the orbit method as developed by Nelson and Venkatesh. In loc. cit. they set ...
Desiderius Severus's user avatar
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 ...
Uday's user avatar
  • 2,219
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 ...
Tom Dickens's user avatar
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5 votes
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Convolution of $\ell$-adic sheaves is commutative if the group is commutative

The thing that makes everything easy here is that the horizontal maps in your diagram are isomorphisms. For instance, every commutative square where the horizontal maps are isomorphisms is Cartesian. ...
Will Sawin's user avatar
  • 141k
5 votes
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Fulton's deformation to the normal cone vs Verdier's

The following answer was emailed to me by Claude Sabbah. I have received his permission to post it here. Whenever you need to get some object on $Y$ from an object existing on the normal cone (in ...
Avi Steiner's user avatar
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