https://pzhou.org/ 2026-06-13T01:41:03+00:00 https://pzhou.org/ https://pzhou.org/lib/tpl/dokuwiki/images/favicon.ico text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2022-10-24?rev=1687708402&do=diff 2022-10-24 I created my personal site, after registered this domain name for a year and let it sit around (yes, I registered last year before the application season.) I don't think this is just useful for math. Math is part of my life, not the other way around. This site will live no matter what. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2022-10-25?rev=1687708402&do=diff 2022-10-25 This week we have the Gauge theory workshop in MSRI, one can participate online. I watched the one by Cherkis on AL? spaces and Doan on Fueter equation. Sergei Cherkis's talk on Gravitational Monopole Cherkis and Kapustin constructed certain moduli spaces, in the late 90s. Recently, there are classifications on the math side about these spaces, one of the authors is also my friend Chen Gong, also advisor of my friend. These spaces are instanton or monopole moduli spaces with 4 re… text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2022-10-27?rev=1687708402&do=diff 2022-10-27 * Discussion with Davis * Geordie's Presentation * About Application Davis Discussion with Davis Lazowski, about how to understand the twisted D-module on $\P^1$. It seems one can easily have the correspondence for $\C^*$ equivariant D-module on $\C^2$, $$ E = x \d_y, \quad F = y \d_x \quad, H = x\d_x - y d_y $$ Then its action preserves the $\C^*$-weight. For for the weight $k$ of $O(\C^2)$-mod, this $E,F,H$ will act on it. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2022-11-07?rev=1687708402&do=diff 2022-11-07 * Watched the youtube talk by Teleman at CMSA this year, on Kapustin-Rozanski-Saulina 2-category theory. * Reading the paper by Peter on DAHA and A-brane * Went to the seminar talk by Ivan Peter Koroteev Brane and Quantization One interesting thing is about this coisotropic brane. This was introduced by Kapustin-Orlov in 2001, then Aldi-Zaslow worked out an example, then Gukov-Witten, Gaiotto-Witten did some work on it. Still, there is no definition. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2022-11-08?rev=1687708402&do=diff 2022-11-08 * Mina's talk at MSRI * Comonadic Descent, again * Real MA equation around singularity * What is Matrix Factorization? text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2022-11-09?rev=1687708402&do=diff 2022-11-09 * The ambiguity of matching MF with Fuk * FLTZ-Morelli already considered the shard sheaves already, and showed that shard sheaves generate. So no need for me to worry about it, but still, things to do in the relative case * Rachel Webb's talk about GW on orbifold: Costello have already proved the closed string, symmetric quotient of product can help one ramp-up genus. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2022-11-10?rev=1687708402&do=diff 2022-11-10 * Reading Sheridan's fano hypersurface paper Sheridan, Fano What is a Frobenius algebra, and why we care? I know it has something to do with TFT, but precisely how? Here I copied down a paragraph in a (computer science paper? almost, text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2022-11-26?rev=1687708402&do=diff 2022-11-26 Saturday * chatted with Lei about log stuff * Some feedback on research statement Log stuff <https://math.berkeley.edu/~ogus/preprints/logrihfordocumenta.pdf> Ogus is writing in a gentle pleasant way. * First, smooth proper complex manifold. Local system corresponds to vector bundle with flat connection. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2022-12-09?rev=1687708402&do=diff 2022-12-09 Friday * Resolution using T-branes How does it work? I thought I knew it, but I don't, even for one strands, no disks. Is it because of the chain complex vs cohomology, or some other weird sign error? Can we do something easier and almost trivial? For example, braiding a bunch of punctures on $\R$? Do we know how it works? OK, there, it should work. Because we have puncture, and there is no dots. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2022-12-10?rev=1687708402&do=diff 2022-12-10 * Construction from skeleton's core Core, not co-core Why do we have matrix factorization from mirror symmetry of pair-of-pants? The most elegant answer, I think, comes from Seidel's immersed Lagrangian trick. Consider the space of weakly bounding cochain, and the natural superpotential function on it. This is most natural, since we don't need to consider any critical points on the A-side at all. See text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2022-12-14?rev=1687708402&do=diff 2022-12-14 Wed * Cross my t and dot my i. T and I branes $\gdef\End{\text{End}}$ Let me be super careful, and state the condition that I need. Let $X$ be a Weinstein manifold. Let $T_p$ be a collection of cocores, one for each critical point of the psh function $\varphi$. Let $I_p$ be the collection of cores. We have $\Hom(I_p, T_q) = \delta_{p,q}$, and similarly $\Hom(T_p, I_q) = \delta_{pq}[-n]$. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2022-12-18?rev=1687708402&do=diff 2022-12-18 Sunday * What is a handle-slide? Oszvath-Szabo has a nice article here. * How do we see if the two Lagrangian torus are equivalent or not? Auroux's slide His 2011 talk slide on the big pictures. We are building 4d TFT, which should assign a number to a closed 4-manifold. a vector space to a closed 3-manifold, and a category to a closed 2-manifold. Moreover, for a cobordism of 3-manifolds $Y_0^3 \leadsto Y_1^3$ by $W^4$, we have a homomorphism from $Z(W): Z(Y_0) \to Z(Y_1)$… text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-01-08?rev=1687708402&do=diff 2023-01-08 Heard a talk by Tim Logvinenko, about generalized braid group action. There are many interesting ideas, very concrete diagramatic. Of course, Kapranov can say, it is all in his old work with Schechtmann on perverse schober on $Sym^n \C$, but I still like this work. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-01-12-smoothing-nodal-curve?rev=1687708402&do=diff 2023-01-12 smoothing nodal curve Discussed with Xiaohan about how to glue and smooth a nodal curve (possibly open). * The story of Kaehler differential. MO * Rachel Webb's intro to quasi-maps. <https://arxiv.org/pdf/1910.07262.pdf> * Okounkov-Pandharipanda, <https://arxiv.org/abs/math/0101147> (I don't know what is this about) text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-01-22-xy-2?rev=1687708402&do=diff 2023-01-22 $xy^2$ What happens when you try to do Fukaya-Seidel category with the function $f = xy^2$ on $\C^2$? Well, you would say first, let's compute the regular fiber, which is $(\C^*)$, parameterized by $y$. Then, you ask, what is the monodromy. I don't think there is anything special, so let's say, the monodromy is trivial. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-01-25?rev=1687708402&do=diff 2023-01-25 I am in a group of mathematician. I watched this beautiful moving sequences of youtube videos. * by Michael Hopkins * by Jim Simons * and by the advices Jim Simons says “Don't give up”. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-01-27?rev=1687708402&do=diff 2023-01-27 End of AIM workshop * [AK]: Vector Bundle on $\P^2$ * [EG]: Affine Springer Fiber * [WL]: Ruling and Stratification Vector Bundles on $\P^2$ It is always a good idea to share your thoughts, it might induce more sparks. We follow Knutson and Sharpe. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-02-01?rev=1687708402&do=diff 2023-02-01 complete intersection model. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-02-08?rev=1687708402&do=diff 2023-02-08 * Localization for KRLW algebra over a downstairs skeleton. * Realizing the skeleton of hypertoric quotient. * Comparison of Honda-Tian-Yuan and Mak-Smith. localization Let's consider baby things first. Can we glue up $cNH_2$ from $NH_2$ and $NH_1 \boxtimes NH_1$ over $(S^1 \times S^1) / S_2$. The latter space is a stratified manifold. