Research

Research Interests: 

 I am broadly interested in the interaction between analytic approaches, geometric structures, and topological invariants. In terms of subjects, my research involves differential topology, global analysis, index theory, noncommutative geometry, and symplectic geometry. 

Research Papers: 

1. Morse-Bott-Smale cohomology and the Witten deformation (arXiv; draft): 

Given a torus-action on a closed manifold, we are able to construct an invariant version of the Morse-Bott-Smale chain complex by a certain type of Morse-Bott functions. The construction is inspired by Austin and Braam. The most interesting fact is, this chain complex has a perfect one-to-one correspondence with the invariant version of Witten's instanton complex (not just quasi-isomorphic) when the metric is locally Euclidean. The key step is to concentrate harmonic forms around critical components. 

2. Semi-characteristics for noncompact manifolds (draft): 

The Euler characteristic of an odd dimensional closed manifold is zero, and thus there is a nonvanishing vector field on this manifold. However, if we want two independent vector fields on this manifold, what should be the correct topological obstruction to these two vector fields? Atiyah gave an answer to compact cases, and I am going to explore it in a noncompact case admitting a Lie group action, based on a Hodge theory developed by Tang, Yao, Zhang. 

Ongoing Projects: 

1. Influences of horizontal directions to the Witten complex (ongoing).

2. Blow-up groupoids, the Witten Laplacian and warped products (ongoing).

3. TTY cohomology and semi-characteristics of symplectic manifolds (ongoing).

4. The fibration of the complement of a knotted-manner variety and the K-group of each fiber (ongoing).