• Communications of the Korean Mathematical Society
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• v.29 no.2
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• pp.285-293
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• 2014

Let M be a closed, symplectic connected Riemannian manifold and f a symplectomorphism on M. We prove that if f is $C^1$-stably weak shadowable on M, then the whole manifold M admits a partially hyperbolic splitting.

• Bulletin of the Korean Mathematical Society
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• v.39 no.2
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• pp.277-282
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• 2002

We Show that When ($M,\;\omega$) is a closed, simply connected, symplectic manifold for all $\gamma\;\in\;\pi_1(Ham(M),\;id)$ the following inequality holds: $\parallel\gamma\parallel\;{\geq}\;sup_{\={x}}\;|A(\={x})|,\;where\;\parallel\gamma\parallel$ is the coarse Hofer's norm, $\={x}$ run over all extensions to $D^2$ of an orbit $x(t)\;=\;{\varphi}_t(z)$ of a fixed point $z\;\in\;M,\;A(\={x})$ the symplectic action of $\={x}$, and the Hamiltonian diffeomorphisms {${\varphi}_t$} of M represent $\gamma$.

• Journal of the Korean Mathematical Society
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• v.58 no.2
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• pp.451-471
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• 2021

First, we define phase tropical hypersurfaces in terms of a degeneration data of smooth complex algebraic hypersurfaces in (ℂ∗)n. Next, we prove that complex hyperplanes are homeomorphic to their degeneration called phase tropical hyperplanes. More generally, using Mikhalkin's decomposition into pairs-of-pants of smooth algebraic hypersurfaces, we show that a phase tropical hypersurface with smooth tropicalization is naturally a topological manifold. Moreover, we prove that a phase tropical hypersurface is naturally homeomorphic to a symplectic manifold.

• East Asian mathematical journal
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• v.35 no.3
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• pp.357-363
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• 2019

Let the circle act on a compact almost complex manifold M with a non-empty discrete fixed point set. To each fixed point, there are associated non-zero integers called weights. A positive weight w is called primitive if it cannot be written as the sum of positive weights, other than w itself. In this paper, we show that if every weight is primitive, then the Todd genus Todd(M) of M is positive and there are $Todd(M){\cdot}2^n$ fixed points, where dim M = 2n. This generalizes the result for symplectic semi-free actions by Tolman and Weitsman [8], the result for semi-free actions on almost complex manifolds by the author [6], and the result for certain symplectic actions by Godinho [1].

• Bulletin of the Korean Mathematical Society
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• v.40 no.4
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• pp.593-601
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• 2003

Let X be a 4-manifold obtained by gluing two symplectic 4-manifolds Xi, i = 1, 2, along embedded surfaces. Using the gradient flow of a functional on 3-dimensional Seiberg-Witten theory along the cylindrical end, we study the Seiberg-Witten equations on X and have a product formula of Seiberg-Witten invariants on X from the ones on Xi, i = 1, 2.

• Kyungpook Mathematical Journal
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• v.53 no.4
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• pp.603-614
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• 2013

In the article we show that nondieomorphic symplectic 4-manifolds which admit marked Lefschetz fibrations can share the same monodromy group. Explicitly we prove that, for each integer g > 0, every knot surgery 4-manifold in a family {$E(2)_K{\mid}K$ is a bered 2-bridge knot of genus g in $S^3$} admits a marked Lefschetz fibration structure which has the same monodromy group.

• Bulletin of the Korean Mathematical Society
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• v.35 no.3
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• pp.563-570
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• 1998

We makes an explicit description of compact 4-dimensional nilmanifolds as principal torus bundles and show that they are sysmplectic. We discuss some consequences of this and give in particular a Seibebrg-Witten-invariant proof of a Grovmov-Lawson theorem that if a compact 4-dimensional nilmanifold admits a metric of zero scalar curvature, then it is diffeomorphic to 4-tours, $T^4$.

• Journal of the Korean Mathematical Society
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• v.53 no.3
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• pp.489-501
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• 2016

In this paper, we study the rigidity under $K{\ddot{a}}hler$ deformation of the complex structure of odd Lagrangian Grassmannians, i.e., the Lagrangian case $Gr_{\omega}$(n, 2n+1) of odd symplectic Grassmannians. To obtain the global deformation rigidity of the odd Lagrangian Grassmannian, we use results about the automorphism group of this manifold, the Lie algebra of infinitesimal automorphisms of the affine cone of the variety of minimal rational tangents and its prolongations.

• Communications of the Korean Mathematical Society
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• v.29 no.4
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• pp.569-579
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• 2014

We consider a condition under which the projectivization $P(E^k)$ of a complex k-bundle $E^k{\rightarrow}M$ over an even-dimensional manifold M can have the hard Lefschetz property, affected by [10]. It depends strongly on the rank k of the bundle $E^k$. Our approach is purely algebraic by using rational Sullivan minimal models [5]. We will give some examples.

• The Pure and Applied Mathematics
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• v.3 no.1
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• pp.83-94
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• 1996

Classical mechanics begins with some variants of Newton's laws. Lagrangian mechanics describes motion of a mechanical system in the configuration space which is a differential manifold defined by holonomic constraints. For a conservative system, the equations of motion are derived from the Lagrangian function on Hamilton's variational principle as a system of the second order differential equations. Thus, for conservative systems, Newtonian mechanics is a particular case of Lagrangian mechanics.(omitted)