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-02-10?rev=1687708402&do=diff 2023-02-10 * Reading about symmetric product, by Dykerhoff-Jasso-Lekili and Auroux's ICM. * Idea about generation Upstairs Skeleton How to understand it? Downstairs, we have $Sym^2(\C^*)$, which we can shrink to $Sym^2(S^1)$. Upstairs, I just don't know what to put over the diagonal. We know the fiber of the diagonal is $\C^* \times \C_u$, and the superpotential is basically the variable $u$, so if we just ask for the fiberwise skeleton, it is zero. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-02-13?rev=1687708402&do=diff 2023-02-13 Vivek gave a talk, and talked about stuff during dinner. * What you can do with skein-on-brane, and higher genus open Gromov-Witten invariant. skein, moduli of brane $\gdef\lcal{\mathcal L}$ skein Let $\lcal \In \C$ be a $2 \dim_\R$ Legendrian in a 5 dimensional contact manifold. Let $L_\infty = \lcal \times \R \In \C \times \R$ be corresponding Lagrangian in the symplectization. One can compute curves bounded by $L_\infty$ and Reeb chords. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-02-24?rev=1687708402&do=diff 2023-02-24 VGIT take 2, this time with superpotential on the B-side. On one hand, this story is well understood by other people, not me; on the other hand, the story of window on A-side is not so well understand, and we don't know what the mirror of taking quotient is. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-02-25?rev=1687708402&do=diff 2023-02-25 Again, considering the VGIT and LG problem. Read a bit BFK in the morning. The graded MF is really something. Another thing that I realized is, at least for HMS of toric GIT, the choice of a cocharacter in $T_B$ corresponds to a 1PS in $T_A$. One probably should consider compactifying the base. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-02-26?rev=1687708402&do=diff 2023-02-26 Let me do some concrete stuff tonight. * Write up and finish the complete intersection project. * write up the legendrian thickening stuff * write up the matrix factorization for conifold case. who goes to what. * dream about, how to do A-side GIT quotient. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-02-27?rev=1687708402&do=diff 2023-02-27 Discussion again with Vivek Toric Stuff Suppose we have fibration with singularity over torus on the A-side, and dual torus action on the B-side, can we say taking mirror is like taking fiber? Step 0: no superpotential, no divisor. With group action. done. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-03-04-l4l5?rev=1687708402&do=diff 2023-03-04, Branter's lecture note L4 and L5 We finished some elementary discussion of monad, in the usual category. L4: infinity category $\gdef\bDelta{\mathbf{\Delta}}$ $\gdef\colim{\text{colim}}$ We start with simplex category $\Delta$, with objects $[0], [1], \cdots$, and order preserving morphisms. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-03-04?rev=1687708402&do=diff 2023-03-04 * learn what is Barr-Beck (don't worry about Lurie, I am not $\infty$ yet). Barr-Beck condition $\gdef\colim{\text{colim}}$ Let me follow Branter's note . And, Akhil Matthew's Serre Criterion for affiness. L2 Category warm-up. * What is a left-adjoint functor? $\otimes$, $i^*$ (restriction to open set) text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-03-06?rev=1687708402&do=diff 2023-03-06 Monday Pavel Putrov came and give a talk, about Kapustin-Witten equation. Witten's path integral What does Witten want to do? Analyltic continuation of Chern-Simons. We want to get finite dimensional approximation. Intersection of holomorphic Lagrangian in a Hitchin system. Then, we do Floer theory. Or, should I do holomorphic Floer theory? text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-03-07?rev=1687708402&do=diff 2023-03-07 More crazy ideas from Vivek Categorical Integrate out In physics, we always talk about path integral, and about integrating out some 'heavy' mode, and be left with an easier theory. In Fukaya category, we can do similar things. The simplest example is Knorerr periodicity, where you can have extra $(\C^, x^2)$ factor, and doesn't change your category. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-03-08?rev=1687708402&do=diff 2023-03-08 discussed with Hayata and Nicolo Liouville Isotopy True statements: * Given a Liouville domain, the space of Liouville structure is convex, hence contractible. * Given any two Liouville structures on the same domain, we can find a canonical path connecting them. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-03-10?rev=1687708402&do=diff 2023-03-10 * Comonad * todo list Comonad $\gdef\ccal{\mathcal{C}}$ $\gdef\dcal{\mathcal{D}}$ $\gdef\lra{\leftrightarrow}$ $\gdef\xto{\xrightarrow}$ $\gdef\Om{\Omega}$ What a comonad? Given two categories $$ L: \ccal \lra \dcal: R$$ We can form the comonad $\Om = LR \in End(\dcal)$, which have $$ \epsilon: \Om \to 1_\dcal, \quad L \eta R: \Om \to \Om \circ \Om $$ text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-03-12?rev=1687708402&do=diff 2023-03-12 * Comonad * What is the ind-completion of Fukaya category? Gluing Fukaya Categories from localization Suppose we have a B-side category, and we have a bunch of open subsets, then we can glue quasi-coherent sheaves. I would just take direct sum of each pieces, and cancel the over-count by over-count. That's that. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-03-13?rev=1687708402&do=diff 2023-03-13 * Talk by Paul W text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-03-16?rev=1687708402&do=diff 2023-03-16 * Talking with Yan * Talking with Ivan text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-03-21?rev=1687708402&do=diff 2023-03-21 Darn it, time really flies. No recollection whatsoever what happened last few days. Maybe doesn't matter. I am recently very interested in the relationship between solving differential equation and doing Floer theory. Generalized Riemann-Hilbert Correspondence text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-03-22?rev=1687708402&do=diff 2023-03-22 A simple exercise in skeleton. If I cannot do it in 20 minutes, then I go to sleep. Skeleton Consider the simplest case, $\C^* \RM \{1\}$. Can you realize its skeleton as a circle of radius $\epsilon$ around $1$, together with some circle attached to it? Let's parametrize $\C^*$ using $e^{\rho + i \theta}$, then we have a Kahler potential, that is periodic in $\theta$. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-03-23?rev=1687708402&do=diff 2023-03-23 * Discussion with Vivek * Discussion with Gus With Vivek About turning holomorphic disk into Morse Tree Let $M$ be a smooth manifold, and $L \In T^*M$ be a Lagrangian graph, i.e $\pi: L \to M$ is 1-to-1, say $L = \Gamma_{df}$ for some smooth function $f: M \to \R$. Then, we can compute $\Hom(M, L)$ using Morse theory. There is a rough matching between holomorphic strips and gradient lines on $M$ and $L$. This was Fukaya. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-03-24?rev=1687708402&do=diff 2023-03-24 * plan the papers * plan the revision text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-03-29?rev=1687708402&do=diff 2023-03-29 6 hours. * skeleton for hypertoric base mfd. * implement the idea of path conormal. * exact WKB, what's going on. Skeleton Reading Gammage-Mcbreen-Webster again, I notice that they indeed would require unimodular. But their phrasing of unimodularity is a bit strange. they require the projections to these coordinates. