Overview & Keywords of My Research Interests
My research focuses on exploring the deep connections between symplectic geometry (A-side), algebraic geometry (B-side), and representation theory through Homological Mirror Symmetry (HMS).
Embedded Files
Key Points
The Z2-graded Fukaya categories of surfaces (A-side) are equivalent to the singularity categories of the mirror Landau-Ginzburg models with zero loci given by normal crossing surface singularities (B-side) [AAEKO13, PS21, ...].
In the local model case, the objects on the B-side can be represented and classified via Burban-Drozd's category of triples, whose objects are decorated representations on the hexagonal quiver. The category is representation-tame, and indecomposable objects fall into two types: the band-type and the string-type [BD17a].
In [Rho23], the decorated representations on the hexagonal quiver were related to the geometry of the underlying singular surface. Its globalization is still in progress.
The singularity categories on the B-side are equivalent to the categories of matrix factorizations [Orl03, PV11, Orl12], which are more directly connected to the Fukaya categories via Cho-Hong-Lau's localized mirror functors [CHL17].
In [CJKR22], we related the band-type objects of rank 1 to immersed loops in the Fukaya category of pair-of-pants surface. Some symplectic-geometric techniques were developed to allow the localized mirror functor to work well on the immersed loops.
In [CR24], we related all band-type objects to immersed loops with a local system in the Fukaya category. The corresponding canonical form of matrix factorizations was suggested, with many applications of geometric insights to algebraic operations.
The Z-graded Fukaya categories of graded surfaces with stops (A-side) are equivalent to the perfect derived categories of the corresponding gentle algebras [HKK17, LP20].
In representation theory, gentle algebras are well-known to be closed under derived equivalences and derived-tame, and indecomposable objects in their derived categories are classified into the band-type and the string-type [SZ03, BD17b and references therein]. Their relation with surface models was discovered in [Sch15], and the correspondence of the band- and string-type objects and graded immersed loops and arcs was established in [OPS18].
In [LP20], the authors constructed the surfaces corresponding to gentle algebras and used their partially wrapped Fukaya categories to obtain a criterion for (graded) gentle algebras to be derived equivalent.
On the B-side, certain non-commutative orders (called Auslander orders) over nodal singular curves (e.g. gluing of projective lines) were constructed in [BD11]. Their derived categories of coherent sheaves have a tilting complex, whose endomorphism algebra provides a broad class (but not all) of (ungraded) gentle algebras. Combined with the A-side relation with gentle algebras described above, they complete in these cases both the A- and B-sides of HMS.
They were generalized in [LP18] to Auslander orders over nodal stacky curves (e.g. gluing of weighted projective lines), which establish HMS of (fully) wrapped Fukaya categories of arbitrary genus punctured surfaces (but still with some restrictions on line fields on the surfaces). In [BD18], the authors again generalized the way of gluing weighted projective lines, which leads to a much broader class of gentle algebras and corresponding graded surfaces with stops. See my notes on the summary on the B-side.
Papers
∎ HMS of Indecomposable Cohen-Macaulay Modules for Some Degenerate Cusp Singularities [CJKR22]
with Cheol-Hyun Cho, Wonbo Jeong and Kyoungmo Kim (2022, arxiv)
∎ Homological Mirror Symmetry and Geometry of Degenerate Cusp Singularities [Rho23]
(Ph.D. Thesis, 2023, pdf)
∎ Canonical Form of Matrix Factorizations from Fukaya Category of Surface [CR24]
with Cheol-Hyun Cho (2024, arxiv)
Papers in Progress
∎ Geometric Fukaya Categories of Surfaces with Immersed Curves
∎ Formal Generators of Fukaya Categories of Orbifold Surfaces and Semi-gentle Algebras
with Severin Barmeier, Cheol-Hyun Cho, Kyoungmo Kim, Sibylle Schroll and Zhengfang Wang
∎ Intrinsic Geometry of Normal Crossing Surface Singularities and Matrix Factorizations
Invited Talks
∎ KIAS Topology Group Seminar, Korea Institute for Advanced Study, 2021.07.
∎ IBS-CGP Symplectic Monday Seminar, IBS Center for Geometry and Physics, 2021.09.
- Mirror symmetry correspondence between modules and Lagrangians (video)
∎ QSMS Topology Geometry Autumn Workshop, Quantum Structures in Modules and Spaces, 2021.10.
∎ Yonsei Geometry Seminar, Yonsei University, 2021.11.
∎ BRL Geometry of Submanifolds Seminar, Pusan National University, 2022.05.
∎ QSMS-BK21 Symplectic Seminar, Seoul National University, 2022.10.
- Representations of gentle algebras and partially wrapped Fukaya categories (notes)
∎ QSMS-BK21 Symplectic Seminar, Seoul National University, 2022.12.
- Morita theory for derived categories (notes)
∎ QSMS-BK21 Symplectic Seminar, Seoul National University, 2023.01.
- Geometric models for graded skew-gentle algebras (notes)
∎ IBS-CGP Symplectic Monday Seminar, IBS Center for Geometry and Physics, 2023.03.
- Homological mirror symmetry of degenerate cusp singularities and their representations (slides, video)
∎ Rookies Pitch, Seoul National University, 2023.05.
- An introduction to homological mirror symmetry (video (in Korean))
∎ Workshop for Young Symplectic Geometers, Pusan National University, 2023.06.
∎ KIAS Topology Seminar, Korea Institute for Advanced Study, 2023.07.
∎ QSMS-BK21 Symplectic Seminar, Seoul National University, 2023.07.
- Associative Yang-Baxter equation and Fukaya categories (notes)
---------------------------------- (Ph.D. completed & moved from Korea to Germany) ----------------------------------
∎ CRC Retreat, Bad Sassendorf, 2023.09.
- Project C3: A symplectic view of matrix factorizations (slides)
∎ Oberseminar Algebra und Geometrie, Universität Paderborn, 2023.10.
- An introduction to Fukaya category of surfaces: with applications via HMS (notes I, II)
∎ BIREP Seminar, Universität Bielefeld, 2023.11.
- Homological mirror symmetry correspondence on an affine model (slides)
∎ Seminar on Poisson Geometry and Integrable Systems, Universität Paderborn, 2023.12 ~ 2024.02
- I. Lagrangian mechanics (notes) + II. Hamiltonian mechanics (notes) + III. Integrability of rigid body problems (notes)
∎ CHARMS Summer School, Université de Versailles Saint-Quentin-en-Yvelines, 2024.05.
- Homological mirror symmetry of gentle algebras (B-side) (notes, video)
(as a mini course talk of lecture series on “An A and B perspective on geometric models for gentle algebras” by Sibylle Schroll)
∎ CRC Graduate Seminar, Universität Paderborn, 2024.06.
- An introduction to homological mirror symmetry (notes)
∎ Seminar on Poisson Geometry and Integrable Systems, Universität Paderborn, 2024.07.
- Integrability of high dimensional rigid bodies I ~ III (notes)
∎ Cologne Representation Theory Day 2, Universität zu Köln, 2024.10.
- Geometric Fukaya category of surfaces and its applications to representation theory (notes)
∎ Singularities and Geometric Structures, University of Miami, 2024.11.
- Matrix factorizations from Fukaya categories of surfaces (notes)
∎ Oberseminar Algebra und Geometrie, Universität Paderborn, 2024.12.
- Cohen-Macaulay sheaves on normal crossing surface singularities
Page updated
Google Sites
Report abuse