Selected works

1. Every finite graph arises as the singular set of a compact 3-d calibrated area minimizing surface, Comm. Pure Appl. Math. 77 (2024), no. 9, 3670–3707., preprint available at https://arxiv.org/abs/2106.03199

2. On a conjecture of Almgren: area-minimizing surfaces with fractal singularities, to appear in Annals of Mathematics, preprint available at https://arxiv.org/abs/2110.13137

3. Area-minimizing submanifolds are not generically smooth, to appear in Duke Mathematical Journal, preprint available at https://arxiv.org/abs/2206.08315

4. Ph.D. dissertation, The Hasse Principle for Geometric Variational Problems: An Illustration via Area-minimizing Submanifolds, preprint available at https://arxiv.org/abs/2508.21045

P.S. There is some other research work available on arxiv: https://arxiv.org/a/liu_z_17.html. They either are superseded by the works of others or my own works.

Research acknowledgment: I want to thank my advisor Camillo De Lellis for giving me the questions in 1,2,3. Camillo De Lellis and Robert Bryant pointed out the ray construction, which formed the basis of 1. The main construction in 2 dates back to Frank Morgan and Dana Mackenzie's work. Leon Simon's work inspired the author to pursue 2. Frank Morgan, Mark Haskins, and Tommaso Pacini first pointed out the two types of obstructions in 3 in the Euclidean space. Gary Lawlor's work set the analytic foundation for 3.  Camillo De Lellis suggested using Yongsheng Zhang's work to deal with 3. Communications with Michael Freedman partially inspired 4. Almgren's example and Frank Morgan's works convinced me that 4 is true. Yongsheng Zhang's work has remained foundational to all of 1,2,3,4. Last but not least, William Allard, Hubert Bray, and Robert Bryant taught me geometric measure theory and Riemannian geometry, all of which are used in 1,2,3,4. William Allard's story inspired 4, Huber Bray's story inspired 2, conversations with Robert Bryant inspired 1,2,3. Donghao Wang and Kai Xu are instrumental in helping me navigate differential and algebraic topology, which are essential to 1,2,3,4.  

This leaves open the question of what my contribution to the work above is.

Seminar talks

MIT Geometric Analysis Seminar, April 2025

Brown University Geometric Analysis, April 2025

NC State Geometric Analysis Seminar, September 2024

Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics, May 2024

Columbia University Analysis Seminar, March 2024

University of Pennsylvania Geometry Seminar, February 2024

Lehigh University Geometric Analysis Seminar, January 2024

University of Maryland Geometric Analysis Seminar, January 2024

Stanford Geometric Analysis Seminar, January 2024

Cornell Analysis Seminar, January 2024

Yau Mathematical Sciences Center, January 2024

The UC-Berkeley Differential Geometry Seminar, May 2023

Caltech Geometry & Topology Seminar, April 2023

University of Chicago - Geometric Analysis Seminar, October 2022

UMN PDE seminar, January 2022

Geometric Analysis and Topology Seminar, NYU Courant, January 2022

Simons Collaboration on Special Holonomy in Geometry, Analysis, and Physics, September 2021

Fellowships

Centennial Fellowship, Princeton, 2019-current

My proudest achievement

I've been fortunate enough to have raised a question in topology that gets answered in a manuscript by the leading experts on this topic, Diarmuid Crowley and Mark Grant.