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-03-30?rev=1687708402&do=diff 2023-03-30 * Fukaya category of a singular space Fukaya category of a singular space Suppose you have a singular affine space $Y_0$, as the unique fiber of some fibration $W: Y \to \C$, where the singularity is not so bad. There are two ways to define the Fukaya category of the singular space. We can either do $$ Fuk(Y_0) = Fuk(Y_1) / \text{vanishing cycle} $$ Or we can do $$ Fuk(Y_0) = Fuk(Y \times \C_\eta, \eta W)$$ text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-04-06?rev=1687708402&do=diff 2023-04-06 Long time no see. When we do algebraic Hamiltonian reduction, we only use line. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-04-07?rev=1687708402&do=diff 2023-04-07 What should I accomplish today? Let me see. * Skeleton? * Frobenius? * Mina paper. Yes,let's do this. Found the best place to type paper, Vista park parking lot! text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-04-08?rev=1687708402&do=diff 2023-04-08 Evening office take too, this time, communitiy center parking lot. I become more and more like a homeless researcher, haha! Well, this 'suburban' place has no evening coffee shop, that's the reason. Now that I am almost done with this paper, should I go in and finish the last 5%? I should. otherwise it will bug me, and every hour that it left unfinished, it bugs me more. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-04-09?rev=1687708402&do=diff 2023-04-09 in the mindset of a start-up, just do the valuable thing. * finish up the work with Yixuan * finish up the work with Danny. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-04-15?rev=1687708402&do=diff 2023-04-15 Sat What can I do in one hour? text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-04-17?rev=1687708402&do=diff 2023-04-17 Mon Two adjoints Say $X = \C$ and $U = \C^*$, $j: U \to \C$. Do you know what is $j^*$ and $j^!$ of $O_X$? Actually, I don't really care about the $j^!$ (since it is a right-adjoint). <https://arxiv.org/pdf/1607.02064.pdf> This paper discusses general stuff. Very readable, except the notation needs some getting used to. The setup is as following : $X$ is some stable $\infty$-category. No need to assume compactly generated. $U$ is some subcategory which is both reflexive and cor… text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-04-18?rev=1687708402&do=diff 2023-04-18 Tuesday Don't waste time, just write the paper. * reflexive and coreflexive projector Suppose $i: D \into C$ is a full subcategory, and the inclusion preserves colimit, then we have the right-adjoint of $i$. Denoted as $R_D$. Suppose $D$ is generated by one object $d$, $$ R_D = d \otimes_{End(d)} Hom(d, -) : C \to D $$ everything is derived. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-04-22?rev=1687708402&do=diff 2023-04-22 Saturday * Paper revision * Nagao and Nakajima. Transition of Conifold, and DT transformation. Revision TJM Mirror symmetry intertwines equivalences transformation between A-side and B-side. In the case of $\C^N / \C^*$, we know how the B-side works (via toric window), and how the mirror symmetry work, via (CCC), so here is how the A-side will work. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-04-28?rev=1687708402&do=diff 2023-04-28 notes on dimer Let $X$ be a toric CY 3-fold, and this will be our A-side. Let $N_X$ and $M_X$ be 1PS and character lattice. $\Sigma_X$ lives in $N_X$. Let $P_\Sigma$ denote the compact polytope, being the convex hull of $0$ and the ray generators of $\Sigma$. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-04-30?rev=1687708402&do=diff 2023-04-30 * More on quiver gauge theory from toric CY3 * Editing the paper quiver gauge theory from toric CY3 Let's read what did Treumann-Williams-Zaslow do. Input data is a bipartite graph $\Gamma$ on a torus, Output data is a spectral curve. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-05-01-signs?rev=1687708402&do=diff 2023-05-01 signs I need to worry about signs. This is done in Seidel's book, thank goodness. At least I know if I try hard enough, I can understand it. section 11 of Seidel What is orientation? it is not a framing, but just an element in the top exterior power of the tangent space. Or, equivalently, top ext of the cotangent space. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-05-02?rev=1687708402&do=diff 2023-05-02 * chat with Denis Denis Nesterov Over lunch discussion. wall crossing What is wall crossing? You are varying stability conditions. There are some 'stack', and you want to restrict to some one nice substack, versus another nice substack, where there are large overlaps. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-05-06?rev=1687708402&do=diff 2023-05-06 Sat * Grading * Orientation * Index Reading Colin-Honda-Tian Let $\Sigma$ be a Liouville domain, with $\hat \Sigma$ be its Liouville completion. Let $D = [0,1] \times \R$, be the base disk with two punctures. Let $\hat X = \hat \Sigma \times D$. Let $s \in \R$ and $t \in [0,1]$, so $t$ is the Reeb parameter, and $s$ is the gradient flow parameter. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-05-07?rev=1687708402&do=diff 2023-05-07 Sun * SBim, character sheaves, mixed geometry * some Weinstein and Contact geometry from this HDHF paper. sheaves on $B \backslash G / B$ * paper by Ben-Zvi and Nadler: <https://arxiv.org/pdf/0904.1247.pdf> * paper by Ho and Li on mixed geometry: <https://arxiv.org/pdf/2202.04833.pdf> * paper by Ho-Li on HOMFLY-PT and Hilb on $\C^2$: text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-05-18?rev=1687708402&do=diff 2023-05-18 Thu Question to myself: * what is the wrapped Fukaya category? Can we not wrap the Lagrangian, just work with the path space between Lagrangians? What is the Floer equation? text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-05-30?rev=1687708402&do=diff 2023-05-30 visit to U Oregon. talking with Ben Elias Singular Soergel Bimodule It is very useful to consider singular Soergel module. Question: is crossing the same as merge then split? What is $x_1, x_2$? Consider $Fl(0,1,2)$, the complete flag in $\C^2$ text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-06-06?rev=1687708402&do=diff 2023-06-06 * Action by correspondence, history of quiver Hecke algebra * Lie superalgebra Quiver Hecke algebra Let $\Gamma$ be the quiver with a single dot. The $A_1$ quiver. Corresponding to the single root of $sl_2$. We consider the moduli stack of quiver representation, it is classified by the dimension vector, and in this case it is just $d \in \Z_{\geq 0}$. Let $X_d = [pt / GL(d)]$. Map from a space $X$ to this stack (ok, I like it better than BGL(d)) is the same as having a princi… text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-06-07?rev=1687708402&do=diff 2023-06-07 What is the B-side? For the simplest case, what is the $gl(1 | 1)$? In the case where there is no puncture, and just k strand, we get the fermionic cylindrical Nil-Hecke algebra. $fcNH_k$. The nilCoxeter algebra, with 0 q-grading, but minus Maslov grading. Nontrivial differential. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-06-10?rev=1687708402&do=diff 2023-06-10 * lattice and B-field lattice and B-field Let $\Sigma$ be a Riemann surface. Let $Q$ be a quiver on $\Sigma$, consist of vertices and directed edges. (No edge is contractible to a point, and no two edges are homotopic. ) Fix a hermitian vector bundle $E$ over $\Sigma$ with unitary connection $\nabla$. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-06-11?rev=1687708402&do=diff 2023-06-11 * Categorification of what action? KWWY, just Lie algebra Here they defined a parabolic Coulomb branch. Recall what is a Coulomb branch, you first construct the BFN space. No? No. You first construct the $V(K)/G(K)$ stack, then you consider $V(O) / I_P$, where $I_P \In G(O)$ is such that at the center of the disk, we restrict the group to be in $P$. Why do we want that? Well, $G(O)$ is the gauge transformation group. So, $I_P$ is saying, you can do whatever gauge transformation … text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-06-14?rev=1687708402&do=diff 2023-06-14 * Reading Teleman * chatting with Spencer The role of Coulomb branches in 2D gauge theory What's the input data? A compact Lie group (what's the difference between this and a complex reductive group?) and a polarisable quaternionic representation $E$ (ok, this is saying we can write $E =T^*V$, but we don't have a canonical choice of $V$, and we shouldn't fixiate on a choice.) text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-06-15?rev=1687708402&do=diff 2023-06-15 * Discussion with all Teleman's recipe If you point a gun at me, and ask me the gluing formula, I would say: * introduce two identical copies of the pure Coulomb branch algebra (I don't know why two?) * ok, you have two sets monopole operators, and the equivariant coefficient ring. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-06-17?rev=1687708402&do=diff 2023-06-17 * equivariant cohomology and localization equivariant Let $G$ be a compact connected Lie group, acting on a smooth manifold $M$. Suppose we have another $G$-manifold $N$, and a map $\pi: M \to N$, then I want to integrate along the fiber $$ \pi_*: H^*(M) \to H^{*-d}(N). $$ This should be a module map of $H^*(N)$, where we view $H^*(M)$ as a module over $H^*(N)$. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-06-18?rev=1687708402&do=diff 2023-06-18 * still working on Teleman's shift operation Teleman's shift Let $G$ be compact Lie group. the pure gauge theory case is fine, just equivariant homology of the based loop space. Now, you have a representation $V$, and $G$ acts on $V$. If you are BFN, you can consider a sphere, with a principal $G$ bundle over it, which is made from a $S^1 \to G$ a cluching function. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-06-20?rev=1687708402&do=diff 2023-06-20 * muse on euler class euler class Yesterday I read about BFN's construction. One thing that strikes me is the appearance of 'euler class'. I have never really understood euler class. the input is some bundle over some space, and the output is some cohomology class on the space itself. The only natural thing that I can think of, is the intersection of the zero section with its own perturbation. However, thinking in terms of perturbation is not very intrinsic. this thing should … text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-06-21?rev=1687708402&do=diff 2023-06-21 I don't like the feeling of 'having to do something', such as, writing this paper. I need to persuade myself again and again to write it. The only reason that I have is, I need to write this paper, so I am done with it. That's not a good feeling. text/html 2023-06-27T18:32:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-06-27?rev=1687890732&do=diff 2023-06-27 Back to action. I still want to pin down the two patches gluing philosophy, why it works. BFN approach of matter diversion on topology Let's tell the story just one more time. We have $GL_1(\C)$ acting on $\C$ in the standard way. We have two stacks, one is $\C(K) / GL_1(K)$, one is $\C(O) / GL_1(O)$. It is a set with some group action. text/html 2023-07-08T18:09:30+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-07-07?rev=1688839770&do=diff 2023-07-07 yesterday, I was stuck on abelianization. One question is, given $G$ acting on $V$, and $T$ a maximal torus in $G$, what can we say about the Coulomb branch spaces $M(T,V)$ and $M(G,V)$? First, consider the case where $V=0$. We know $M(T,0) = T^\vee \times Lie(T)$, or $$ A(T, 0) = H_*(T(O) \RM T(K) / T(O) ) = \C[T^\vee] \times H_T^*(pt). $$ What does a $T(O)-T(O)$ orbit look like in $T(K)$? well, it is indexed by the pole order $T(K) \to \Z$. And for each pole order, the action… text/html 2023-07-16T06:02:47+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-07-15?rev=1689487367&do=diff 2023-07-15 Let me write down the statement, and call it a day. Of course, I still have many doubts about the thing we do, and about equivariant localization. But, let's keep going. text/html 2023-07-16T17:53:28+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-07-16?rev=1689530008&do=diff 2023-07-16 * planning for the Coulomb branch planning Let me think about my desired statement first. What do I want to state in this section. Yes, I need to discuss about the geometry of the Coulomb branch in this section. What is my point? Don't be bind up by the setup, do your own setup. If I were to rewrite this paper, I would do this: immediately after the introduction, I will do the most important computation, 2 strands nilhecke, and the interaction ones. They can be stated without any… text/html 2023-07-20T20:09:45+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-07-20?rev=1689883785&do=diff 2023-07-20 the example on abelian Coulomb branch $\gdef\acal{\mathcal A}$ $\gdef\tcal{\mathcal T}$ $\gdef\K{\mathcal K}$ $\gdef\O{\mathcal O}$ The next simplest example is the abelian gauge theory with matter. Let $G = GL_1$ and $V=\C$. We find that $$ Spec \acal^+(G,V) = \{(u,v,z) \in \C^3 \mid u v = z\}, \quad Spec \acal^\times(G,V) = \{(u,v,z) \in \C^2 \times \C^* \mid u v = 1-z\}. $$ where one can see \cite[Section 4.1]{BFN} for details. text/html 2023-07-21T19:20:26+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-07-21?rev=1689967226&do=diff 2023-07-21 Have to say that the papers by Nakajima is really high level, not just stating the definition, proposition and proof, but like talking. But I really need to learn, what are all these instanton counting, vertex function about, before I can understand what's going on. text/html 2023-07-23T08:14:20+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-07-22?rev=1690100060&do=diff 2023-07-22 Reading BFM BFM They want to understand what is the coherent Satake category, and the first step is to understand the K-theory, and the graded version. Do you have any intuitive reason that, horizontal Hilbert scheme, or universal centralizer, or open Toda lattice would show up? text/html 2023-07-25T06:32:29+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-07-24?rev=1690266749&do=diff 2023-07-24 * reading Chriss-Ginzburg the whole morning * watched a youtube video 8 traits of successful people * try to fix the annoying keyboard on a macbook pro, which turns out to be not my (or my wife's) fault. * found an interesting lecture note of Teleman on rep theory. Never really understood what is character formula. text/html 2023-07-25T23:48:42+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-07-25?rev=1690328922&do=diff 2023-07-25 * problem with torsion in $\pi_1(G)$. Torsion So, what's the difference between $GL_2, PGL_2, SL_2$? $GL_2$ is the father of all, $PGL_2$ and $SL_2$ each take out some abelian part out of it. $PGL_2$ totally killed the central subgroup, by quotienting out it, whereas $SL_2$ tries to take a slice, that intersects minimally with it. text/html 2023-07-27T05:59:44+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-07-26?rev=1690437584&do=diff 2023-07-26 Declaration: I was lost in abstract definitions for too long. I will now compute examples, examples, and examples. Not only because abstraction makes me sleepy, but also examples makes my hands dirty and my mind happy. * Reading Ginzburg's paper text/html 2023-07-27T23:05:39+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-07-27?rev=1690499139&do=diff 2023-07-27 * what to say next Review of BFN * What is the desired statement? We want to say that the general Coulomb branch maps to $T/W$. Yes, and the fiber near a point is given by the centralizer of the particular torus element, and the matter representation. Why do you only keep those representation whose weight pairs zero with the group or Lie algebra