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Jongil Park

Publications and source records attributed to Jongil Park.

At least 19 recordsLinked to original sources
arXiv2026

The Algebraic Montgomery-Yang Problem

We completely resolve the algebraic Montgomery - Yang problem, a conjecture of Kollar stating that every rational homology projective plane with quotient singularities and a simply-connected smooth locus has at most three singular points. The crux of our proof is a new lattice theoretic constraint, obtained by combining Donaldson's diagonalization theorem with the distinguished spin^c structure on the smooth locus whose determinant line bundle is the canonical bundle. Together with the orbifold Bogomolov - Miyaoka - Yau inequality, this constraint rules out all remaining cases in the problem and completes the proof.

math.AG↗
arXiv2025

Towards Efficient Post-Training via Fourier-Driven Adapter Architectures

We propose a novel framework, termed Fourier-Activated Adapter (FAA), for parameter-efficient fine-tuning of large pre-trained language models. By incorporating random Fourier features into lightweight adapter modules, FAA decomposes intermediate representations into complementary low- and high-frequency components, enabling frequency-aware modulation of semantic information. This design allows the model to selectively emphasize informative frequency bands during adaptation while preserving the representational capacity of the frozen backbone. Extensive experiments on GLUE, E2E NLG, and instruction-tuning benchmarks demonstrate that FAA consistently achieves competitive or superior performance compared to existing parameter-efficient fine-tuning methods, while maintaining low computational and memory overhead. Ablation studies further verify the effectiveness of frequency-aware activation and adaptive weighting mechanisms, highlighting FAA as a robust and efficient approach for post-training large language models.

cs.CL↗
arXiv2024

On lens spaces bounding smooth 4-manifolds with $\boldsymbol{b_2=1}$

We study which lens spaces can bound smooth 4-manifolds with second Betti number one under various topological conditions. Specifically, we show that there are infinite families of lens spaces that bound compact, simply-connected, smooth 4-manifolds with second Betti number one, yet cannot bound a 4-manifold consisting of a single 0-handle and 2-handle. Additionally, we establish the existence of infinite families of lens spaces that bound compact, smooth 4-manifolds with first Betti number zero and second Betti number one, but cannot bound simply-connected 4-manifolds with second Betti number one. The construction of such 4-manifolds with lens space boundaries is motivated by the study of rational homology projective planes with cyclic quotient singularities.

math.GT↗
arXiv2024

On rational homology projective planes with quotient singularities of small indices

In this article, we study the effects of topological and smooth obstructions on the existence of rational homology complex projective planes that admit quotient singularities of small indices. In particular, we provide a classification of the types of quotient singularities that can be realized on rational homology complex projective planes with indices up to three, whose smooth loci have trivial first integral homology group.

math.GT↗
arXiv2024

Algebraic Montgomery-Yang problem and smooth obstructions

Let $S$ be a rational homology complex projective plane with quotient singularities. The algebraic Montgomery-Yang problem conjectures that the number of singular points of $S$ is at most three if its smooth locus is simply-connected. In this paper, we leverage results from the study of smooth 4-manifolds, including the Donaldson diagonalization theorem and Heegaard Floer correction terms, to establish additional conditions for $S$. As a result, we eliminate the possibility of a rational homology complex projective plane of specific types with four singularities. Moreover, we identify large families encompassing infinitely many types of singularities that satisfy the orbifold BMY inequality, a key property in algebraic geometry, yet are obstructed from being a rational homology complex projective plane due to smooth conditions. Additionally, we discuss computational results related to this problem, offering new insights into the algebraic Montgomery-Yang problem.

math.GT↗
arXiv2023

A topological characterization of symplectic fillings of Seifert 3-manifolds

In this paper, we investigate a relation between rational blowdown surgery and minimal symplectic fillings of a given Seifert 3-manifold with a canonical contact structure. Consequently, we determine a necessary and sufficient condition for a minimal symplectic filling of a Seifert 3-manifold satisfying certain conditions to be obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding weighted homogeneous surface singularity. Furthermore, as an application of the main results, we prove that every minimal symplectic filling of a large family of Seifert 3-manifolds with a canonical contact structure is in fact realized as a Milnor fiber of the corresponding weighted homogeneous surface singularity in the Appendix.

math.GT↗
arXiv2022

On symplectic fillings of small Seifert $3$-manifolds

In this paper, we investigate the minimal symplectic fillings of small Seifert 3-manifolds with a canonical contact structure. As a result, we classify all minimal symplectic fillings of small Seifert 3-manifolds satisfying certain conditions. Furthermore, we also demonstrate that every such a minimal symplectic filling is obtained by a sequence of rational blowdowns from the minimal resolution of the corresponding weighted homogeneous complex surface singularity.

