Kyungmin Rho - Overview

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).

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.