element? text/html 2023-07-31T08:24:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-07-30?rev=1690791840&do=diff 2023-07-30 * abelian with matter is the trouble Coulomb branches Compare with pure abelian gauge theory, the Coulomb branch of general cases needs two direction modifications, with matter representations, and with non-abelian gauge group. In the abelian case, how do we deal with matter? We specify the multiplication table for the 'monopole operators', a vector space basis over the base coefficient ring. It is a bit brutal force. text/html 2023-08-01T06:43:58+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-07-31?rev=1690872238&do=diff 2023-07-31 * obsidian? Obsidian I saw some of my classmates using it, and I would like to give a try, say 30 min. * <https://obsidian.md/download> , download * ok, need to create a 'vault' (ok, cool name, folder, box, repository...). should I do a remote one? let's see how expensive is that. Google says, it is 10 USD / month. forget it, I will just store it in icloud, right? text/html 2023-09-03T19:07:42+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-08-01-academia-or-not?rev=1693768062&do=diff 2023-08-01 academia or not The application season is coming. But I do not want to apply again. I think I will get rejected again. I will just write papers, write codes, and do stuff for people to make money. That being said, I am free and I am going to work on stuff that I am really interested in. text/html 2023-08-06T04:10:38+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-08-05?rev=1691295038&do=diff 2023-08-05 * write up the notes that are useful for myself. * why quiver gauge theory has anything to do with Kac-Moody algebra? The stuff that I typed up below, are so incoherent and dreamy, that I don't know what am I talking about. So they should be either cleaned up or deleted. text/html 2023-08-31T07:36:19+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-08-30?rev=1693467379&do=diff 2023-08-30 Let's think about hypertoric variety, Gale duality. Given a vector space in $\R^N$, we have $$ V \to \R^N \to (V^\perp)^* $$ $$ V^\perp \to (\R^N)^* \to V^* $$ Great. Given $\eta \in (V^\perp)^*$ and $\xi \in V^*$, we look at the fiber $V_\eta$ and $(V^\perp)_\xi$, they are partitioned by the restriction of the sign partitions in $\R^N$ and $(\R^N)^*$. text/html 2023-09-01T19:56:34+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-08-31?rev=1693598194&do=diff 2023-08-31 So I talked with YX a bit on 'what is the weight grading', in mixed sheaves. It is just the eigenvalue of the Frobenius action, on etale sheaves. Then, why the endomorphism of the big tilting has that weight grading. text/html 2023-09-03T20:13:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-09-02?rev=1693771996&do=diff 2023-09-02 I read Tom Braden's paper on mixed category for toric varieties. At least one idea is useful, namely where does the extra grading come from. The idea is that, given a vector space with a unipotent operator, then you can split the vector space into Jordan blocks. Each Jordan block has a Z-indexed-filtration, with the index well-defined upto a global shift. text/html 2023-09-05T07:10:06+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-09-04?rev=1693897806&do=diff 2023-09-04 It is useful to recap what I did today, or these days. I have been thinking about Koszul duality a lot these days. * where does the mysterious grading come from. What is mixed category? The terminology is weird that, mixed constructible sheaf does not form a mixed category. you need to take a (not full) subcategory. text/html 2023-09-06T16:39:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-09-05?rev=1694018396&do=diff 2023-09-05 It is so annoying that I cannot pin this done. Let me try again. Let's say, just in the hypertoric case (which I know I am cheating, but whatever). You have two spaces, why there is a matching of Fukaya categories for two seemingly unrelated spaces? Even just for $T^*P^{n-1}$ and $\C^2 / \mu_n$ resolution, why? text/html 2023-09-07T08:42:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-09-06?rev=1694076130&do=diff 2023-09-06 Hey, I made some progress today. About Koszul duality, at least, I have a concrete conjecture. Let's state it as 'The symplectic Koszul duality for category O'. It goes as following: take a 3d N=4 gauge theory, some compact group $G$ acting on some representation $T^*N$. You can form two LG A-models, let's call that $(M_{H, \alpha}, W_{H,\beta})$ and $(M_{C, \beta}, W_{C, \alpha})$, where $\alpha$ and $\beta$ are some parameters. Then you prove that the two wrapped Fukaya categories … text/html 2023-09-07T19:43:43+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-09-07?rev=1694115823&do=diff 2023-09-07 Now, what is the expectation? We should use the KLRW algebra as the benchmark to tell me what grading on the endo of the T-brane I should get. Basically, dot has positive grading 2, crossing with a puncture has grading 1. First question, you claim that, you have an $S^1$-family of symplectic form, show me. Consider $\C^2 / \mu_n$, with weight $(1,-1)$, $n=2$. We consider the coordinate ring $$ \C[x,y]^{\mu_2} = \C[xy, x^2, y^2] $$ when you blow-up, you put in the ratio coordinate $u… text/html 2023-09-14T00:10:30+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-09-12?rev=1694650230&do=diff 2023-09-12 The new exciting thing today is, the raising and lowering operator. raising and lowering Let $E$ be the functor for adding strands, and $F$ the functor for removing strand, then we should * define the functor $F$ * check that $F$ and $E$ are adjoint text/html 2023-09-16T22:41:03+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-09-16?rev=1694904063&do=diff 2023-09-16 The removing strand operator is not that simple: taking intersections, and putting in the object. There must be some interesting differentials correcting it. Let $L^k$ be a k-tuple of Lagrangians in $\Sym^k(\Sigma)$, avoiding a stop. Let $E$ be the raising operator, i.e., $F$ the lowering operator. Let $F$ be adding a brane, by adding a T-brane, and $E$ be $Hom(T, -)$. (note the change of notation). text/html 2023-09-21T06:15:51+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-09-20?rev=1695276951&do=diff 2023-09-20 so you found a secret rule to define the differential, ok, good for you! how to prove that it works? they are guesses, though very solid ones. 1. you need to define a functor; 2. you need to prove an excision lemma, things are only dependent on the boundary, which reduces this question to the many stop case. text/html 2023-09-25T07:13:27+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-09-24?rev=1695626007&do=diff 2023-09-24 1. after discussion with M yesterday, I realized I need more terms in the differential, ok, not bad. I should write up some examples, for other people to get it. 2. I don't know how it is related with representations of $gl(1|1)$. I think it is about Alexander polynomial. (is it about oriented knot?) text/html 2023-09-29T06:00:30+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-09-28?rev=1695967230&do=diff 2023-09-28, Thursday How does sl2 work? We know that, by KLRW algebra, the planar version, modulo some stop, there is only one way to realize the categorification. I bet, we can do a purely downstairs theory, even define it. So, how do we define it? Suppose, we say that hom betewen T-branes follows KLRW algebra, then what? text/html 2023-10-02T08:09:13+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-10-01?rev=1696234153&do=diff 2023-10-01 fk, a month has passed. what do I want? I want raising and lowering operator, which is adding and removing strand operator, which is a special case of gluing an extra guy and put some extra strands operator. text/html 2023-12-06T21:42:32+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-12-05?rev=1701898952&do=diff 