Categorical resolutions and stability conditions on Kuznetsov component of nodal cubics (in preparation)
Let Y be a 1-nodal cubic hypersurface. We show that the Kuznetsov component Ku(Y) admits a categorical resolution Ku(Y). For Y of dimension 3, 4, and 5, the known stability conditions on Ku(Y) deform and descent to a stability condition on Ku(Y). Furthermore, we show that the categorical resolution also induces morphisms between the corresponding Bridgeland moduli spaces of Ku(Y) and Ku(Y), which are resolutions of singularities. For the low-dimensional moduli spaces, this recovers the geometric resolutions of the Fano schemes of lines for nodal cubic 3-folds and 4-folds, and the Fano scheme of planes for nodal cubic 5-folds.
Let Y be a cubic 7-fold. If Y is smooth, its Kuznetsov component Ku(Y) is a Calabi–Yau 3-category, and we construct an embedding of Ku(Y) into the bounded derived category of coherent Clifford modules over P⁴. When Y is a general cubic 7-fold singular along a line, we construct an explicit crepant categorical resolution of Ku(Y) by the derived category of a smooth Calabi–Yau 3-fold, recovering a result of Favero–Kelly.
Stability Conditions & Moduli Spaces on Kuznestsov Component of Cubic 5-Folds (arXiv: 2509.21454)
We study the Kuznetsov component of cubic fivefolds via their quadric fibration model, and construct a family of Serre-invariant Bridgeland stability conditions on it. For every primitive numerical class, we prove that the associated Bridgeland moduli space contains a non-empty smooth locus, on which the restriction to a general hyperplane section preserves stability. As a consequence, we obtain Lagrangian immersions into hyper-Kähler varieties arising as moduli spaces on the Kuznetsov component of cubic fourfolds, generalizing a geometric construction of Illiev-Manivel, which realizes the Fano surface of planes of the cubic fivefold as a Lagrangian subvariety in a hyper-Kähler fourfold.
Categorical Resolutions & Kuznetsov Component of Cubic 5-Folds (PhD second year report)
Loosely organised notes on crepant categorical resolutions of nodal/cuspidal singularities, and a conjecture about the Kuznetsov component of nodal/smooth cubic 5-folds.
Fano 3-Folds: Classification and Semi-Orthogonal Decompositions (PhD first year report)
Notes on prime Fano 3-folds, K3 surfaces, derived categories, and stability conditions.
Notes prepared for a series of talks on derived categories of coherent sheaves in the homological mirror symmetry reading group. (Latest update: 22 Aug 2022).
Deformation Quantisation (MMathPhys dissertation)
An exposition of Kontsevich's proof of the formality theorem and the classification of deformation quantisation on a Poisson manifold, with an account of the physical background and the deformation theory via differential graded Lie algebras and L-infinity algebras.
Path Integral Quantisation of Bosonic Strings (String Theory I mini-project)
A brief introduction to the path integral and BRST quantisation of bosonic strings.
Dispersionless KdV & Inverse Scattering in WKB Approximation (summer project, 2021)
A refinement of the work by V. V. Geogjaev (1984). We studied the dispersionless limit of the KdV equation as an example of the inverse scattering problem of the Schrödinger equation in the WKB approximation.
Algebraic Surfaces (2023)
Growing notes centred around the Enriques classification and the Mori program of algebraic surfaces. Materials are mainly drawn from Chapter V of Hartshorne's Algebraic Geometry and the lectures of Topics in Algebraic Geometry (MA9M6).
Commutative Algebra II (2022)
Handwritten revision outline for 4th-year commutative algebra course (MA4J8). Contains statements of important definitions and theorems.
Linear Functional Analysis (2021)
Summary notes for 3rd-year functional analysis courses (B4.1 & B4.2). The main theory is devoted to studying the bounded linear operators between normed vector spaces. Some extra material on the Hahn-Banach theorem and dual spaces are included for the sake of self-containedness.
Category Theory (2021)
Incomplete notes on the basic concepts of category theory.
Algebraic Curves (2021)
Summary notes for the algebraic curves course (B3.3). Closely follow the book Complex Algebraic Curves by Frances Kirwan.
(Revised and expanded in March 2023)
Undergraduate Algebra (2020-2021)
These partially completed notes intend to cover the mainline of undergraduate algebra courses (A3, B2.2, B3.1, C2.2), including the fundamentals of commutative rings, fields, and modules.
有限群的表示论 (Representation Theory of Finite Groups) (2020)
Summary notes for the representation theory course (B2.1) in Chinese. 主要内容包括半单模的结构，特征标理论，以及诱导表示及其应用。
Differential Equations II (2020)
Summary notes on the 2nd-year courses Differential Equations II (A6), Integral Transforms, and Calculus of Variations. Proofs and calculation details are omitted.
Further Quantum Mechanics (2020)
Handwritten notes on the 2nd-year course Further Quantum Mechanics. The main topics are approximation methods and atomic physics.
Complex Analysis (2018-2019)
Notes for the 2nd-year course complex analysis (A2) and some more. It includes the basic materials, such as Cauchy integrations, Laurent series and isolated singularities, and residue calculus, and some off-syllabus topics such as the Weierstrass Factorisation Theorem and the proof of the Riemann Mapping Theorem.
分析学基础 (Introductory Analysis) (2018-2019)
前数学系时代笔记，包括朴素集合论，实数理论，极限，连续性。
线性代数 (Linear Algebra) (2017-2018)
高中时代的数学入门，从求解线性方程组到外积的泛性质，基础和应用并重的笔记。主要跟随 Sheldon Axler's Linear Algebra Done Right 以及其他一些兼顾应用的书。
在此郑重感谢一位不愿署名的知乎网友用 LaTeX 排版了前三章：线性代数 (排版)