Let $\Phi(p,q)$ be a rainbow-style front projection of the Legendrian $(p,q)$-torus knot. Let $\mathcal{M}_1(p,q)$ be its moduli of constructible sheaves of microlocal rank one. Proposition. $\mathcal{M}_1(p,q)$ is naturally identified with an “open positroid variety” in $\mathrm{Gr}_p(\mathbf{C}^{p+q})$, divided by its action of $(\mathbf{C}^*)^{p+q}$ To explain why, exchange a sheaf on a rainbow front, such as for …
I went to a talk of David Jordan who was explaining what factorization algebras are, and how they can be used to quantize character varieties. As far as I understood, a factorization structure on a manifold is some locally constant cosheaf of algebras in which the corestriction is an equivalence whenever the inclusion of open …
Yesterday I went to an extremely interesting talk of Rafe Mazzeo about the ends of the moduli of Higgs bundles. Roughly speaking, what was happening is that he described a new compactification — a sort of blowup of the usual compactification at the boundary — in which the limiting behavior corresponds to the Higgs field …
Let me recall what an ideal web is, then throw out a partially-baked question regarding their relationship to exact Lagrangians. The notion of ideal web appears in unpublished work of Goncharov (see here: http://people.math.umass.edu/~tevelev/AGNES_vids/Goncharov.mp4), but builds on a lot of pictures familiar from earlier work of e.g. Postnikov and relates those pictures to his work …
Let $M$ be a one-dimensional manifold. A thorn on $M$ is a ray in $T^* M$, or equivalently a point of $T^{\infty} M$. Thorns $S \subset T^{\infty} M$ are in “general position” if the projection $\pi:S \to M$ is injective. If $S$ is in general position then we define a quiver $Q_S$: 1. The vertices of $Q_S$ are …
Sections 8.3 and 8.4 of this http://arxiv.org/pdf/1303.3253v4.pdf (which were in a book that appeared in my mailbox) are a short (dense) summary of Stokes moduli spaces and their different incarnations. The paper surely has loads of gems, but is about as scattered and unstructured as this blog.
I’m uploading this page summarizing a talk I gave. trefoil-hitchin-onepage
On Monday, a bunch of people convened in my office to try and extract some GMN out of Harold. It turned out to be the most insanely productive day since the SQUARE. Harold had the following perspective on the GMN procedure. It has two steps: Choose a point in the Hitchin base. Solve certain flow equations; this gives …
A moment of clarity, inspired by a talk of Harold Williams and subsequent conversations: A cluster chart on a Poisson algebraic variety $X$ is just an open torus $T = Spec\, \mathbb{C}[x_1^\pm,\ldots, x_n^\pm]$ so that the Poisson structure on $X$ restricts to an integral coordinate Poisson structure on $T$, i.e., $\omega(x_i, x_j) = n_{ij} \delta_{ij}$ …
“More” because the discussion here is also about regular singularities. Let $X,Y,Z \in \mathrm{GL}_2$ have $XYZ = 1$. They determine a rank 2 local system on a 3-times punctured $\mathbf{P}^1$ — $X$ the monodromy around $0$, $Y$ the monodromy around $1$, $Z$ the monodromy around $\infty$. The simultaneous conjugacy class of the triple $(X,Y,Z)$ is determined …