2023-12-05 well, two months passed. what did I learn today? with Alexei's talk yesterday and the discussion today. what I text/html 2023-12-07T15:13:29+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-12-06?rev=1701962009&do=diff 2023-12-06 going to revise the VGIT paper. Here, the base is one dimensional, fiber is easy. We can take the easy way out, just prove enough for this case. Or, we can do full proof. I don't want to be super general, but just for this case. I want to consider skeleton on the total space, and skeleton on a sub-level set. text/html 2023-12-07T20:24:08+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2023-12-07?rev=1701980648&do=diff 2023-12-07 browsing papers 1. <https://arxiv.org/pdf/2312.01949.pdf> Integral HMS. Would be interesting to learn how the HH go, and how to go from Fukaya category to curve counting, and what is mirror map. 2. text/html 2024-01-07T01:04:35+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2024-01-05?rev=1704589475&do=diff 2024-01-05 In the simplest setting, we have mirror symmetry for $Coh(\C^* \times \C^2)$. Next, we are going to take symmetric power. Do you remember what happens when two eigenvalues collide? No, don't do Hermitian matrices, that will never be nilpotent. What if you have a matrix that looks like $( (1,1), (0, 1+x) )$. What is the eigenvector for e.v. $1+x$? How about $(1,x)$? Eigenvector for $\lambda = 1$, is $(1,0)$. So you see, when $x \to 0$, the two eigenspaces also collide! text/html 2024-01-16T04:21:43+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2024-01-14?rev=1705378903&do=diff 2024-01-14 a category with a notion of 'equal', or 'isomorphism', or 'quasi-isomorphism'. * category of set? then isomorphism, bijection * category of vector space? isomorphism * of chain complex of abelian groups? Well, quasi-isomorphism may not be. text/html 2024-05-21T05:18:29+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2024-05-20?rev=1716268709&do=diff 2024-05-20 What's new? * What is the central charge formula on the B-side? What's the relationship with stability condition? SOD? * What's Xin Jin's story? How does the open torus embed? text/html 2024-06-11T07:18:05+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2024-06-10?rev=1718090285&do=diff 2024-06-10 koszul duality physics side the story is about 3d mirror symmetry. from a 3d N=4 SYM theory, you have Coulomb branch and Higgs branch as low energy effective field theory. Consider boundary conditions. If we did A-twist in the bulk, we have A-twist in the boundary. 2d A-model with a target space that has a G-action. text/html 2024-06-30T06:40:16+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2024-06-29?rev=1719729616&do=diff 2024-06-29 braiding functor / bimodule If I have an $A-B$-bimodule $K$, and I have a right $A$-mod $M$, then I can do $$ M \otimes_A K$$ to get a right $B$-mod. And conversely, given a right $B$-mod $N$, I can do $$ N \otimes_B K^\vee = Hom(K, N) $$ to get a right $A$-mod, assuming $K$ has enough dualizablity (whatever that means). text/html 2024-07-22T06:32:33+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2024-07-21?rev=1721629953&do=diff 2024-07-21 reading Etingof on quantum group quantum group quantum group is just a fancy name for Hopf algebra. A Hopf algebra is just an algebra with $(\Delta, \epsilon, S)$, coproduct, counit and antipode satisfying a bunch of axioms. $\epsilon$ and $S$ are determined by $\Delta$. For the algebra of function $O(G)$, the coproduct is interesting. It determines $\epsilon$ to be the restriction to $e$. But then, given a function $f$, we pullback, and restrict to the anti-diagonal. fine, tha… text/html 2024-07-23T22:44:13+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2024-07-23?rev=1721774653&do=diff 2024-07-23 * Goal: understand Yetter-Drinfeld module, Drinfeld double and then Andy Manion's comment paper What is YD-module Let $H$ be a Hopf algebra, and $Rep(H)$ is the cat of finite dim rep. Let $(Y, \varphi)$ be such central element, then we have the following rep $$ \varphi: H \otimes Y \to Y \otimes H $$ As written this is a rep H morphism. We may restrict to $ \{1\} \otimes Y$, and get a linear map $$ \tau: Y \to Y \otimes H. $$ This has the potential to be a (right) co-action of $… text/html 2024-08-24T17:42:59+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2024-08-24-b-side-for-knots?rev=1724521379&do=diff 2024-08-24 B-side for knots Mina has the whole categorifications on A-side and B-side, whereas Cautis-Kamnitzer also had some earlier B-side construction. I need to understand the relations. text/html 2024-09-10T16:38:18+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2024-09-08?rev=1725986298&do=diff 2024-09-08 Let's summarize what can be written. * HMS for K-theoretic Coulomb branches HMS for K-theoretic Coulomb branches basically we want to prove half, or even less than half (no generation result). so, this can only be called as, symplectic realization of multiplicative KLRW algebra. text/html 2024-09-11T19:56:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2024-09-11-quiver-hecke-algebra?rev=1726084615&do=diff 2024-09-11 quiver hecke algebra After so many years, let me read up on the 'convolution algebra' presentation for quiver hecke algebra, because after all, hecke algebra originates from convolution on flag variety, and quiver hecke algebra is a generalization. text/html 2024-09-12T22:09:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2024-09-12?rev=1726178950&do=diff 2024-09-12 reading the classic papers, Khovanov-Lauda, Rouquier, Varagnolo-Vasserot, to dig out how to setup the convolution algebra (in general) I don't even understand why these things can be realized using Fukaya category and Floer theory, besides 'physics motivation'. What's the math motivation? text/html 2024-11-03T05:47:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2024-11-02?rev=1730612832&do=diff text/html 2024-12-05T09:51:10+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2024-12-04?rev=1733392270&do=diff 2024-12-04 When migrating the server, I forgot to backup the yiye website, lost a lot of memories, sad. I also dumped a lot of my phd notes to dumpster anyway when I move away from Chicago, it is inevitable to let go. If I consider coherent sheaf category as the category of B-branes, then how does it depends on the complexified Kahler parameters? One can either do it in the physical way, reading Witten, Hori etc; or one can do it in the math way, thinking about deformation of the Coh category… text/html 2024-12-06T06:53:49+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2024-12-05?rev=1733468029&do=diff 2024-12-05 What is that hemisphere partition function? Consider a gauged linear sigma model with superpotential, namely a reductive complex Lie group $G$ acting on $V$ preserving volume form and a superpotential $W: V \to \C$. To get $\Z$-graded MF category, we choose certain non-negative $R$-charge (could be in $\Q$) so that $W$ has weight $2$. text/html 2024-12-09T07:37:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2024-12-06?rev=1733729874&do=diff 2024-12-06 So Mauricio asks, what is the 'A-side' window? I thought I know the answer, a cheap one, not in the form of a generator, but just in terms of a subcategory generated by certain classes of objects. And I am not sure if it is right. mirror to non-abelian GLSM text/html 2024-12-21T08:28:25+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2024-12-20?rev=1734769705&do=diff 2024-12-20 What is the disk with three stops? Disk with two stops, $k$ strands T-brane, has endomorphism algebra $NH_k$, with $q$ grading for crossing $q^{-2}$. Correspondingly, we have $(\pi: BB \to BG)_* \C_{BB}$, whose endomorphism involves $\pi^!