math.GT↗
arXiv2019

A Lefschetz fibration on minimal symplectic fillings of a quotient surface singularity

In this article, we construct a genus-$0$ or genus-$1$ positive allowable Lefschetz fibration on any minimal symplectic filling of the link of non-cyclic quotient surface singularities. As a byproduct, we also show that any minimal symplectic filling of the link of quotient surface singularities can be obtained from a sequence of rational blowdowns from its minimal resolution.

math.GT↗
arXiv2018

Milnor fibers and symplectic fillings of quotient surface singularities

We determine a one-to-one correspondence between Milnor fibers and minimal symplectic fillings of a quotient surface singularity (up to diffeomorphism type) by giving an explicit algorithm to compare them mainly via techniques from the minimal model program for 3-folds and Pinkham's negative weight smoothing. As by-products, we show that: -- Milnor fibers associated to irreducible components of the reduced versal deformation space of a quotient surface singularity are not diffeomorphic to each other with a few obvious exceptions. For this, we classify minimal symplectic fillings of a quotient surface singularity up to diffeomorphism. -- Any symplectic filling of a quotient surface singularity is obtained by a sequence of rational blow-downs from a special resolution (so-called the maximal resolution) of the singularity, which is an analogue of the one-to-one correspondence between the irreducible components of the reduced versal deformation space and the so-called $P$-resolutions of a quotient surface singularity.

math.AG↗
arXiv2017

On dissolving knot surgery $4$-manifolds under a $\mathbb{CP}^2$-connected sum

In this article we prove that, if $X$ is a smooth $4$-manifold containing an embedded double node neighborhood, all knot surgery $4$-manifolds $X_K$ are mutually diffeomorphic to each other after a connected sum with $\mathbb{CP}^2$. Hence, by applying to the simply connected elliptic surface $E(n)$, we also show that every knot surgery $4$-manifold $E(n)_K$ is almost completely decomposable.

math.GT↗
arXiv2015

Smoothly embedded rational homology balls

In this paper we prove the existence of rational homology balls smoothly embedded in regular neighborhoods of certain linear chains of smooth $2$-spheres by using techniques from minimal model program for 3-dimensional complex algebraic variety.

math.GT↗
arXiv2015

Lefschetz fibrations on knot surgery $4$-manifolds via Stallings twist

In this article we construct a family of knot surgery $4$-manifolds admitting arbitrarily many nonisomorphic Lefschetz fibration structures with the same genus fiber. We obtain such families by performing knot surgery on an elliptic surface $E(2)$ using connected sums of fibered knots obtained by Stallings twist from a slice knot $3_1 \sharp 3^*_1$. By comparing their monodromy groups induced from the corresponding monodromy factorizations, we show that they admit mutually nonisomorphic Lefschetz fibration structures.

math.GT↗
arXiv2013

Families of nondiffeomorphic 4-manifolds with the same Seiberg-Witten invariants

In this article, we show that, at least for non-simply connected case, there exist an infinite family of nondiffeomorphic symplectic 4-manifolds with the same Seiberg-Witten invariants. The main techniques are knot surgery and a covering method developed in Fintushel and Stern's paper (Geometry and Topology, 1999).

math.GT↗
arXiv2011

A complex surface of general type with $p_g=0$, $K^2=2$ and $H_1=\mathbb{Z}/4\mathbb{Z}$

We construct a new minimal complex surface of general type with $p_g=0$, $K^2=2$ and $H_1=\mathbb{Z}/4\mathbb{Z}$ (in fact $π_1^{\text{alg}}=\mathbb{Z}/4\mathbb{Z}$), which settles the existence question for numerical Campedelli surfaces with all possible algebraic fundamental groups. The main techniques involved in the construction are a rational blow-down surgery and a $\mathbb{Q}$-Gorenstein smoothing theory.

math.AG↗
arXiv2011

Surfaces of general type with $p_g=1$ and $q=0$

In this paper we construct a new family of simply connected minimal complex surfaces of general type with $p_g=1$, $q=0$, and $K^2=3, 4, 5, 6, 8$ using a $\mathbb{Q}$-Gorenstein smoothing theory. We also reconstruct minimal complex surfaces of general type with $p_g=1$, $q=0$, and $K^2=1, 2$ using the same method.

math.AG↗
arXiv2009

Lefschetz fibration structures on knot surgery 4-manifolds

In this article we study Lefschetz fibration structures on knot surgery 4-manifolds obtained from an elliptic surface E(2) using Kanenobu knots $K$. As a result, we get an infinite family of simply connected mutually diffeomorphic 4-manifolds coming from a pair of inequivalent Kanenobu knots. We also obtain an infinite family of simply connected symplectic 4-manifolds, each of which admits more than one inequivalent Lefschetz fibration structures of the same generic fiber.

math.GT↗