Algebraic geometry. I am interested in moduli problems in algebraic geometry and representation theory, the techniques for studying them and their applications, such as:
stability conditions
exact equivalences (autoequivalences, Fourier-Mukai transforms, tilting equivalences)
counting invariants
Machine reasoning. I am also interested in applying category theory and ideas from algebra and geometry to theoretical aspects of autonomous systems and machine reasoning, including:
autonomous learning
problem solving
inductive reasoning and analogical reasoning
Currently, my research is partially supported by an AFOSR grant (FA9550-24-1-0268).
Papers
31. From data to concepts via wiring diagrams, with Nima Jafari**.
In preparation.
30. Stability of the cotangent bundle on some surfaces, with Tristan Collins, Yun Shi, and Shing-Tung Yau.
In preparation.
29. Higher rank DT/PT wall-crossing in Bridgeland stability, 2025, with Marcos Jardim, Antony Maciocia, and Cristian Martinez.
Preprint. arXiv:2503.20008
28. Bridgeland/weak stability conditions under spherical twist associated to a torsion sheaf, 2024, with Tristan Collins, Yun Shi, and Shing-Tung Yau.
Preprint. arXiv:2411.18554
27. Weak stability conditions as limits of Bridgeland stability conditions, 2024, with Tristan Collins, Yun Shi, and Shing-Tung Yau.
Canad. J. Math., accepted. arXiv:2410.02466
26. Quantifying analogy of concepts via ologs and wiring diagrams, 2024.
Preprint. arXiv:2402.01020
25. Stability for line bundles and deformed Hermitian-Yang-Mills equation on some elliptic surfaces, 2023, with Tristan Collins, Yun Shi, and
Shing-Tung Yau. Submitted. arXiv:2306.05620
24. Intersection numbers on fibrations and Catalan numbers, 2022, with Rimma Hämäläinen*** and Edward Morales***.
Exp. Math., Vol. 33 (4), pp. 768-775, 2024. doi:10.1080/10586458.2023.2293292. arXiv:2210.02001
23. Geometric stability conditions under autoequivalences and applications: elliptic surfaces, 2022, with Cristian Martinez.
J. Geom. Phys., Vol.194, 104994, 2023. arXiv:2210.01261
22. Stable pairs of 2-dimensional sheaves on 4-folds, 2021, with Amin Gholampour and Yunfeng Jiang.
Pure Appl. Math. Q., to appear. arXiv:2110.01342
21. A note on Bridgeland stability conditions and Catalan numbers, 2020, with Karissa Wong*.
Involve, Vol. 15 (3), pp. 427-432, 2022. arXiv:2012.12851
20. Weight functions, tilts, and stability conditions, 2020.
Submitted. arXiv:2007.06857
19. Stability and Fourier-Mukai transform on elliptic threefolds.
Proceedings of the ICCM, 2018.
18. Fourier-Mukai transforms and stable sheaves on Weierstrass elliptic surfaces, 2019, with Wanmin Liu and Cristian Martinez.
Bull. Braz. Math. Soc. (N.S.), Vol. 55, 47, 2024. arXiv:1910.02477 (This supersedes arXiv:1710.04652.)
17. A relation between higher-rank PT stable objects and quotients of coherent sheaves.
Kyoto J. Math, Vol. 61 (4), pp. 815-842, 2021. arXiv:1810.01380
16. A framework for torsion theory computations on elliptic threefolds, 2018, with David Angeles** and Courtney van der Linden**.
J. Aust. Math. Soc., Vol. 110 (2), pp. 175-193, 2021. arXiv:1809.10662
15. Fourier-Mukai transforms of slope stable torsion-free sheaves on Weierstrass elliptic threefolds.
J. Algebra, Vol. 604, pp. 40-86, 2022. arXiv:1710.03771
14. Fourier-Mukai transforms of slope stable torsion-free sheaves on a product elliptic threefold.
Adv. Math, Vol. 358, 106846, 2019. arXiv:1710.02963
13. Duality spectral sequences for Weierstrass fibrations and applications, with Ziyu Zhang.
J. Geom. Phys., Vol. 123, pp. 362-371, 2018. arXiv:1706.01923
12. Preservation of semistability under Fourier-Mukai transforms, with Ziyu Zhang.
Geom. Dedicata, Vol. 193 (1), pp. 89-119, 2018. arXiv:1510.00310
11. t-structures on elliptic fibrations.
