I'm a PhD student in algebraic topology, with a special interest in chromatic homotopy theory. Currently I'm studying unstable vn-periodic homotopy theory, with a focus on the structural properties of the Bousfield–Kuhn functors.
In my master's thesis, I studied the application of manifold calculus to the theory of Vassiliev knot invariants. Joint with Danica Kosanović and Peter Teichner, we are working on a geometric interpretation of the manifold calculus tower associated to the embedding space of knots, using the theory of capped gropes.
The theory of Goodwillie calculus plays important roles in the above two projects. I find this theory very elegant and would like to discover more of its applications!
A key notion in Goodwillie calculus are excisive functors. Together with Hana Jia Kong, Ang Li and Mingcong Zeng, we are working towards a model of N∞-spectra using excisive functors where the N∞-algebra structure does not necessarily come from a G-universe.
Goodwillie's cosimplicial model for the space of long knots and its applications. Dec 09, 2020. arXiv 2012.04036.
We work out the details of a correspondence observed by Goodwillie between cosimplicial spaces and good functors from a category of open subsets of the interval to the category of spaces. Using this, we compute the first page of a integral Bousfield--Kan homotopy spectral sequence associated to the space of long knots arising from manifold calculus. Based on the methods in [Con08], we give a combinatorial interpretation of the differentials mapping into the diagonal terms, by introducing the notion of (i,n)-marked unitrivalent graphs.
Vassiliev invariants via manifold calculus [PDF]
This is my master's thesis. In the thesis, we give an alternative proof of a theorem by Budney–Conant–Koycheff–Sinha, which says that the manifold calculus tower of the space of long knots induces Vassiliev invariants. We calculate and interpret the first differentials ending at the diagonal of the integral homotopy spectral sequence associated to the manifold calculus tower. In an expository part, we give a detailed explaination of the correspondence between cosimplical spaces and "good" functors on the category of open subsets of the unit interval.
The Alexander polynomial [PDF]
This is my bachelor's thesis. The aim of the thesis is to understand basic concepts of knot theory and various constructions of the Alexander polynomial of a knot. We provided a detailed calculation of the Alexaner polynomial of torus knots and twists knots.