When I first started learning about irregular perverse sheaves, I struggled to find a clear answer to the question: “what does an irregular perverse sheaf look like, in the simplest cases? What should be my intuition?” If I don’t know about the motivation from differential equations, how can I picture them? This is not yetContinue reading “Sabbah-Mochizuki-Kedlaya’s Hukuhara-Levelt-Turrittin Theorem”

# Author Archives: brianhepler

## Deligne’s regular solution in dimension 1

In this post, I want to recall the elements of the regular Riemann-Hilbert Correspondence (but only in dimension 1, on our small disk around the origin). We’ll talk more about the category of Meromorphic Connections on with singularities at 0, and how they’re just a different way of phrasing linear ODEs whose solutions have singularitiesContinue reading “Deligne’s regular solution in dimension 1”

## Irregular Riemann-Hilbert Correspondence: introduction to the problem

One of the most successful bridges between analysis and algebraic geometry is the classical Riemann-Hilbert (R-H) correspondence between regular holonomic D-modules and perverse sheaves on complex manifolds, where D is the sheaf of differential operators with holomorphic coefficients (proved independently by Kashiwara and Mebkhout in 1984). This correspondence is a far-reaching generalization of Hilbert’s 21st Problem asking about the existenceContinue reading “Irregular Riemann-Hilbert Correspondence: introduction to the problem”

## Some Microlocal Computations

A couple of posts ago, I mentioned the “microlocal” category , and I talked about a couple of neat things you could do with it. Looking back, I’m finding myself unsatisfied with the level of detail in my examples (as well as the number of examples!), so I figured I’ll make a post solely dedicatedContinue reading “Some Microlocal Computations”

## Symplectic Basics III: The Cotangent Bundle

Now that we’ve gotten a little comfortable with the idea of a symplectic vector space, it shouldn’t take a huge leap of the imagination to say there’s a similar notion for (smooth) manifolds, too: they’re called symplectic manifolds (surprise). A (real, smooth) symplectic manifold of dimension consists of a pair , where is the manifold, andContinue reading “Symplectic Basics III: The Cotangent Bundle”

## Symplectic Basics II : The Lagrangian Grassmanian

Before we proceed with Lagrangian stuff, I should really talk about a fundamental property of (finite dimensional, real!) symplectic vector spaces: proposition 1: They’re always even dimensional! proof: Let be a symplectic vector space of dimension m. I claim then that m is even. Choose some basis of , so that we get a matrix representative of , i.e., forContinue reading “Symplectic Basics II : The Lagrangian Grassmanian”

## Symplectic Basics I: Symplectic Linear Algebra

Last post I mentioned some types of subsets of the cotangent bundle, associated to the bundle’s natural symplectic structure (i.e., the isotropic, involutive, and Lagrangian subsets). What was I talking about? Back to basics! Today, I want to talk about some “symplectic linear algebra.” A symplectic vector space is a pair , where is aContinue reading “Symplectic Basics I: Symplectic Linear Algebra”

## Working Microlocally

So, by now, we have some ideas about what these object are, as (co)directions of non-propagation of sections of the various cohomology sheaves of the complex . But that’s really only scratching the surface, as all we’ve done so far is fiddle around with the definition of microsupport. For a general complex of sheaves (XContinue reading “Working Microlocally”

## Propagation of Sections

I mentioned last post that one should think of the microsupport of a complex of sheaves in terms of some loose idea of “propagation.” I want to talk about that a little more now. The gist I want you to walk away with is “the microsupport characterizes (co)directions of non-propagation.” Whatever that means. Let’s startContinue reading “Propagation of Sections”

## Microsupport and Propagation

So, last time, I briefly mentioned a sheaf-theoretic “Local Morse Datum” for a smooth (Morse) function at a (non-degenerate) critical point : which gives the integral cohomology of the “local Morse datum” of at , considered as a pair of spaces. Confusing, I know. It seems like complete overkill at this point, and it is.Continue reading “Microsupport and Propagation”