Kyoto J. Math., Vol. 56 (4), pp. 701-735, 2016. arXiv:1509.03216
10. Stability and Fourier-Mukai transforms on higher dimensional elliptic fibrations, with Wu-Yen Chuang.
Comm. Anal. Geom., Vol. 24 (5), pp. 1047-1084, 2016. arXiv:1307.1845
9. Torsion pairs and filtrations in abelian categories with tilting objects.
J. Algebra Appl, Vol. 14, 1550121, 2015. arXiv:1302.2991
8. Some examples of tilt-stable objects on threefolds, with Yogesh More.
Comm. Algebra, Vol. 44 (3), pp. 1280-1301, 2016. arXiv:1209.2749
7. Stability and Fourier-Mukai transforms on elliptic fibrations.
Adv. Math, Vol. 255, pp. 86-118, 2014. arXiv:1206.4281
6. On some moduli of complexes on K3 surfaces.
Preprint. arXiv:1203.1558
5. Representing stable complexes on projective spaces, with Ziyu Zhang.
J. Algebra, Vol. 400, pp. 185-218, 2014. arXiv:1202.4948
4. Polynomial Bridgeland stable objects and reflexive sheaves.
Math. Res. Lett., Vol. 19 (4), pp.873-885, 2012. arXiv:1112.4511
3. Mini-walls for Bridgeland stability conditions on the derived category of sheaves over surfaces, with Zhenbo Qin.
Asian J. Math., Vol. 18 (2), pp.321-344, 2014. arXiv:1103.4352
2. Moduli of PT-semistable objects II.
Trans. Amer. Math. Soc., Vol. 365, pp. 4539-4573, 2013. arXiv:1011.6306
1. Moduli of PT-semistable objects I.
J. Algebra, Vol. 339 (1), pp. 203-222, 2011 . arXiv:1011.5688
*: high school student
**: undergraduate student
***: Master's student
Research Talks
2025:
Third Workshop on Multimodal AI (MultimodalAI'25), London, UK, September 2025. (short talk & poster)
NCTS Seminar in Algebraic Geometry, National Taiwan University, June 2025. (online)
Linkou Chang Gung Memorial Hospital, Taiwan, May 2025.
National Cheng Kung University, Taiwan, Algebraic Geometry meets Machine Learning, April 2025.
University of Kansas, AMS Special Session on Enumerative Invariants of Moduli Spaces in Algebraic Geometry, March 2025.
2024:
University of Rosario, Colombia, SIAM Student Chapter Seminar, October 2024. (online)
UT San Antonio, AMS Special Session on A Showcase of Algebraic Geometry at Undergraduate Institutions, September 2024.
San Francisco, Joint Mathematics Meeting, AMS Special Session on Combinatorial Insights into Algebraic Geometry, II, January 2024.
2023:
CSU Bakersfield, CSU Mathematical Sciences Conference, November 2023.
AG@PUI Virtual Seminar, October 2023.
University of Missouri-Columbia, Geometry and Topology Seminar, May 2023.
2022:
National Taiwan University, TIMS Seminar, December 2022.
State University of Campinas, Brazil, Algebraic Geometry Seminar, November 2022. (two talks)
2021:
University of Kansas, Kansas Geometry Day, August 2021. (online)
CUHK-Harvard-YMSC Joint Differential Geometry Seminar, April 2021. (online)
State University of Campinas, AG arXiv club, March 2021. (online)
FD seminar - online seminar on representation theory of finite-dimensional algebras, January 2021.
2019:
Hong Kong University of Science and Technology, Seminar on Geometry, May 2019.
Claremont Colleges, Algebra/Number Theory/Combinatorics Seminar, May 2019.
Harvard University, Workshop on Mirror Symmetry and Stability, March 2019.
National Taiwan University, CASTS Seminar in Algebraic Geometry, January 2019.
2018:
Taipei, ICCM annual meeting, December 2018.
California State University, Northridge, Algebra/Number Theory/Discrete Math Seminar, March 2018.
POSTECH, South Korea, Frontiers of Algebraic Surfaces, February 2018.
National Cheng Kung University, Taiwan, Algebraic Geometry Seminar, January 2018.
2017:
Stanford University, Algebraic Geometry Seminar, October 2017.
Center for Geometry and Physics, IBS, Pohang, South Korea, June 2017 (two 2-hour talks).
Seoul National University, Workshop on Moduli Theory and Derived Category, June 2017. (two talks)
Leibniz University Hannover, Research seminar Algebraic Geometry, June 2017.
