Siu-Cheong Lau

MS

 

Room 230, Department of Mathematics and Statistics,

Boston University

111 Cummington Mall, Boston, MA 02215

Email: 1lau1@math.bu.edu (remove the two 1’s)

About me

I am a faculty member in the Department of Mathematics and Statistics at Boston University.

Research interest

My main research interest lies in complex algebraic geometry and symplectic geometry, and more specifically mirror symmetry.  With my dear collaborators, we developed a constructive theory of mirror symmetry which leads to interesting results on SYZ, open Gromov-Witten invariants, mirror maps, homological mirror symmetry and modular forms.


Recently, I am particularly interested in quiver stack coming from noncommutative deformations of Lagrangian immersions, and the application of quiver geometry and near-algebra in quantum models and machine learning.


I am very excited to observe a beautiful connection between quiver/stack/moduli/category theory and the relatively young subject of neural networks. Such a connection is rather surprising to myself. Here is what I feel:


Neural networks and deep learning are models of classical projection and generalization of quantum physics, which is a core part of the ultimate truth. It will gradually give birth to a revolutionary branch of mathematics in companion with the modern developments of computational models and quantum computing, just like Calculus in companion with Newton's discovery of gravity and laws of motion.

Pieces of beauty

The above figure shows a hexagon tiling and some related graphs.  It appears in the study of SYZ mirror symmetry, and is related to Riemann theta functions, modular forms, Gromov-Witten theory, and general-type varieties.  It comes from my joint work with Atsushi Kanazawa.

The above figure shows several beautiful tessellations of the plane.  Polygon countings in the figure have miraculous relations with periods of elliptic curves.  It comes from my joint work with Cheol-Hyun Cho and Hansol Hong.  In my joint work with Jie Zhou, we relate the countings with modular forms.

Publications

42. Mirror Symmetry for Quiver Algebroid Stacks, with Junzheng Nan and Ju Tan, arxiv:2206.03028.

41. SYZ mirror symmetry for del Pezzo surfaces and affine structures, with Yu-Shen Lin and Tsung-Ju Lee, arxiv:2206.01681.

40. Quantum Finite Automata and Quiver Algebras, with George Jeffreys, Proceedings of MaxEnt 2022, arxiv:2203.07597.

39. Noncommutative Geometry of Computational Models and Uniformization for Framed Quiver Varieties, with George Jeffreys, arxiv:2201.05900.

38. Kähler Geometry of Quiver Varieties and Machine Learning, with George Jeffreys, arxiv:2101.11487, to appear in Foundations of Computational Mathematics.

37. Big Quantum cohomology of orbifold spheres, with Lino Amorim, Cheol-Hyun Cho and Hansol Hong, arxiv:2002.11180.

36. Equivariant SYZ mirror symmetry, to appear in ICCM proceedings 2019.

35. T-equivariant disc potentials for toric Calabi-Yau manifolds, with Hansol Hong, Yoosik Kim and Xiao Zheng, arXiv:1912.11455.

34. T-equivariant disc potential and SYZ mirror construction, with Yoosik Kim and Xiao Zheng, arXiv:1906.11749.

33. Immersed two-sphere and SYZ with applications to Grassmannians, with Hansol Hong and Yoosik Kim, to appear in Journal of Differential Geometry.

32. On the Complex Affine Structures of SYZ Fibration of Del Pezzo Surfaces, with Tsung-Ju Lee and Yu-Shen Lin, to appear in Advances in Theoretical and Mathematical Physics.

31. Gluing Localized Mirror Functors, with Cheol-Hyun Cho and Hansol Hong, arXiv:1810.02045, to appear in Journal of Differential Geometry

30. Affine elliptic surfaces with type-A singularities and orbi-conifolds, arXiv:1802.08891.

29. A note on disk counting in toric orbifolds, with Kwokwai Chan, Cheol-Hyun Cho, Naichung Conan Leung and Hsian-Hua Tseng, SIGMA 16 (2020), 055, 15 pages.

28. Open Gromov-Witten invariants and mirror maps for semi-Fano toric manifolds, with Kwokwai Chan, Naichung Leung and Hsian-Hua Tseng, to appear in Pure and Applied Mathematics Quarterly.

27. SYZ mirror symmetry for hypertoric varieties, with Xiao Zheng, arXiv:1804.05506, to appear in Communications in Mathematical Physics.

26. Mirror of Atiyah flop in symplectic geometry and stability conditions, with Yu-Wei Fan, Hansol Hong and Shing-Tung Yau, Advances in Theoretical and Mathematical Physics 22 (2018), no. 5.

25. Local Calabi-Yau manifolds of affine type A and open Yau-Zaslow formula via SYZ mirror symmetry, with Atsushi Kanazawa, arXiv:1605.00342, to appear in Journal of Geometry and Physics.

24. Geometric transitions and SYZ mirror symmetry, with Atsushi Kanazawa, arXiv:1503.03829, to appear in Pacific Journal of Mathematics.

23. Localized mirror functor constructed from a Lagrangian torus, with Cheol-Hyun Cho and Hansol Hong, arXiv:1406.4597, to appear in Journal of Geometry and Physics.

22. Noncommutative homological mirror functor, with Cheol-Hyun Cho and Hansol Hong, to appear in Memoirs of the AMS.

21. Moduli of Lagrangian immersions with formal deformations, Proceedings of the Gokova Geometry-Topology Conference 2017, 9-36, Int. Press, Somerville, MA, 2018.

