Research

Here is my Google Scholar profile, and here is my research statement (updated 09/09/2022).

1. Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius (with Facundo Mémoli and Osman Berat Okutan). Algebraic & Geometric Topology. To appear (accepted in 2022). arXiv:2001.07588.

2. The Gromov-Hausdorff distance between spheres (with Facundo Mémoli and Zane Smith). Geometry & Topology 27-9 (2023), 3733--3800. arXiv:2105.00611.

3. Weisfeiler-Lehman meets Gromov-Wasserstein (with Samantha Chen, Facundo Mémoli, Zhengchao Wan, and Yusu Wang). International Conference on Machine Learning (ICML), pages 3371--3416. PMLR, 2022. arXiv:2202.02495.

4. The Weisfeiler-Lehman Distance: Reinterpretation and Connection with GNNs (with Samantha Chen, Facundo Mémoli, Zhengchao Wan, and Yusu Wang). ICML workshop: Topology, Algebra, and Geometry in Machine Learning (2023). To appear. arXiv:2302.00713.

5. Classical Multidimensional Scaling on Metric Measure Spaces (with Facundo Mémoli). Submitted. arXiv:2201.09385.

6. Gromov-Hausdorff distances, Borsuk-Ulam theorems, and Vietoris-Rips complexes (with Henry Adams et al). Submitted. arXiv:2301.00246.

7. Some results about the Tight Span of spheres (with Facundo Mémoli, Zhengchao Wan, Qingsong Wang, and Ling Zhou). arXiv:2112.12646.

8. Reverse Bernstein Inequality on the Circle (with Jürgen Jost, Parvaneh Joharinad, and Rostislav Matveev). arXiv:2302.10122.

The followings are my (current) favorite open problems:

Q1) Classify the homotopy types of the Vietoris-Rips complex VR(S^n;r) for n>=2 and r>0. Equivalently (by [1] above), classify the homotopy types of open r-neighborhood B_r(S^n,E) of S^n where E=L^\infty(S^n) or the tight span (=injective envelope) of S^n.

Q2) Prove/Disprove the following: The Gromov-Hausdorff distance between m-sphere S^m and (m+1)-sphere S^(m+1) is exactly (1/2)*Arccos(-1/(m+1)) for m>=3. Note that m=1,2 cases are already proven true in [2] above.

Q3) Clarify the relationship between the distortion of the cMDS embedding (=sum of the negative eigenvalues of the cMDS operator) and the sectional curvatures of the input space.