University of Edinburgh, Edinburgh Geometry Seminar, May 2017.
UC Santa Barbara, Algebraic Geometry Seminar, May 2017.
University of Arizona, Algebraic Geometry Seminar, February 2017.
American Institute of Mathematics, Workshop on Stability and Moduli Spaces, January 2017.
National Taiwan University, TIMS Seminar in Algebraic Geometry, January 2017.
2016:
University of Missouri-Columbia, I-70 Algebraic Geometry Symposium, November 2016.
University of Missouri-Columbia, Geometry/Topology Seminar, June 2016.
Center for Geometry and Physics, IBS, South Korea, Algebraic Geometry Seminar, May 2016. (two 2-hour talks)
Rutgers University, Geometry, Symmetry and Physics Seminar, May 2016.
Chinese University of Hong Kong, MIST: Mathematics Inspired by String Theory, February 2016.
University of Illinois at Urbana-Champaign, Algebraic Geometry Seminar, February 2016.
2015:
University of Minnesota, Algebraic Geometry Seminar, October 2015.
University of Hong Kong, Geometry Seminar, August 2015.
Salt Lake City, AMS Summer Institute in Algebraic Geometry (contributed talk), July 2015.
National Taiwan University, NCTS Reading Seminar in Algebraic Geometry, July 2015.
Purdue University, Algebraic Geometry Seminar, March 2015.
University of Illinois at Urbana-Champaign, Algebraic Geometry Seminar, February 2015.
2014:
University of Missouri-Columbia, Geometry/Topology Seminar, September 2014.
Chinese University of Hong Kong, IMS Geometry Seminar, July 2014.
Hong Kong University of Science and Technology, Seminar on Pure Mathematics, July 2014.
University of Edinburgh, Edinburgh Geometry Seminar, May 2014.
Max Planck Institute for Mathematics, Bonn, Seminar Algebraic Geometry, May 2014.
2013:
National Taiwan University, NCTS Seminar on Algebraic Geometry, August 2013.
--. National Tsing Hua University, Colloquium, Department of Mathematics, February 2013.
--. Yeosu, South Korea, Symposium on Projective Algebraic Varieties and Moduli, February 2013.
National Taiwan University, TIMS Seminar in Algebraic Geometry, February 2013.
National Taiwan University, TIMS Seminar in Algebraic Geometry, January 2013.
--: unable to attend due to military service
2012:
University of Missouri-Columbia, Geometry/Topology Seminar, December 2012.
University of Akron, AMS Special Session on Geometry of Algebraic Varieties, October 2012.
Kansas State University, M-Seminar, Mirror Symmetry & Tropical Geometry Research Center, April 2012.
University of Kansas-Lawrence, AMS Special Session on Geometry of Moduli Spaces of Sheaves, April 2012.
University of Kansas-Lawrence, AMS Special Session on Algebraic Geometry and its Applications, April 2012.
University of Missouri-Columbia, Geometry/Topology Seminar, March 2012.
2011:
Georgia Southern University, AMS Special Session on Algebraic Geometry, March 2011.
2010:
University of Missouri-Columbia, Geometry/Topology Seminar, September 2010. (two talks)
University of British Columbia, Algebraic Geometry Seminar, January 2010.
Expository Talks
7. An introduction to polynomial stability conditions,
Seminar for Algebraic Geometry Group, State University of Campinas, Brazil, July 2024.
6. Algebra, geometry, and machines,
Department of Mathematics Open House, California State University, Northridge, March 2024.
5. A Graph Theory Approach Towards Classifying Calabi-Yau Spaces and Beyond,
RSP Summer Grant Presentations, California State University, Northridge, Februrary 2020.
4. An Interplay Between Algebra and Geometry,
Matador Math Society, California State University Northridge, April 2017.
3. Examples of Torsion Pairs, and t-structures on elliptic fibrations,
Learning Seminar in Algebraic Geometry (2 lectures), National Center for Theoretical Sciences, Taipei, Jun 2015.
2. From Algebra to Geometry, I & II,
Undergraduate Geometry Seminar, Chinese University of Hong Kong (2 lectures), Jul 2014.
1. Techniques in Derived Categories, I & II,
IMS Geometry Seminar, Chinese University of Hong Kong (2 lectures), Jul 2014.
Control Theory
Writings
The Pole Placement Problem and Degeneracy of Autoregressive Systems, NICTA Technical Report, 2005.
Talks
RSISE, Australian National University, Systems and Control Seminar, June 2005.