$. Recall that for sheaves (not coherent sheaves), $\pi^! = \pi^* [\dim_\R fiber]$. This explains why we have those negative cohomological degrees. text/html 2024-12-26T06:15:54+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2024-12-23?rev=1735193754&do=diff 2024-12-23 Let's keep on thinking about disk with three stops. There are a few things in common with disk with two stops and one feature in the middle, like a hole, a puncture. One need to decide if the T-branes are on the left or on the right. In the case of punctures, or several punctures, we are looking at how the standard flag in $\C^k$ map to the target flag $0 \In \C$. text/html 2024-12-26T07:05:41+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2024-12-25?rev=1735196741&do=diff 2024-12-25 It's a good place to work and study, the Joshua Tree Field Station (very cool hotel). I am still trapped by the disk with 3 stops, no punctures. But do you understand the disk with 2 stops, how the gluing works? one disk has $k_1$ strands; another disk has $k_2$ strands, both with $2$ stops, put them together, get $k=k_1+k_2$ strands. text/html 2025-01-02T07:23:53+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-01-01?rev=1735802633&do=diff 2025-01-01 I watched a bunch of Tim Logvinenko's video, talking about generalized braid category and its representations. The origin of the story is trying to understand Caustic-Kamnitzer's construction, namely, what acts on the ambient space's coh category rather than the slice's. text/html 2025-01-04T07:14:42+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-01-02?rev=1735974882&do=diff 2025-01-02 I still want to understand what Cautis-Kamnitzer-Licata did. Coherent Geometric Satake One motivation is the geometric of geometric Satake, which says $Rep(G)^{fd}$ and $Perv_{G^\vee(O)}(Gr_{G^\vee})$ are related, simple representation $V(\lambda)$ goes to IC sheaf $IC_\lambda$. The guess is, if $\lambda$ is miniscule, then $Gr_\lambda$ is smooth projective, and the graded dg category $Coh_{\C^*}(Gr_\lambda)$ can be used to do categorification of $V(\lambda)$. Recall $\C^*$ acts o… text/html 2025-01-06T10:12:32+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-01-05?rev=1736158352&do=diff 2025-01-05 Today, I did * more study on the slices of affine Grassmannian * Return to CKL Vasily Krylov's note I run into a note by Vasily Krylov's note on slices. $\gdef\Gr{\mathcal{Gr}}$ There are two key new ideas, * one is that $\Gr$ is $Bun_G(\P^1)$ with a trivialization on $\P^1 \RM 0$ text/html 2025-01-06T10:31:21+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-01-06?rev=1736159481&do=diff 2025-01-06 We had a long detour into Cautis-Kamnitzer's constructions, at least we had some familiarity with flag variety. But, I still do not know about disk with 3 or more stops. * Disk with 2 stops and $k$-strands, is assigned to $D-mod(pt/GL(k) )$ which is derived equivalent to $NH_k-Mod$. text/html 2025-01-08T07:29:12+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-01-07?rev=1736321352&do=diff 2025-01-07 * Mina relates Cherns-Simon's partition function $CS(\Sigma \times S^1)$ with our rank of $K$-theory formula. * Lecture note on Ringel-Hall algebra. text/html 2025-01-10T18:51:06+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-01-09?rev=1736535066&do=diff 2025-01-09 * colimit of algebras colimit Let's be really naive and simple. What is a colimit? Suppose you have a category $\mathcal{C}$, many objects, morphisms. And then, there is a push-out diagram $A \gets B \to C$. This gang of objects talks with everyone, for example, someone called $X$. $A$ talks with $X$, get a set $Hom(A,X)$, so does $B$, get $Hom(B,X)$. It might be big or small, we don't know, depends on $X$'s relation with all of them. Oh, and don't forget $Hom(C,X)$. So, how do … text/html 2025-01-15T20:58:35+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-01-15?rev=1736974715&do=diff 2025-01-15 I have been thinking about Liouville sector. The condition on stop is very harsh, in the sense that the Liouville flow need to preserve the boundary. I am not sure if GPS themselves constructed these required structures. * Construct Liouville structure? text/html 2025-01-23T00:55:49+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-01-21?rev=1737593749&do=diff 2025-01-21 I was discussing with Yixuan yesterday. He mentioned a few works by Siu-Cheong Lau are noteworthy. * Localized mirror functor. No, not just probing the space using a compact immersed Lagrangian, but with deformation, allowing immersed Lagrangians. text/html 2025-01-24T03:04:55+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-01-23?rev=1737687895&do=diff 2025-01-23 * Discussion with Xin Jin about her proof Xin Jin's result Let $G$ be a (adjoint or simply connected?) group. We consider $J^G_\gfrak$, the regular universal centralizer. What's the definition? * I guess one definition is, take $T^* G$, consider the text/html 2025-02-16T21:11:48+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-02-16?rev=1739740308&do=diff 2025-02-16 A new approach on Fukaya-Seidel category Let $X$ be a smooth affine complex manifold, and let $f: X \to \C$ be a holomorphic function. From this data, we should get a category (up to equivalences). At this moment, we did not specify the Kahler form (which is not important), just as Riemannian metric is not important for homology of a manifold. It does not matter which car you use to drive from A to B. text/html 2025-02-20T07:12:20+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-02-19?rev=1740035540&do=diff 2025-02-19 Gaiotto-Moore-Witten What is this algebra of infra-red from GMW about? What did Doan-Rezchikov do about comparing Fukaya-Seidel and GMW? What did Kapranov-Kontsevich-Soibelman do? What does the critical lines look like? Why do we need web? text/html 2025-02-27T09:05:42+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-02-26?rev=1740647142&do=diff 2025-02-26 The 3d GMW theory. In Kapranov-Kontsevich-Soibelmann paper, 10 years ago, they mentioned that it is possible to consider marked polytope in $\R^3$. There is a $E_3$-algebra controlling the deformation of $E_2$-algebra, and there can be coefficient enhancing all these. I want to understand what precisely is the statement. text/html 2025-03-14T18:22:52+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-03-14?rev=1741976572&do=diff 2025-03-14 Cech what is cech There are three things that has to do with the Cech cover * one is the computation of global cohomology, when we do $\Hom(\C_X, -)$, we can do a Cech resolution of $\C_X$ by take open subjects $j_{U_j}: U_j \to X$, and do $j_{U_j,!} \C_{U_j} \to \C_X$, and consider pulling back. The same can be for coherent sheaves $j_! \mathcal{O}_U \to \mathcal{O}_X$, the hom from $j_!