Email: hq238@cam.ac.uk
I am a postdoc at University of Cambridge, under the supervision of Sungkyung Kang. Before that, I obtained a Ph.D. from Brandeis University, and my advisor is Daniel Ruberman. My research interests are low dimensional topology, and in particular, gauge theory and homotopy theory.
Math Project
A surgery formula for Seiberg-Witten invariants (arxiv:2409.02265)
- We prove a surgery formula for the ordinary Seiberg-Witten invariants of smooth 4-manifolds with b^1=1. This formula can be used to find exotic smooth structures on nonsimply connected 4-manifolds, and gives a lower bound of the genus of an embedding surface in nonsimply connected 4-manifolds. In forthcoming work, we will extend these results to give a surgery formula for the families Seiberg-Witten invariants.
Surgery formulas for Seiberg-Witten invariants and family Seiberg-Witten invariants (arxiv:2411.10392)
- We prove a surgery formula for the ordinary Seiberg-Witten invariants, and a surgery formula for the families Seiberg-Witten invariants of families of 4-manifolds obtained through fibrewise surgery. Our formula expresses the Seiberg-Witten invariants of the manifold after the surgery, in terms of the original Seiberg-Witten moduli space cut down by a cohomology class in the configuration space. We use these surgery formulas to study how a surgery can preserve or produce exotic phenomena.
Exotic diffeomorphism on $4$-manifolds with $b_2^+ = 2$ (arxiv:2409.07009)
- While the exotic diffeomorphisms turned out to be very rich, we know much less about the $b^+_2 =2$ case, because parameterized gauge-theoretic invariants are not well defined. In this paper we present a method (that is, comparing the winding number of parameter families) to find exotic diffeomorphisms on simply-connected smooth closed $4$-manifolds with $b^+_2 =2$, and as a result we obtain that $2\CP^2 # 10 \overline{\CP^2}$ admits exotic diffeomorphisms.
Dehn twist on a sum of two homology $4$-tori (arxiv:2410.02461)
- Kronheimer-Mrowka shows that the Dehn twist along a 3-sphere in the neck of two K3 surfaces is not smoothly isotopic to the identity. Their result requires that the manifolds are simply connected and the signature of one of them is 16mod32. We generalize the Pin(2)-equivariant family Bauer-Furuta invariant to nonsimply connected manifolds, and construct a refinement of this invariant. We use it to show that, if X1,X2 are two homology tori such that the determinants r1,r2 of them are odd, then the Dehn twist along a 3-sphere in the neck of X1#X2 is not smoothly isotopic to the identity.
Engineering Project
An Embedded project to solve Rubik’s cube, built in Java and implemented by a robot constructed from LEGO bricks. I use maching learning to recognize colors on the cube.