I am a PhD student at University of California, Santa Cruz since 2024. I received my Master’s and Bachelor’s degree from the 3+2 program jointly run by University of Wisconsin-Madison and Southern University of Science and Technology.
My undergraduate thesis was mentored by Yifei Zhu and Botong Wang.
Currently I am interested in the crossover field of algebraic topology and algebraic geometry. I also have a broad interest in other topics related to topology, algebra and geometry.
Academic Activities
- 2024 Summer: Summer school on motivic homotopy theory at IWoAT 2024.
- 2023 Summer: Directed reading with Laurentiu Maxim, read first half of the book Intersection Homology & Perverse Sheaves, with Applications to Singularities.
- 2022 Summer: Directed reading with Botong Wang on de Rham theorem and Cech-de Rham complex.
- 2021 Fall: Participated in Undergrad Topology Seminar in SUSTech. Gave a talk on Mayer-Vietoris sequence of de Rham cohomology.
Writings and Notes
Cohomology Theories for Manifolds and Their Equivalences (2023 Spring)
This is the undergraduate graduation thesis for bachelor’s degree in SUSTech, mentored by Yifei Zhu and Botong Wang. In this paper, the classical proof of equivalence between de Rham cohomology, singular cohomology and Cech cohomology is given using smooth approximation, de Rham map and Cech-de Rham complex. Next, the theory of sheaf cohomology is given, and this paper demonstrates that the proofs can be elegantly reformulated under this framework. Moreover, the derived-functor based definition of sheaf cohomology indicates that it can be interpreted as local-to-global obstacle. This can serve as a bridge between topology and geometry structures. An example of topology of underlying manifold affecting the global information of its structural sheaf is given in the end.
Spectra, Categories, Composition Law and Segal Condition (2022 Spring)
This paper is for the final project of Algebraic Topology II in UC Berkeley, taught by Constantin Teleman. This paper gives a method to associate a spectra to a category with some additional structure. Since each spectra corresponds to a generalized cohomology theory, in this way we can build a correspondence between these categories and generalized cohomology theories. The approach used in this paper is generally based on the paper by G. Segal, which uses the construction of Γ-space as the bridge in between.
Mayer-Vietoris Sequence of de Rham Cohomology (2021 Fall)
This is a note of a talk for Undergrad Topology Seminar in SUSTech. The main reference for the seminar is Differential Form in Algebraic Topology by Bott and Tu. Mayer-Vietoris sequence (and its generalized form) is the main homological algebraic tool for dealing with local to global problem in de Rham cohomology. Proof of this theorem heavily rely on partition of unity and smooth bump function, i.e. depending on certain “softness” of smooth structure.
Other Skills
Linux user for 8+ years (favorite distrobution: NixOS). Having some experience in programming in Scala, Java, C++, Python, Lua, HTML/CSS/JS, Matlab, Mathematica, Scheme Lisp. Familar with standard tools like Git, Emacs and Vim.
Participated in Mathematics Contest in Modeling in the year 2020 and 2021 as programmer and the best result is Honorable Participant.