arXiv subjects

Hansol Hong

Publications and source records attributed to Hansol Hong.

At least 19 recordsLinked to original sources

arXiv2026

Mirror functor for deformed preprojective algebras

We study localized homological mirror symmetry associated to an immersed Lagrangian brane $\mathbb{L}$, possibly equipped with a higher rank flat bundle, of a symplectic manifold $X$. Under a certain finiteness assumption on the Floer theory of $\mathbb{L}$, we deduce a quasi-equivalence $\mathcal{D}\mathrm{Fuk}_\mathbb{L}(X) \cong \mathcal{D}_{\mathrm{fd}}(\tilde{\mathcal A}_\mathbb{L})$ using Koszul duality, where $\tilde{\mathcal{A}}_{\mathbb{L}}$ is the dual differential graded quiver algebra called the extended localized mirror. We apply this to obtain some HMS results for plumbings of cotangent bundles of spheres, and to reproduce known results of split-generation of compact objects. In the second part of the paper, we consider bulk deformation cycles of $X$ that have non-trivial intersections with $\mathbb{L}$. This gives rise to noncommutative deformations of the mirror. When applied to (framed) plumbings, we obtain mirror functors to deformed preprojective algebras (or Nakajima quiver varieties at a general complex moment-map level). For the ADHM and affine $ADE$-type immersions, our construction produces mirror functors to the noncommutative spaces studied by Kapustin-Kuznetsov-Orlov, Baranovsky-Ginzburg-Kuznetsov and Kawamata.

math.SG↗

arXiv2025

Closed-string mirror symmetry for dimer models

For all punctured Riemann surfaces arising as mirror curves of toric Calabi--Yau threefolds, we show that their symplectic cohomology is isomorphic to the compactly supported Hochschild cohomology of the noncommutative Landau--Ginzburg model defined on the NCCR of the associated toric Gorenstein singularities. This mirror correspondence is established by analyzing the closed-open map with boundaries on certain combinatorially defined immersed Lagrangians in the Riemann surface, yielding a ring isomorphism. We give a detailed examination of the properties of this isomorphism, emphasizing its relationship to the singularity structure.

math.SG↗

arXiv2025

MS2toImg: A Framework for Direct Bioactivity Prediction from Raw LC-MS/MS Data

Untargeted metabolomics using LC-MS/MS offers the potential to comprehensively profile the chemical diversity of biological samples. However, the process is fundamentally limited by the "identification bottleneck," where only a small fraction of detected features can be annotated using existing spectral libraries, leaving the majority of data uncharacterized and unused. In addition, the inherently low reproducibility of LC-MS/MS instruments introduces alignment errors between runs, making feature alignment across large datasets both error-prone and challenging. To overcome these constraints, we developed a deep learning method that eliminates the requirement for metabolite identification and reduces the influence of alignment inaccuracies. Here, we propose MS2toImg, a method that converts raw LC-MS/MS data into a two-dimensional images representing the global fragmentation pattern of each sample. These images are then used as direct input for a convolutional neural network (CNN), enabling end-to-end prediction of biological activity without explicit feature engineering or alignment. Our approach was validated using wild soybean samples and multiple bioactivity assays (e.g., DPPH, elastase inhibition). The MS2toImg-CNN model outperformed conventional machine learning baselines (e.g., Random Forest, PCA), demonstrating robust classification accuracy across diverse tasks. By transforming raw spectral data into images, our framework is inherently less sensitive to alignment errors caused by low instrument reproducibility, as it leverages the overall fragmentation landscape rather than relying on precise feature matching. This identification-free, image-based approach enables more robust and scalable bioactivity prediction from untargeted metabolomics data, offering a new paradigm for high-throughput functional screening in complex biological systems.

q-bio.QM↗

arXiv2024

Orbifold Kodaira-Spencer maps and closed-string mirror symmetry for punctured Riemann surfaces