$ is computed by a colimit, so $j_! O_U$ itself is a projective limit, it represent the fun… text/html 2025-03-16T19:09:06+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-03-15?rev=1742152146&do=diff 2025-03-15 proving isomorphism reducing bar resolution How to show a chain map is a quasi-isomorphism? easy, just provide the map in the reverse direction. So, you have a functor and you want to show that it is fully faithful. You got two objects, $X,Y$, send them over to $FX, FY$, there is an injective map $$ \Hom(X,Y) \to \Hom(FX, FY) $$ and we know the image. We probably also know the inverse chain map, just map the top of the chain complex back. small to big back to small is OK, big to… text/html 2025-03-21T06:23:57+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-03-20?rev=1742538237&do=diff 2025-03-20 Dold-Kan Dyckerhoff introduced categorified Dold-Kan correspondence. Before categorification, given a simplicial abelian group, we can make it into a chain complex by taking 'totalization'. Given a chain complex (in positive degree, differential decreases degree by 1) text/html 2025-05-11T17:03:03+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-05-11-super?rev=1746982983&do=diff 2025-05-11 super Here is a list of references for representation of super Lie algebra $gl(m|n)$. * Cat O of $gl(m|n)$ by J. Brundan * A sketch of super Lie algebra by V. Kac * GSM 144 text/html 2025-07-03T09:00:08+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-07-02?rev=1751533208&do=diff 2025-07-02 i want to translate, wedrich-dyckerhoff, into our own language. First, what is Beck-Chevalley stuff? I am reading page 10 of DW. There are two ways to go from $(2,1)$ to $(1,2)$ partition on the line, one is break-then-merge, the other is merge-then-break. The two ways are different, obviously. And even better, there is a relation between the two ways, how? We have adjunctions, in our setting, splitting is the right adjoint (splitting is restriction) of merge. (we can have various … text/html 2025-09-05T22:25:43+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-09-05?rev=1757111143&do=diff 2025-09-05 how to describe perverse schober on $(D, 0)$? it is a machine that, input a disk with stop / singular Lagrangian skeleton / holomorphic function on an open subset, and output a category; input a morphism of object, output a functor; finally, input a 2-morphisms between 1-morphism, and output a natural transformation. text/html 2025-10-23T22:50:36+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-10-23?rev=1761259836&do=diff 2025-10-23 to do: * what's holomorphic Floer theory? In relation to NAHT. * what's the idea / statement of geometric langlands? * what zastava? no, I am not the kind of mathematician that writes proofs, I am rather the one that discovers structures and building bridges. text/html 2025-11-05T04:06:08+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2025-11-04?rev=1762315568&do=diff 2025-11-04 I need to write more stuff down, otherwise it is only in my head, mysterious even to my collaborator. reading the paper that is based on my phd work, makes me a bit happy and sad. happy that my work is useful, sad that I am still in the jobless state. But at least, I am doing math and discussing math with interesting people. text/html 2026-01-28T22:52:35+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2026-01-28?rev=1769640755&do=diff 2026-01-28 Talking with Hansol about conic fibration over some hypersurface. He mentioned Seidel's paper on Lefschetz fibration's suspension, which I am reading now. Seidel's Paper <https://arxiv.org/pdf/0907.2063> text/html 2026-02-27T21:46:35+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2026-02-24?rev=1772228795&do=diff 2026-02-24 I am recording this past discussion with Ben G. Here (n+1)-dim TFT will be given by some n-category, we use $nA$ or $nB$ as the corresponding category. Let $G$ and $G^\vee$ be Langlands dual groups. Assume relative Langlangds says there are some equivalence of 3-cats $$ 3A_G \cong 3B_{G^{\vee}}, \quad 3IndPerv(BG) \cong 3IndCoh(BG^\vee). $$ where objects in $3A_G$ comes from Ham G-space $M = T^*X$, where $X$ are $G$-spherical varieties, which means Borel acts with finitely many obj… text/html 2026-02-27T20:56:51+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2026-02-27?rev=1772225811&do=diff 2026-02-27 Talked with Kifung and Conan about how 3d MS will transport brane to brane. There are some partial success, but still more to be understood. One interesting thing is that they somehow don't need to specify which side is 3dA-side and which is 3d B-side. Their prescription is somewhat simple, just take fiber product of Lagrangians in (possibly shifted) (real or complex) symplectic manifold, then apply Fuk or Coh to the fiber product. text/html 2026-03-01T17:25:38+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2026-02-28-2?rev=1772385938&do=diff 2026-02-28 (2) Previously, we have said that $M_H$ and $M_C$ are $n=0$-shifted symplectic stack. And there is no natural way to shift the $n$ around, so naturally the home for 3d MS is about 0-shifted symp stack. Now, suppose we are given two hol'c symp Lagrangians, then their intersection is $-1$-shifted symp mfd. $$ X \times_{T^*X} X = T^*[-1] X = Spec_X(Sym(T[1]X)) $$ text/html 2026-03-01T03:57:26+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2026-02-28?rev=1772337446&do=diff 2026-02-28 Here is some vague thought about 3d mirror symmetry (following Ben G and Justin H). Let $G$ and $G^L$ be Langlands dual varieties. Assume $(G, M)$ is S-dual to $(G^L, M^L)$, where $M$ is some nice G-Ham space, same for $M^L$. Example $$ M = pt, \quad M^L= Whit_{G^L}(T^*G^L) $$ where $Whit_{G^L}$ is one-side symp reduction by $U$ with generic character. text/html 2026-03-12T05:42:37+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2026-03-11-should-i-be-a-bs-artist?rev=1773294157&do=diff 2026-03-11 should I be a BS artist? The first time I heard about this phrase, it is used by Prof Z about some work by Prof B. It is half joking, half disapproval. One either needs to painstakingly verify lots of details, or one can wave ones' way out. Through my academic career, I have seen many handwaver, but also more solid prover. text/html 2026-03-21T23:46:46+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2026-03-21?rev=1774136806&do=diff 2026-03-21 In order to prove some Liouville pair is a Weinstein pair, we need to know that the stop is good. If our space is like $\R \times \R_-$, one factor of space is cutting off some factor, but leaving some other factors intact. How does the fiber look like? It would be just the stop-fiber from the relevant factor, times the entire space from the non-partipating factor. So to show the Weinstein-ness, one just need to show that the two factors are. Now, where is the participating factor?… text/html 2026-03-23T18:18:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2026-03-23?rev=1774289936&do=diff 2026-03-23, let there be I want a wiki / blog tool, that is online and easy to use and share. overleaf is good, but not good at sharing or updating. notability is good and smooth, but not good at sharing. text/html 2026-03-28T13:24:11+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2026-03-27?rev=1774704251&do=diff 2026-03-27 In the nicest setting, max of smooth psh function is still psh, but with kink when the dominant term switch over. To solve this problem, people developed softmax, which is a smearing of max function. When we softmax a bunch of psh function, the outcome is smooth and psh. text/html 2026-04-13T08:00:05+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2026-04-12-a-new-possibility?rev=1776067205&do=diff 2026-04-12 a new possibility our goal is to prove bar gluing, namely colimit of a bar diagram is the desired Fukaya category. VS provides a new method, let's see how it works. we added two stops to the picture. and split the picture into left middle and right. middle can map to left and right. text/html 2026-04-27T15:57:25+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2026-04-27?rev=1777305445&do=diff 2026-04-27 new space and new function. I want to study $[1]-(1)$ quiver. In the sense of how to see it as framed zastava space. text/html 2026-05-22T07:02:30+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2026-05-21?rev=1779433350&do=diff 2026-05-21 Yuji proposed an interesting construction of category on a disk with stops. The bulk is decorated with some category $C$, and stops are decorated with something else, like $D_1,\cdots, D_n$. Then we have functors $D_i \to C$. We want to take some sort of global section on this. text/html 2026-06-12T04:46:11+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/2026-06-11?rev=1781239571&do=diff 2026-06-11 I am thinking about proving multiplicatve - multiplicative HMS, with Spencer, and with quantization. text/html 2023-07-16T17:08:42+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/blog/start?rev=1689527322&do=diff blog