Journal of the
Korean Mathematical Society
JKMS
ISSN(Print) 0304-9914
ISSN(Online) 2234-3008
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2023-11-01
Low rank orthogonal bundles and quadric fibrations
Insong Choe, George H. HitchingAbstract : Let $C$ be a curve and $V \to C$ an orthogonal vector bundle of rank $r$. For $r \le 6$, the structure of $V$ can be described using tensor, symmetric and exterior products of bundles of lower rank, essentially due to the existence of exceptional isomorphisms between $\mathrm{Spin} (r , \mathbb C)$ and other groups for these $r$. We analyze these structures in detail, and in particular use them to describe moduli spaces of orthogonal bundles. Furthermore, the locus of isotropic vectors in $V$ defines a quadric subfibration $Q_V \subset \mathbb P V$. Using familiar results on quadrics of low dimension, we exhibit isomorphisms between isotropic Quot schemes of $V$ and certain ordinary Quot schemes of line subbundles. In particular, for $r \le 6$ this gives a method for enumerating the isotropic subbundles of maximal degree of a general $V$, when there are finitely many.
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2022-05-01
Erratum to ``Static and related critical spaces with harmonic curvature and three Ricci eigenvalues'' [J. Korean Math. Soc. 57 (2020), No. 6, pp. 1435--1449]
Jongsu KimAbstract : In this erratum, we offer a correction to [J. Korean Math. Soc. 57 (2020), No. 6, pp. 1435--1449]. Theorem 1 in the original paper has three assertions (i)-(iii), but we add (iv) after having clarified the argument.
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2024-01-01
Periodic surface homeomorphisms and contact structures
Dheeraj Kulkarni, Kashyap Rajeevsarathy, Kuldeep SahaAbstract : In this article, we associate a contact structure to the conjugacy class of a periodic surface homeomorphism, encoded by a combinatorial tuple of integers called a marked data set. In particular, we prove that infinite families of these data sets give rise to Stein fillable contact structures with associated monodromies that do not factor into products to positive Dehn twists. In addition to the above, we give explicit constructions of symplectic fillings for rational open books analogous to Mori's construction for honest open books. We also prove a sufficient condition for the Stein fillability of rational open books analogous to the positivity of monodromy for honest open books due to Giroux and Loi-Piergallini.
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2022-09-01
Spectral decomposition for homeomorphisms on non-metrizable totally disconnected spaces
Jumi OhAbstract : We introduce the notions of symbolic expansivity and symbolic shadowing for homeomorphisms on non-metrizable compact spaces which are generalizations of expansivity and shadowing, respectively, for metric spaces. The main result is to generalize the Smale's spectral decomposition theorem to symbolically expansive homeomorphisms with symbolic shadowing on non-metrizable compact Hausdorff totally disconnected spaces.
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2023-03-01
Invariant graph and random bony attractors
Fateme Helen Ghane, Maryam Rabiee, Marzie ZajAbstract : In this paper, we deal with random attractors for dynamical systems forced by a deterministic noise. These kind of systems are modeled as skew products where the dynamics of the forcing process are described by the base transformation. Here, we consider skew products over the Bernoulli shift with the unit interval fiber. We study the geometric structure of maximal attractors, the orbit stability and stability of mixing of these skew products under random perturbations of the fiber maps. We show that there exists an open set $\mathcal{U}$ in the space of such skew products so that any skew product belonging to this set admits an attractor which is either a continuous invariant graph or a bony graph attractor. These skew products have negative fiber Lyapunov exponents and their fiber maps are non-uniformly contracting, hence the non-uniform contraction rates are measured by Lyapnnov exponents. Furthermore, each skew product of $\mathcal{U}$ admits an invariant ergodic measure whose support is contained in that attractor. Additionally, we show that the invariant measure for the perturbed system is continuous in the Hutchinson metric.
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2023-11-01
The automorphism groups of Artin groups of edge-separated CLTTF graphs
Byung Hee An, Youngjin ChoAbstract : This work is a continuation of Crisp's work on automorphism groups of CLTTF Artin groups, where the defining graph of a CLTTF Artin group is connected, large-type, and triangle-free. More precisely, we provide an explicit presentation of the automorphism group of an edge-separated CLTTF Artin group whose defining graph has no separating vertices.
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2023-11-01
Finite quotients of singular Artin monoids and categorification of the desingularization map
Helena Jonsson, Volodymyr Mazorchuk, Elin Persson Westin, Shraddha Srivastava, Mateusz Stroinski, Xiaoyu ZhuAbstract : We study various aspects of the structure and representation theory of singular Artin monoids. This includes a number of generalizations of the desingularization map and explicit presentations for certain finite quotient monoids of diagrammatic nature. The main result is a categorification of the classical desingularization map for singular Artin monoids associated to finite Weyl groups using BGG category $\mathcal{O}$.