20. Quantum corrections and wall-crossing via Lagrangian intersections, to appear in ICCM Notices.

19. Generalized SYZ mirror transformation, to appear in ICCM proceedings.

18. Lagrangian Floer potential of orbifold spheres, with Cheol-Hyun Cho, Hansol Hong and Sang-Hyun Kim, Advances in Mathematics 306 (2017), 344-426.

17. Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for P^1_{a,b,c}, with Cheol-Hyun Cho and Hansol Hong, Journal of Differential Geometry 106 (2017), no. 1, 45-126.

16. Open Gromov-Witten invariants, mirror maps and Seidel representations for toric manifolds, with Kwokwai Chan, Naichung Conan Leung and Hsian-Hua Tseng, Duke Mathematical Journal 166 (2017), no. 8, 1405-1462.

15. Non-Kaehler SYZ mirror symmetry, with Li-Sheng Tseng and Shing-Tung Yau, Communications in Mathematical Physics 340 (2015), no.1, 145-170.

14. Modularity of open Gromov-Witten potentials of elliptic orbifolds, with Jie Zhou, Communications in Number Theory and Physics 9 (2015), no.2, 345-385.

13. Gross fibration, SYZ mirror symmetry, and open Gromov-Witten invariants for toric Calabi-Yau orbifolds, with Kwokwai Chan, Cheol-Hyun Cho and Hsian-Hua Tseng, Journal of Differential Geometry 103 (2016), no.2, 207-288.

12. Gross-Siebert's slab functions and open GW invariants for toric Calabi-Yau manifolds,
Mathematical Research Letters 22 (2015), no.3, 881-898.

11. Genaralized SYZ and homological mirror symmetry, with Cheol-Hyun Cho,
Handbook for Mirror Symmetries of Calabi-Yau and Fano Manifolds.

  1. 10.Open Gromov-Witten invariants and SYZ under local conifold transitions,
    Journal of the London Mathematical Society (2) 90 (2014), no. 2, 413-435.

  2. 9.Toric, global, and generalized SYZ,
    International Congress of Chinese Mathematicians 2013.

  3. 8.Lagrangian Floer superpotentials and crepant resolutions for toric orbifolds, with Kwokwai Chan, Cheol-Hyun Cho and Hsian-Hua Tseng, Communications in Mathematical Physics 328 (2014), no. 1, 83-130.

  4. 7.Enumerative meaning of mirror maps for toric Calabi-Yau manifolds, with Kwokwai Chan, Naichung Conan Leung and Hsian-Hua Tseng, Advances in Mathematics 244 (2013), 605-625.

  5. 6.Open Gromov-Witten invariants and superpotentials for semi-Fano toric surfaces, with Kwokwai Chan, International Mathematics Research Notices (2014), no. 14, 3759-3789.

  6. 5.Open Gromov-Witten invariants on toric manifolds,
    Oberwolfach Reports 9 (2012), no.2, 1265-1267.

  7. 4.Mirror maps equal SYZ maps for toric Calabi-Yau surfaces, with Baosen Wu and Naichung Leung, Bulletin of the London Mathematical Society 44 (2012), no.2, 255-270.

  8. 3.A relation for Gromov-Witten invariants of local Calabi-Yau threefolds, with Baosen Wu and Naichung Leung, Mathematical Research Letters 18 (2011), pp. 943-956.

  9. 2.SYZ mirror symmetry for toric Calabi-Yau manifolds, with Kwokwai Chan and Naichung Leung,
    Journal of Differential Geometry 90 (2012), pp. 177-250.

  10. 1.Conformal geometry and special holonomy, with Naichung Leung,
    Recent Advances in Geometric Analysis 11 (2010), pp. 195-209.

Teaching (old and update stopped; I have switched to Blackboard)

Notes of selected talks

I hope you will find these notes helpful.  Comments are welcome.

Quiver algebroid stacks and its applications
BU-Keio-Tsinghua workshop 2022.

Equivariant disc potentials for SYZ torus fibrations
International Congress of Chinese Mathematicians 2019.

Moduli theory of Lagrangian immersions and mirror symmetry
Kyoto University, and International Congress of Chinese Mathematicians 2017.

A short trip to tropical geometry
Talk to high school students, The Chinese University of Hong Kong, 2016.

SYZ, tiling and modularity
Simons Center, 2015, and UC Berkeley, 2016.

Generalized SYZ and homological mirror symmetry
ICM Satellite Conference 2014, Taiwan.

Open GW invariants and Seidel elements of toric manifolds
Simons Center, 2012.

SYZ mirror symmetry for toric CY manifolds
The Chinese University of Hong Kong, 2011.

Awards

Best Paper Award (2019), ICCM


Best Paper Silver Award (2017), ICCM


Distinguished Paper Award (2017), ICCM


Certificate of Teaching Excellence (2014), Harvard University


Doctoral Thesis Gold Award (2012), New World Mathematics Award at ICCM

Hobbies besides mathematics

Chinese flute


Novels (金庸,古龍,黃易, 三體...)

In the broad light of day mathematicians check their equations and their proofs, leaving no stone unturned in their search for rigour. But, at night, under the full moon, they dream, they float among the stars and wonder at the miracle of the heavens. They are inspired. Without dreams there is no art, no mathematics, no life.

Michael Atiyah