Thesis
Higher Stacks and Derived Critical Loci
2023-05, Toulouse, Bachelor’s thesis
It’s my Bachelor’s thesis supervised by Prof. Nuiten in Toulouse which consists of three parts. In the first part we give a concise introduction to Toën and Vezzosi’s work on homotopical algebraic geometry, a particular example of which is the derived algebraic geometry. Our main objects to study in this part are simplicial presheaves and we will show higher stacks are simplicial presheaves satisfying the descent for hypercovers. From this we can also obtain the concept of derived stacks. In the second part we focus on derived critical loci which are actually a derived intersection. We will show this construction is dual to BV-formalization in mathematical physics and serves a model for (-1)-shifted symplectic structure. Finally there is an appendix about homotopical algebra necessary for reading this thesis especially the first part. In this appendix we will talk about abstract homotopy theory in detail especially homotopy (co)limits and simplicial model categories. Download thesis here
Notes
MSc. Seminar on Perverse Sheaves
2024 04 to 2024-07,
This master seminar is organized by Dr. Stefan Dawydiak in Bonn and here is the outline. My part is to introduce the concept of t-structures for perverse sheaves. Notes for talk 10
Some Geometry of Local Systems
2024-04
It’s to understand the theory of local systems. We talk about some different characterizations of local systems e.g. locally constant sheaves, representations of the fundamental group, flat connections, D-modules and crystals. And we try to generalize them from the classical case to a higher case to see whether there still exist similar relations for these definitions. Finally we get a concept of the (derived) moduli stack of local systems, which has two versions i.e. Betti and de Rham. We also consider (derived) symplectic structures on this stack. In the notes there are some questions I want to figure out in my future study. Download notes here
The Moduli of Triangles
2022-05
The goal is to understand stacks by the moduli problem of triangles. To do this we make the concept of a continuous family of triangles precise and prove the moduli space classifying oriented triangles is fine but isosceles triangles having non-trivial automorphisms prevent the moduli space classifying non-oriented triangles from being fine. When viewing moduli functors as stacks, the latter moduli functor will be equivalent to the quotient stack of some permutation group. Download notes here
Localizations and Homotopy Theories
2022-05
This student project is to understand the localization of categories from the viewpoint of homotopy theories i.e. model categories. We study homotopy theories of topological spaces and chain complexes, and discuss their model categorical structures. We also use the calculus of fractions to describe derived categories. Download notes here
Galois Theory
2022-01, Lecture notes of the course given by Prof. Yongquan Hu
The course is mainly about Galois theory and topics covered contain finite Galois extensions, compass and straightedge construction, solvability of algebraic equations, infinite Galois theory, Galois cohomology, Kummer theory and so on. During the course, I helped Prof. Hu type the lecture notes. Download notes here
A Short Intrduction to Representable Functors
2021-04, a presentation in the class of ‘Elementary Algebraic Geometry’
We give a short introduction to representable functors and prove the category of variety functors is equivalent to the category of k-algebras of finite presentation. Download notes here