https://pzhou.org/ 2026-05-22T21:16:58+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/notes/2022-10-25-mina-aganagic-categorification-of-knot?rev=1687708402&do=diff 2022-10-25, Mina Aganagic, categorification of knot Talk at MSRI gauge theory workshop. Khovanov homology and Jones polynomial Given a knot or link in $\R^3$, Jones produces a 1-variable polynomial $J_K(q)$. Khovanov upgraded that polynomial to a bi-graded vector space, and taking Euler characteristics in one grading, put formal variable $q$ in the other grading, recovers Jones polynomial. The construction is very much functorial, compatible with knot cobordism. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/notes/2022-10-27-bridgeland-on-joyce-structure?rev=1687708402&do=diff 2022-10-27, Bridgeland on Joyce structure Talk given in the M-seminar. Key words: Joyce structure, $\tau$-function, complex hyperkahler structure. text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/notes/2022-10-28-arnav-hyperkahler-mirror-symmetry?rev=1687708402&do=diff 2022-10-28, Arnav, hyperkahler mirror symmetry Talk at MSRI workshop text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/notes/2022-11-08-cherednik-on-hecke?rev=1687708402&do=diff 2022-11-08, Cherednik on Hecke 1: Hecke algebra in rep theory very funny comment about real math and imaginary math, one can see he is an concrete guy with feet on ground (i.e. real axis). * What is a zonal spherical function? From wiki, it is a text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/notes/2022-12-29-teleman-s-paper-on-2d-3d-tft?rev=1687708402&do=diff 2022-12-29 Teleman's paper on 2d-3d TFT <https://ems.press/content/serial-article-files/12429> * $U_1$ theory first $U_1$ theory He really starts from the very very beginning. The abelian theory. But, how does $G(O)$ acts on $G(K)/G(O)$, when $G = \C^*$. I guess it must be trivial. Similarly, how does the compact group $G$ acts on the based loop space $\Omega G$? I guess it must be by conjugation. But, how do I reconcile with the two descriptions? text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/notes/2023-04-11-about-khovanov-homology?rev=1687708402&do=diff 2023-04-11 about Khovanov Homology 1. Khovanov's paper, 'categorification of Jones polynomial'. This was in 1999, like 20 years ago. about 1500 citations. wow, this is pretty clear. Why we want to improve on this? text/html 2026-02-09T10:28:40+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/notes/2026-02-09?rev=1770632920&do=diff A Natural Transformation that vanishes on generator objects Motivation $\gdef\cal{\mathcal }$ $\gdef\ccal{\mathcal C}$ $\gdef\dcal{\mathcal D}$ Let $\cal C, \dcal$ be pre-triangulated $A_\infty$ (or dg) categories, $F, G: \ccal \to \dcal$ exact $A_\infty$ functors. We assume $\cal C_{gen}$ is a full subcategory of $\cal C$ that generate $\cal C$ in the sense that $Tw(\cal C_{gen}) \cong \cal C$. text/html 2026-03-18T16:53:31+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/notes/2026-03-17?rev=1773852811&do=diff Batyrev-Borisov Construction This note is my attempt to understand how the mirror construction works, and how it compares with the monomial-divisor toric correspondence. Batyrev-Borisov construction Let $V$ be a real vector space, $V^\vee$ its dual. For $x \in V, y \in V^\vee$, we use $(x,y)$ for their pairing. For $A, B \In V$, we use $A+B$ for their Minkowski sum, use $Conv(A \cup B)$ for convex hull. text/html 2026-03-20T19:29:00+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/notes/2026-03-20?rev=1774034940&do=diff Product vs Convex Hull Basic Facts Let $V_1^\Z, V_2^\Z$ be lattices of rank $n_1, n_2$, and $V_i = V_i^\Z \otimes \R$. Suppose $\Delta_i$ is a reflexive polytope in $V_i$, with polar dual polytope $\Delta_i^\vee$. Consider the product space $V_1 \times V_2$ and view $\Delta_1$ as a sub-polytope $\Delta_1 \times 0$, etc. Then we have dual reflexive polytopes $$ (\Delta_1 + \Delta_2)^\vee = Conv(\Delta_1^\vee \cup \Delta_2^\vee) =: \Delta_1^\vee \diamond \Delta_2^\vee. $$ text/html 2026-03-19T02:59:03+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/notes/double-cover-cy?rev=1773889143&do=diff Double Cover CY Start with dual nef partitions on $\Delta$ and $\nabla$. Attempt 1 Now, pick generic section $f_i \in O(2 \Delta_i)$, and $g_i \in O(2 \nabla_i)$. Put bundle $\oplus_i O(-\Delta_i)$ on $X_\Delta$, do LG model $$ W_\Delta = \sum_i ev_{O(-\Delta_i)} (f_i) $$ text/html 2026-03-15T20:42:19+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/notes/learning-hitchin-fibration?rev=1773607339&do=diff Hitchin Fibration – Learning Roadmap The Hitchin fibration is one of the central structures in modern geometry. It connects: * Higgs bundles * integrable systems * non-abelian Hodge theory * geometric Langlands * mirror symmetry This page collects a recommended path to learn the subject, with references. text/html 2026-02-25T04:13:23+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/notes/more-about-cy-completion?rev=1771992803&do=diff More about CY completion In this note, I want to record how to do the CY completion for a smooth and proper category $C$. Let $S: C \to C$ denote the Serre functor, which has the property that $$ Hom(x, y) = Hom(y, Sx)^\vee. $$ Monadic Construction text/html 2026-04-15T16:26:04+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/notes/quantum-group-at-roots-of-unity?rev=1776270364&do=diff Quantum Group at roots of unity One main obstacle for doing Mina's story for general 3-manifold, rather than just $\Sigma \times \R$, is that, the general $U_q(sl_2)$-representation has too many simple objects. The usual RT / WZW / CS theory, by pass theory problem, by working with integral, rather than text/html 2026-02-27T21:11:32+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/notes/some-prediction-about-mirror-symmetry?rev=1772226692&do=diff Some prediction about mirror symmetry Forget about 3d MS for a second. Toric HMS suppose to be easy. pure gauge case We are supposed to have $T^*(BG_m)$ 3d dual to $T^*(G_m^\vee)$. BenG told me, * $G_m / G_m$ should be mirror to $G_m^\vee$ text/html 2023-06-25T15:53:22+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/notes/start?rev=1687708402&do=diff Notes text/html 2026-02-24T00:58:42+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/notes/what-can-you-say-about-toric-cy-variety?rev=1771894722&do=diff What can you say about toric CY variety? Let $X$ be a toric CY variety, by which we mean $X$ has a toric fan $\Sigma \In N_\R$, with ray generators $v_\rho$ for each ray $\rho$, lying on a affine hyperplane of distance $1$. That is, there is $m \in M$, such that $\la v_\rho, m \ra = 1$ for all $\rho$. That $m$ gives me a distinguished function $W: X \to \C$ (modulo a constant factor). text/html 2026-02-12T22:49:14+00:00 Anonymous (anonymous@undisclosed.example.com) https://pzhou.org/notes/what-is-affine-lie-algebra-hat-sl_2?rev=1770936554&do=diff $\gdef\sl{\mathfrak{sl}}$ $\gdef\hf{\mathfrak{h}}$ $\gdef\asl{\widehat{\mathfrak{sl}}}$ $\gdef\ahf{\widehat{\mathfrak{h}}}$ What is affine Lie algebra $\asl_2$ As a vector space we have $$\asl_2 = sl_2[t,t^{-1}] \oplus \C K \oplus \C d$$ As Lie algebra, we have: $$ [X t^m, Y t^n] = [X, Y] t^{m+n} + m \delta_{m+n=0} K \quad \forall X, Y \in sl_2 $$ $$ [d, X t^m] = m X t^m, \quad d= t \d_t $$