Long time no post. Boalch proved in some of his papers that the wild character variety on the punctured $\mathbb A^1$ comes from a quasi-Hamiltonian reduction of a “higher fission space” $_G A_H$ by actions of $G$ and $H$ (some Levi subgroup in $G$). The quotient $_G A_H / (G \times H)$ thus carries a Poisson structure, …
This is an amplification of the previous post to the case where $C$ is a curve in an arbitrary linear system $\Delta$ on an arbitrary toric surface $X_\Sigma$. In other words, its purpose is to explain why the result of that computation was “obvious.” Write $\Lambda_1$ for the subset of $\Lambda_\Sigma$ corresponding to the 1-dimensional …
Maybe you guys understood this already, but it just clicked for me today: the Poisson structure I was claiming had to exist on the knotty character varieties is exactly given by the Sabloff duality. Recall the setup. Let $\Lambda$ be a Legendrian, $F, G$ objects in the sheaf or whatever else category. We write $F_+$, …
It’s definitely not a good thing to have to go to this website to see if anything new has happened — for example, I just missed several weeks of content because I thought nothing was happening! Wordpress is definitely up to the task of sending email alerts to users when there are new posts/comments (indeed, …
OK, I had a good idea how to prove Aug = Sh in all dimensions, at least for any front diagram which admits a pixellation (maybe this means any front diagram after an appropriate isotopy), without much mucking about. Recall that “pixellation” just means that we draw our front diagram as a grid diagram; that is the …
The previously promised meditation on the action functional: 1. Let $\Lambda \subset J^1(X)$ be a Legendrian. Let $O_\Lambda$ be the space of open strings from $\Lambda$ to itself. Say I try to compute its cohomology; channeling a competent initiate of Floer theory, I do this by studying the action functional. (By “the action functional”, I mean …
I just suffered a moment of terrifying clarity. Theorem: an augmentation determines a sheaf Proof: work on $J^1(X)$; assume we’ve defined the augmentation category and localized it over $X$ (I think the fact that the localization can be done follows formally from M. Sullivan’s paper on Morse trees). To an augmentation $\alpha$ of the DGA, I am supposed …
Let $V$ be a two-dimensional vector space and let $L_0,\ldots,L_{n-1}$ be $n$ lines in $V$ obeying $L_i \neq L_{i+1}$ for all $i \in \mathbf{Z}/n$. Suppose $n$ is odd. If $g \in \mathrm{GL}(V)$ has $gL_0 = L_1,gL_1 = L_2,\ldots,gL_{n-1} = L_0$, then $g^n$ must be a scalar matrix and the ratio of the two eigenvalues of …
Chamber decomposition of $\mathrm{GL}_n^{\mathit{rss}}/\mathrm{GL}_n$. Let $\mathbf{z}$ be an $n$-element subset of $\mathbf{C}^*$, e.g. the eigenvalues of a regular semisimple matrix. For generic $\mathbf{z}$, associate a reduced word decomposition of the long element $w_0 \in S_n$, in the following way. Each element $z_k \in \mathbf{z}$ determines a stretched-and-shifted cosine graph, $\mathrm{Re}(z_k \exp(i\theta))$ over $\theta \in [0,\pi]$. The …
Today I gave a talk in the MSRI irregular singularities seminar, and afterward was talking with Valerio Toledano Laredo about isomonodromy. In particular now I actually know what isomonodromy is: Recall that the formal type of an ODE is some $\bigoplus R_i \otimes S_i$ where the $R_i$ are regular connections and the $S_i$ are rank …