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2023-09-01
A conjecture of Gross and Zagier: case $E(\mathbb{Q})_{\rm{tor}} \cong \mathbb{Z}/2\mathbb{Z}$ or $\mathbb{Z}/4\mathbb{Z}$
Dongho Byeon, Taekyung Kim, Donggeon YheeAbstract : Let $E$ be an elliptic curve defined over $\mathbb{Q}$ of conductor $N$, $c$ the Manin constant of $E$, and $m$ the product of Tamagawa numbers of $E$ at prime divisors of $N$. Let $K$ be an imaginary quadratic field where all prime divisors of $N$ split in $K$, $P_K$ the Heegner point in $E(K)$, and ${\rm III}(E/K)$ the Shafarevich-Tate group of $E$ over $K$. Let $2u_K$ be the number of roots of unity contained in $K$. Gross and Zagier conjectured that if $P_K$ has infinite order in $E(K)$, then the integer $ c \cdot m \cdot u_K \cdot |{\rm III}(E/K)|^{\frac{1}{2}}$ is divisible by $|E(\mathbb{Q})_{\rm{tor}} |$. In this paper, we prove that this conjecture is true if $E(\mathbb{Q})_{\rm{tor}} \cong \mathbb{Z}/2\mathbb{Z}$ or $\mathbb{Z}/4\mathbb{Z}$ except for two explicit families of curves. Further, we show these exceptions can be removed under Stein--Watkins conjecture.
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2023-09-01
Summability in Musielak--Orlicz Hardy spaces
Jun Liu, Haonan XiaAbstract : Let $\varphi: \mathbb{R}^n\times[0,\infty)\to[0,\infty)$ be a growth function and $H^{\varphi}(\mathbb{R}^n)$ the Musielak--Orlicz Hardy space defined via the non-tangential grand maximal function. A general summability method, the so-called $\theta$-summability is considered for multi-dimensional Fourier transforms in $H^{\varphi}(\mathbb{R}^n)$. Precisely, with some assumptions on $\theta$, the authors first prove that the maximal operator of the $\theta$-means is bounded from $H^{\varphi}(\mathbb{R}^n)$ to $L^{\varphi}(\mathbb{R}^n)$. As consequences, some norm and almost everywhere convergence results of the $\theta$-means, which generalizes the well-known Lebesgue's theorem, are then obtained. Finally, the corresponding conclusions of some specific summability methods, such as Bochner--Riesz, Weierstrass and Picard--Bessel summations, are also presented.
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2024-01-01
Geometry of the moduli space of Higgs pairs on an irreducible nodal curve of arithmetic genus one
Sang-Bum YooAbstract : We describe the moduli space of Higgs pairs on an irreducible nodal curve of arithmetic genus one and its geometric structures in terms of the Hitchin map and a flat degeneration of the moduli space of Higgs bundles on an elliptic curve.
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Dirichlet eigenvalue problems under Musielak-Orlicz growth
Allami Benyaiche, Ismail Khlifi
J. Korean Math. Soc. 2022; 59(6): 1139-1151
https://doi.org/10.4134/JKMS.j210669 -
Computations and conservativeness of traces of one-dimensional diffusions
Ali BENAMOR, Rafed MOUSSA
J. Korean Math. Soc. 2023; 60(4): 779-797
https://doi.org/10.4134/JKMS.j220258 -
Topologically stable points and uniform limits
Namjip Koo, Hyunhee Lee
J. Korean Math. Soc. 2023; 60(5): 1043-1055
https://doi.org/10.4134/JKMS.j220595 -
On $S$-multiplication rings
Mohamed Chhiti, Soibri Moindze
J. Korean Math. Soc. 2023; 60(2): 327-339
https://doi.org/10.4134/JKMS.j220030
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On nonnil-exact sequences and nonnil-commutative diagrams
Wei Zhao, De chuan Zhou
J. Korean Math. Soc. 2023; 60(6): 1215-1231
https://doi.org/10.4134/JKMS.j220503 -
Cup product on relative bounded cohomology
HeeSook Park
J. Korean Math. Soc. 2023; 60(4): 823-833
https://doi.org/10.4134/JKMS.j220345 -
Computations and conservativeness of traces of one-dimensional diffusions
Ali BENAMOR, Rafed MOUSSA
J. Korean Math. Soc. 2023; 60(4): 779-797
https://doi.org/10.4134/JKMS.j220258 -
Weak Herz-type Hardy spaces with variable exponents and applications
Souad Ben Seghier
J. Korean Math. Soc. 2023; 60(1): 33-69
https://doi.org/10.4134/JKMS.j210764
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