Research Interests:
My research area is algebraic geometry with a particular focus on the derived category of coherent sheaves on quasi-projective varieties. Algebraic geometry studies geometric objects defined by polynomial equations, known as varieties. The derived category associated with the variety provides a linear tool for understanding the nonlinear structure in geometry.
My works explore several interconnected topics. Along one direction, I study Bridgeland stability condition, which is a stability condition defined on the derived category of coherent sheaves on an algebraic variety. Our recent work contributes to the understanding of Bridgeland stable objects. In many cases, the stable objects form a well-behaved topological space called the moduli space. By performing intersection theory on this moduli space, one can derive enumerative invariants of the variety, such as the famous result that a cubic surface contains exactly 27 lines. My other research direction focuses on a "categorified" enumerative theory–motivic Donaldson-Thomas theory.
Publications and Preprints:
Bridgeland/weak Stability Conditions under Spherical Twist Associated to a Torsion Sheaf. (with Tristan Collins, Jason Lo, Shing-Tung Yau) , arXiv 2411.18554, submitted
Moduli of sheaves on fourfolds as derived Lagrangian intersections, (with Nachiketa Adhikari), arXiv:2403.00727, under the first round revision of Nagoya Math. J..
Stability for line bundles and deformed Hermitian-Yang-Mills Equation on some elliptic surfaces. (with Tristan Collins, Jason Lo, Shing-Tung Yau) arXiv:2306.05620, submitted.
Weak stability conditions as limits of Bridgeland stability conditions, (with TristanCollins, Jason Lo, Shing-Tung Yau) arXiv:2410.02466, Can. J. Math. 2025:1-40.
D-critical locus structure on the Hilbert schemes of some local toric Calabi-Yau threefolds. (with Sheldon Katz) arXiv:2009.05529, Math. Res. Lett., 31(5), 1493-1522, 2024.
D-critical loci for length n sheaves on local toric Calabi-Yau 3-folds. (with Sheldon Katz) arXiv:2108.13510, Bull. Lond. Math. Soc., 54(6) 2101-2116, 2022.
Orientation data for coherent sheaves on local P2. arXiv:1809.01776, Math. Res. Lett., to appear.
In preparation:
Stability of the cotangent bundle on some surfaces of general type. (with Tristan Collins, Jason Lo, Shing-Tung Yau), draft available upon request.
Stability of the tangent bundle on Del Pezzo surfaces. (with Tristan Collins, Jason Lo, Shing-Tung Yau), draft available upon request.
Other writings:
Motivic Donaldson-Thomas invariant: construction and computation, ICCM proceedings 2020.
Stability and the deformed Hermitian-Yang-Mills equation. (with Tristan Collins) Surveys in Differential Geometry. 24 (2019), 1-38. arXiv:2004.04831
Symmetry is the sine qua non of shape. Li, Y., Sawada, T., Shi, Y., Steinman, R.M., Pizlo, Z. Book Chapter in Shape Perception in Human and Computer Vision. Advances in Computer Vision and Pattern Recognition. 2013
A Bayesian model of binocular perception of 3D mirror symmetrical polyhedra. Li, Y., Sawada, T., Shi, Y., Kwon, T., Pizlo, Z. Journal of Vision, 11(4):11, 1-20, 2011
Invited Talks:
AWM Research Symposium, Madison, WI, May 2025
Geometry and Physics Seminar, Boston University, Feb 2025
Algebraic Geometry Seminar, University of Maryland, May 2024
GPRT seminar, Northeastern University, Feb 2024
CATS 7 conference, CIRM, France, Oct 2023
AWM Research Symposium, Atlanta, GA, Sep 2023
Brazil-China Joint Mathematical Meeting, Foz do Iguaçu, Brazil, Jul 2023
Algebraic Geometry seminar, Boston College, Feb 2023
AMS sectional meeting, University of Utah, Oct 2022
Algebraic Geometry seminar, Duke University, July 2022
Enumerative Geometry seminar, USCD, Jun 2022 (online)
Geometry and Analysis seminar, Oxford, Jan 2022 (online)
ICCM, University of Science and Technology of China, Dec 2020 (online)
Motivic Geometry Conference, CAS at the Norwegian Academy of Science and Letters, Sep 2020 (online)
Non-commutative Geometry and Related Topics Conference, CMSA Harvard, Mar 2020
Algebraic Geometry Seminar, Duke University, Nov 2019
AMS sectional meeting, UC Riverside, Nov 2019
Algebraic Geometry Seminar, UC Davis, February 2019
Grant:
AMS-Simon Travel Grant 2022-2024