(10/2025) New manuscript released! Check it out here.
(08/2025) I'm thrilled to share that I have been awarded the Simons Dissertation Fellowship in Mathematics!
(06/2025) New manuscript released! Check it out here.
(02/2025) I'm happy to share that I've passed the Area Exam! I talked about a joint work in progress with Daren Chen on involutive Khovanov homology.
(08/2025) I attended the Rutgers Symplectic Summer School.
(07/2025) I attended Categorification in Low Dimensional Topology at Ruhr University Bochum.
(06/2025) I gave a talk in the Reunion Conference at Tsinghua University.
(06/2025) I gave a talk in the Topology Seminar at Peking University.
(06/2025) I gave a talk in the Conference on Modern Developments in Low-Dimensional Topology at ICTP.
(05/2025) I gave a lightning talk in Links in Dimensions 3 and 4 at ICERM. Providence is such a lovely city!
(05/2025) I attended Gauge Theory and Floer Homology in Low Dimensional Topology at SCGP.
(03/2025) I attended the Annual Meeting of Simons Collaboration on New Structures in Low-Dimensional Topology and its satellite meeting.
(02/2025) I gave a talk in Caltech Geometry & Topology Seminar.
(11/2024) I gave a talk in Princeton Topology seminar.
(10/2024) I attended the Yamabe Memorial Symposium at UMN.
(08/2024) I attended the workshop What's your trick? A Non-Traditional Conference in Low-Dimensional Topology virtually.
(07/2024) I gave a talk in the Reunion Conference at Tsinghua University.
(07 /2024) I attended the Simons Collaboration Summer School and the Conference at Rényi Institute. I gave a lightning talk in the conference.
(06/2024) I visited BICMR during mid June to mid September and gave a mini course on equivariant aspects in Floer theory.
(05/2024) I gave a talk in Stanford Topology Seminar.
(03/2024) I attended the Annual Meeting of Simons Collaboration on New Structures in Low-Dimensional Topology and the satellite conference at Princeton.
(02/2024) I gave a talk in the Geometry/Topology Seminar at University of South Florida.
(01/2024) I attended the Geometry and Topology Workshop at UCLA.
(12/2023) I attended the Workshop on Exotic 4-Manifold at Stanford.
(12/2023) I attended the Tech Topology Conference at Georgia Tech.
(08/2023) I gave a lightning talk in the IWoAT Summer School.
(07/2023) I attended the Summer School in Geometry and Topology at Princeton.
(05/2023) I attended the Instantons and Foams workshop at MIT.
(04/2023) I've passed my qualification exam.
(03/2023) I attended the Annual Meeting of Simons Collaboration on New Structures in Low-Dimensional Topology.
(01/2023) I attended the Winter school in singularities and low-dimensional topology at Rényi Institute.
One of my favorite research papers is Involutive Heegaard Floer homology by Hendricks and Manolescu. Inspired by Ciprian's groundbreaking work on Pin(2)-equivariant Seiberg-Witten Floer homology, they developed a similar invariant in Heegaard Floer theory. The j-action is replaced here by the conjugation action of Heegaard diagrams. I love this paper because it represents a basic approach of studying math: find analogue in different branches and explore new phenomena, which I believe that is the most feasible way for non-genius people like me. Its formalism has also led to great topological applications (e.g. The (2,1)-cable of the figure-eight knot is not smoothly slice). Recently, Daren and I defined an analog in Khovanov homology; see here. More or less surprisingly, it turns out to not provide any new information (at least for the "hat" version). It hence reflects some mysterious difference between Khovanov homology and Floer theory.
My favorite open problem is the (categorified) Wrapping Conjecture. Proposed by Eli Grigsby, it predicts that the third grading on annular Khovanov homology can tell you the wrapping number of a knot in the thickened annulus, i.e. the minimal intersection number of the knot with a meridional disk. I learned this problem when I was writing my first paper. I found it interesting because it has the same flavor as many "classical" applications of Floer homology (e.g. knot Floer homology detects Seifert genus and fiberness) and is stated in an equally succint way, but it seems very difficult to attack using "traditional" strategy, argued by Gage Martin. It therefore may also allude some foundamental difference between two theories (or maybe it's just wrong!). Currently I don't have anything interesting to say about this conjecture, but I enjoy thinking about it in my leisure.
Judson had an insightful comment on low-dimensional topologists: "There are two kinds of topologists. Some of them do math for doing topology, and others do topology for doing math". Apparently I am in the latter.
香蕉空间 is a Chinese mathematics website in the style of nLab. You may find it especially helpful if you can read Chinese and are interested in some abstract mathematics (in this case, there is a high chance you already know this website).