When a Weinstein manifold admits an action of a finite abelian group, we propose its mirror construction following the equivariant TQFT-type construction, and obtain as a mirror the orbifolding of the mirror of the quotient with respect to the induced dual group action. As an application, we construct an orbifold Landau-Ginzburg mirror of a punctured Riemann surface given as an abelian cover of the pair-of-pants, and prove its closed-string mirror symmetry using the (part of) closed-open map twisted by the dual group action.

math.SG↗

arXiv2023

Morse superpotentials and blow-ups of surfaces

We study the Landau-Ginzburg mirror of toric/non-toric blowups of (possibly non-Fano) toric surfaces arising from SYZ mirror symmetry. Through the framework of tropical geometry, we provide an effective method for identifying the precise locations of critical points of the superpotential, and further show their non-degeneracy for generic parameters. Moreover, we prove that the number of geometric critical points equals the rank of cohomology of the surface, which leads to its closed-string mirror symmetry due to Bayer's earlier result.

math.SG↗

arXiv2022

Maurer-Cartan deformation of Lagrangians

The Maurer-Cartan algebra of a Lagrangian $L$ is the algebra that encodes the deformation of the Floer complex $CF(L,L;Λ)$ as an $A_\infty$-algebra. We identify the Maurer-Cartan algebra with the $0$-th cohomology of the Koszul dual dga of $CF(L,L;Λ)$. Making use of the identification, we prove that there exists a natural isomorphism between the Maurer-Cartan algebra of $L$ and a suitable subspace of the completion of the wrapped Floer cohomology of another Lagrangian $G$ when $G$ is \emph{dual} to $L$ in the sense to be defined. In view of mirror symmetry, this can be understood as specifying a local chart associated with $L$ in the mirror rigid analytic space. We examine the idea by explicit calculation of the isomorphism for several interesting examples.

math.SG↗

arXiv2021

Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces

We construct special Lagrangian fibrations for log Calabi-Yau surfaces, and scattering diagrams from Lagrangian Floer theory of the fibres. Then we prove that the scattering diagrams recover the scattering diagrams of Gross-Pandharipande-Siebert and the canonical scattering diagrams of Gross-Hacking-Keel. With an additional assumption on the non-negativity of boundary divisors, we compute the disc potentials of the Lagrangian torus fibres via a holomorphic/tropical correspondence. As an application, we provide a version of mirror symmetry for rank two cluster varieties.

math.SG↗

arXiv2021

Noncommutative homological mirror functor

We formulate a constructive theory of noncommutative Landau-Ginzburg models mirror to symplectic manifolds based on Lagrangian Floer theory. The construction comes with a natural functor from the Fukaya category to the category of matrix factorizations of the constructed Landau-Ginzburg model. As applications, it is applied to elliptic orbifolds, punctured Riemann surfaces and certain non-compact Calabi-Yau threefolds to construct their mirrors and functors. In particular it recovers and strengthens several interesting results of Etingof-Ginzburg, Bocklandt and Smith, and gives a unified understanding of their results in terms of mirror symmetry and symplectic geometry. As an interesting application, we construct an explicit global deformation quantization of an affine del Pezzo surface as a noncommutative mirror to an elliptic orbifold.

math.AG↗

arXiv2020

Immersed two-spheres and SYZ with Application to Grassmannians

We develop a Floer theoretical gluing technique and apply it to deal with the most generic singular fiber in the SYZ program, namely the product of a torus with the immersed two-sphere with a single nodal self-intersection. As an application, we construct immersed Lagrangians in $\mathrm{Gr}(2,\mathbb{C}^n)$ and $\mathrm{OG}(1,\mathbb{C}^5)$ and derive their SYZ mirrors. It recovers the Lie theoretical mirrors constructed by Rietsch. It also gives an effective way to compute stable disks (with non-trivial obstructions) bounded by immersed Lagrangians.

math.SG↗

arXiv2020

$T$-equivariant disc potentials for toric Calabi-Yau manifolds

We study the equivariant disc potentials for immersed SYZ fibers in toric Calabi-Yau manifolds. The immersed Lagrangians play a crucial role in the partial compactification of the SYZ mirrors. Morever, their equivariant disc potentials have a close relation with that of Aganagic-Vafa branes. We show that the potentials can be computed by using an equivariant version of isomorphisms in the Fukaya category.

