About

Siu-Cheong Lau is an Associate Professor in the Department of Mathematics & Statistics at Boston University. He is a member of the Geometry and Physics research group. For more information about Professor Lau, please see his personal webpage.

Research topics

• Mathematical analysis
• Pure mathematics
• Mathematics
• Geometry

Selected publications

• Mirror construction of Hecke correspondence between Nakajima quiver varieties

  arXiv (Cornell University) · 2026-01-20

  preprintOpen access1st authorCorresponding

  Nakajima constructed geometric representations of a deformed Kac-Moody Lie algebra using Hecke correspondences between quiver varieties. In this paper, we show that Hecke correspondences, which are holomorphic Lagrangians in products of Nakajima quiver varieties, can be obtained by applying the localized mirror construction to the morphism spaces between families of framed Lagrangian branes supported on the core of a plumbing of two-spheres. Moreover, for a non-ADE quiver, we show that the localized mirror functor is fully-faithful.

  PublisherDOI

• Mirror construction of Hecke correspondence between Nakajima quiver varieties

  ArXiv.org · 2026-01-20

  articleOpen access1st authorCorresponding

  Nakajima constructed geometric representations of a deformed Kac-Moody Lie algebra using Hecke correspondences between quiver varieties. In this paper, we show that Hecke correspondences, which are holomorphic Lagrangians in products of Nakajima quiver varieties, can be obtained by applying the localized mirror construction to the morphism spaces between families of framed Lagrangian branes supported on the core of a plumbing of two-spheres. Moreover, for a non-ADE quiver, we show that the localized mirror functor is fully-faithful.

  PublisherOA PDF

• A Logifold Structure for Measure Space

  Axioms · 2025-08-01

  articleOpen accessSenior authorCorresponding

  In this paper, we develop a geometric formulation of datasets. The key novel idea is to formulate a dataset to be a fuzzy topological measure space as a global object and equip the space with an atlas of local charts using graphs of fuzzy linear logical functions. We call such a space a logifold. In applications, the charts are constructed by machine learning with neural network models. We implement the logifold formulation to find fuzzy domains of a dataset and to improve accuracy in data classification problems.

  PublisherDOI

• Equivariant Partially Wrapped Fukaya Categories on Liouville Sectors

  OpenBU (Boston University) · 2025-12-30

  preprintOpen access

  We develop an equivariant Lagrangian Floer theory for Liouville sectors that have symmetry of a Lie group $G$. Moreover, for Liouville manifolds with $G$-symmetry, we develop a correspondence theory to relate the equivariant Lagrangian Floer cohomology upstairs and Lagrangian Floer cohomology of its quotient. Furthermore, we study the symplectic quotient in the presence of nodal type singularities and prove that the equivariant correspondence gives an isomorphism on cohomologies which was conjectured by Lekili-Segal.

  PublisherDOI

• Equivariant Partially Wrapped Fukaya Categories on Liouville Sectors

  ArXiv.org · 2025-12-30

  articleOpen access

  We develop an equivariant Lagrangian Floer theory for Liouville sectors that have symmetry of a Lie group $G$. Moreover, for Liouville manifolds with $G$-symmetry, we develop a correspondence theory to relate the equivariant Lagrangian Floer cohomology upstairs and Lagrangian Floer cohomology of its quotient. Furthermore, we study the symplectic quotient in the presence of nodal type singularities and prove that the equivariant correspondence gives an isomorphism on cohomologies which was conjectured by Lekili-Segal.

  PublisherOA PDF

• Logifold: A Geometrical Foundation of Ensemble Machine Learning

  2024-11-04 · 2 citations

  articleSenior author

  We present a local-to-global and measure-theoretical approach to understanding datasets. The core idea is to formulate a logifold structure and to interpret network models with restricted domains as local charts of datasets. In particular, this provides a mathematical foundation for ensemble machine learning. Our experiments demonstrate that logifolds can be implemented to identify fuzzy domains and improve accuracy compared to taking average of model outputs. Additionally, we provide a theoretical example of a logifold, highlighting the importance of restricting to domains of classifiers in an ensemble.

  PublisherDOI

• SYZ mirror symmetry for del Pezzo surfaces and affine structures

  Advances in Mathematics · 2024-01-18 · 3 citations

  article1st author
  PublisherDOI

• A logifold structure on measure space

  arXiv (Cornell University) · 2024-05-09

  preprintOpen accessSenior author

  In this paper,we develop a local-to-global and measure-theoretical approach to understand datasets. The idea is to take network models with restricted domains as local charts of datasets. We develop the mathematical foundations for these structures, and show in experiments how it can be used to find fuzzy domains and to improve accuracy in data classification problems.

  PublisherOA PDFDOI

• Gluing localized mirror functors

  Journal of Differential Geometry · 2024-03-01 · 2 citations

  articleSenior author

  We develop a method of gluing the local mirrors and functors constructed from immersed Lagrangians in the same deformation class. As a result, we obtain a canonical mirror functor to the glued category. We apply the method to construct the mirrors of punctured Riemann surfaces and show that our functor derives homological mirror symmetry.

  PublisherDOI

• Mirror Construction for Nakajima Quiver Varieties

  arXiv (Cornell University) · 2024-04-24

  preprintOpen access

  In this paper, we construct the ADHM quiver representations and the corresponding sheaves as the mirror objects of formal deformations of the framed immersed Lagrangian sphere decorated with flat bundles. More generally, we construct Nakajima quiver varieties as localized mirrors of framed nodal unions of Lagrangian spheres in dimension two. This produces a mirror functor from the Fukaya category of a framed plumbing of surfaces to the dg category of complexes of bundles over the corresponding Nakajima quiver varieties. For affine ADE quivers in specific multiplicities, the corresponding (unframed) Lagrangian immersions are homological tori, whose moduli of stable deformations are asymptotically locally Euclidean (ALE) spaces. We show that framed stable Lagrangian branes are transformed into monadic complexes of framed torsion-free sheaves over the ALE spaces. A main ingredient is the notion of framed Lagrangian immersions and their Maurer-Cartan deformations. Moreover, using the formalism of quiver algebroid stacks, we find isomorphisms between the moduli of stable Lagrangian immersions and that of special Lagrangian fibers of an SYZ fibration in the affine $A_n$ cases.

  PublisherOA PDFDOI

Frequent coauthors

• Kwokwai Chan

  33 shared

• Naichung Conan Leung

  28 shared

• Hsian-Hua Tseng

  23 shared

• Cheol-Hyun Cho

  22 shared

• Hansol Hong

  Yonsei University

  19 shared

• George Jeffreys

  Boston University

  6 shared

• Baosen Wu

  Harvard University

  5 shared

• Yu-Shen Lin

  Boston University

  5 shared

Education

• Ph.D., Mathematics

  University of California, Berkeley

  1983

• M.S., Mathematics

  University of California, Berkeley

  1979

• B.S., Mathematics

  University of Hong Kong

  1977