I am Ki Fung Chan (Kifung Chan), a PhD student at the Chinese University of Hong Kong under the supervision of Prof. Conan Leung.
My research interest lies in algebraic geometry, symplectic geometry, representation theory, and 2d and 3d mirror symmetry.
Contact
Email: kfchan@math.cuhk.edu.hk
Office: Rm 505, AB1
Tel: (852) 3943 4298
Research
Publication and Preprints
Quantum cohomology, shift operators, and Coulomb branches, joint work with Kwokwai Chan and Eddie lam. arXiv:2505.23400.
Given a complex reductive group $G$ and a $G$-representation $\bN$, there is an associated quantized Coulomb branch algebra $\sA_{G,\bN}^\hbar$ defined by~\cite{Nak,BFN}. In this paper, we give a new interpretation of $\sA_{G,\bN}^\hbar$ as the largest subalgebra of the equivariant Borel--Moore homology of the affine Grassmannian on which shift operators can naturally be defined.
As a main application, we show that if $X$ is a smooth semiprojective variety equipped with a $G$-action, and $X \to \bN$ is a $G$-equivariant proper holomorphic map, then the equivariant big quantum cohomology $QH_G^\bullet(X)$ defines a quasi-coherent sheaf of algebras on the Coulomb branch with coisotropic support. Upon specializing the Novikov and bulk parameters, this sheaf becomes coherent with Lagrangian support. We also apply our construction to recover Teleman's gluing construction for Coulomb branches~\cite{2drole} and derive different generalizations of the Peterson isomorphism~\cite{peterson}.
3d Mirror Symmetry is Mirror Symmetry, joint work with Naichung Conan Leung. arXiv:2410.03611.
This paper introduces an approach to studying 3d mirror symmetry via (2d) mirror symmetry. The main observations are: 1) 3d brane transforms are given by SYZ-type transforms; 2) the exchange of symplectic and complex structures in 2d mirror symmetry induces the exchange of Kähler and equivariant parameters in 3d mirror symmetry; 3) the functorialities of 2d mirror symmetry control the gluing of 3d mirrors. We formulate and prove these observations in some important cases in this paper, and the paper below.
SYZ Mirrors in non-Abelian 3d Mirror Symmetry, joint work with Naichung Conan Leung. arXiv:2408.09479. (submitted)
This paper demonstrates that the SYZ-type transform in Abelian pure gauge theories extends to non-Abelian gauge theories. We reinterpret the existence of such extensions as a statement in symplectic geometry and establish it using Floer theory. To the best of our knowledge, this result has not appeared in the literature before. Notably, it imposes constraints on non-displaceable Lagrangian branes in the presence of a non-Abelian group action a symplectic manifold (though not necessary on the Lagrangians).
Calibrated Geometry in G2 Manifolds, survey with Naichung Conan Leung. Fields Inst. Commun. 84, 103--111.
Appears as a chapter in Karigiannis, S., Leung, N. C., & Lotay, J. D. (Eds.). (2020). Lectures and surveys on G2-manifolds and related topics. Fields Institute Communications (Vol. 84). Springer. https://doi.org/10.1007/978-1-0716-0577-6
The LB cohomology on compact torsion-free G2 manifolds and an application to "almost" formality. Ann. Global Anal. Geom. 55, No. 2, 325--369.
In this paper, we define the LB cohomology, and use it to prove the almost formality of compact torsion-free G2 manifolds. In particular, this result implies the vanishing of most of the Massey triple products. Note that counterexamples to the full formality of compact torsion-free G2 manifolds are provided in arXiv:2409.04362.
Cohomology on almost complex manifolds and the ddbar lemma, joint with Spiro Karigiannis and Chi Cheuk Tsang. Asian J. Math. 23, No. 4, 561--584.
We study cohomologies on an almost complex manifold (M,J), defined using the Nijenhuis-Lie derivations LJ and LN, induced by the almost complex structure J and its Nijenhuis tensor N, regarded as vector-valued forms on M. We use these cohomologies to construct examples of non-integrable almost complex manifolds.
Ongoing projects
Coulomb branch Nil-Hecke algebra, and variants of Nakajima Hikita conjecture (with Conan Leung)
Convexity of moment maps and the exchange of Kähler and equivariant parameters (with Conan Leung)
Jacobian rings (with Eddie Lam, Jiayu Song and Ju Tan)
Coulomb branch and the Springer resolution
Presentations and Conferences
Quantum cohomology, shift operators, and Coulomb branches. 3 July 2025. Kyoto Univeristy. Notes
Mirror symmetry of nonabelian group actions. 13 June 2025; Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Zoominar. Notes Video
Quantum cohomology, shift operators, and Coulomb branches. 2 June 2025. MIST workshop, IMS-CUHK. Notes
3d Mirror symmetry is Mirror Symmetry 11 April 2025; Harvard CMSA. Notes Video
Convexity of moment maps and 3d mirror symmetry 20 Nov 2024; Lunch seminar; Institute of Basic Science
Coulomb branches and 2d mirror symmetry. 19 Nov 2024; Symplectic Monday; Institute of Basic Science. Notes 1 Notes 2 Video
3d Mirror Symmetry is Mirror Symmetry 23 Oct 2024; Boston University.
3d Mirror Symmetry is Mirror Symmetry. 21 Oct 2024; Informal StringMath Seminar, UC Berkeley. Notes Video
SYZ Mirrors in non-Abelian 3d Mirror Symmetry. 12-20 Aug 2024; MIST workshop, IMS-CUHK. Notes
Mirror Symmetry of a non-Abelian group action. 8-12 July 2024; Summer Schools in Geometry and Physics, KIAS. Slides
SYZ Mirrors in non-Abelian 3d Mirror Symmetry. 5-10 June 2024; MIST workshop, IMS-CUHK. Notes
A report on Hikita conjecture. 22 May 2023. Mirror Symmetry Day. IMS-CUHK
Teaching Experience
Teaching Assistant
CUHK:
Course: MATH 3070 – Topology
Year: 2021, 2022
Course: MATH 3060 – Real Analysis
Year: 2021, 2022
Course: MATH 4060 – Complex Analysis
Year: 2020
Course: MATH 2020 – Advanced Calculus II
Year: 2019, 2020
Course: MATH 1010 – University Mathematics
Course: MATH 1010 – Linear Alegbra
Year: 2018 (Best TA award)