math.SG↗

arXiv2020

Big Quantum cohomology of orbifold spheres

We construct a Kodaira-Spencer map from the big quantum cohomology of a sphere with three orbifold points to the Jacobian ring of the mirror Landau-Ginzburg potential function. This is constructed via the Lagrangian Floer theory of the Seidel Lagrangian and we show that Kodaira-Spencer map is a ring isomorphism.

math.SG↗

arXiv2019

Bulk-deformed potentials for toric Fano surfaces, wall-crossing and period

We provide an inductive algorithm to compute the bulk-deformed potentials for toric Fano surfaces via wall-crossing techniques and a tropical-holomorphic correspondence theorem for holomorphic discs. As an application of the correspondence theorem, we also prove a big quantum period theorem for toric Fano surfaces which relates the log descendant Gromov-Witten invariants with the oscillatory integrals of the bulk-deformed potentials.

math.SG↗

arXiv2018

Gluing Localized Mirror Functors

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 global mirror geometry and a canonical mirror functor. We apply the method to construct the mirrors of punctured Riemann surfaces and show that our functor derives homological mirror symmetry.

math.SG↗

arXiv2018

Moduli of Lagrangian immersions with formal deformations

We introduce a joint project with Cheol-Hyun Cho on the construction of quantum-corrected moduli of Lagrangian immersions. The construction has important applications to mirror symmetry for pair-of-pants decompositions, SYZ and wall-crossing. The key ingredient is Floer-theoretical gluing between local moduli spaces of Lagrangians with different topologies.

math.AG↗

arXiv2017

Mirror of Atiyah flop in symplectic geometry and stability conditions

We study the mirror operation of the Atiyah flop in symplectic geometry. We formulate the operation for a symplectic manifold with a Lagrangian fibration. Furthermore we construct geometric stability conditions on the derived Fukaya category of the deformed conifold and study the action of the mirror Atiyah flop on these stability conditions.

math.SG↗

arXiv2016

Localized mirror functor constructed from a Lagrangian torus

Fixing a weakly unobstructed Lagrangian torus in a symplectic manifold X, we define a holomorphic function W known as the Floer potential. We construct a canonical A-infinity functor from the Fukaya category of X to the category of matrix factorizations of W. It provides a unified way to construct matrix factorizations from Lagrangian Floer theory. The technique is applied to toric Fano manifolds to transform Lagrangian branes to matrix factorizations. Using the method, we also obtain an explicit expression of the matrix factorization mirror to the real locus of the complex projective space.

math.SG↗

arXiv2016

Counting of Holomorphic Orbi-spheres in $\mathbb{P}^1_{2,2,2,2}$ and Determinant Equation

We count the number of holomorphic orbi-spheres in the $\mathbb{Z}_2$-quotient of an elliptic curve. We first prove that there is an explicit correspondence between the holomorphic orbi-spheres and the sublattices of $\mathbb{Z} \oplus \mathbb{Z} \sqrt{-1} (\subset \mathbb{C})$. The problem of counting sublattices of index $d$ then reduces to find the number of integer solutions of the equation $αδ- βγ= d$ up to an equivalence.

math.SG↗

arXiv2015

Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for P^1_{a,b,c}

This paper gives a new way of constructing Landau-Ginzburg mirrors using deformation theory of Lagrangian immersions motivated by the works of Seidel, Strominger-Yau-Zaslow and Fukaya-Oh-Ohta-Ono. Moreover we construct a canonical functor from the Fukaya category to the mirror category of matrix factorizations. This functor derives homological mirror symmetry under some explicit assumptions. As an application, the construction is applied to spheres with three orbifold points to produce their quantum-corrected mirrors and derive homological mirror symmetry. Furthermore we discover an enumerative meaning of the (inverse) mirror map for elliptic curve quotients.

math.SG↗