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Min Kyu Kim

Publications and source records attributed to Min Kyu Kim.

15 recordsLinked to original sources
arXiv2026

Guidelines for Designing AI Technologies to Support Adult Learning

AI-powered educational technologies have demonstrated measurable benefits for learners, but their design and evaluation have largely centered on K-12 contexts. As a result, many AI-supported learning systems remain poorly aligned with the needs, constraints, and goals of adult learners. To better understand how AI systems function in adult education, this paper examines the deployment of several AI learning technologies developed within a multidisciplinary, national research institute in the United States focused on adult learning and online education. Drawing on longitudinal deployment data, we conducted a reflexive thematic analysis to identify recurring challenges and design considerations across systems. These insights were synthesized into a set of 19 design guidelines intended to inform future AI-supported adult learning technologies. We demonstrate the utility of these guidelines through a heuristic evaluation of the deployed systems. Lastly, we present a guideline exploration tool that aids in the ideation of technologies by connecting the guidelines to stakeholder statements surfaced in the analysis process.

cs.CY↗
arXiv2018

Hard Lefschetz Property for Hamiltonian torus actions on 6-dimensional GKM manifolds

In this paper, we study the hard Lefschetz property of a symplectic manifold which admits a Hamiltonian torus action. More precisely, let $(M,ω)$ be a 6-dimensional compact symplectic manifold with a Hamiltonian $T^2$-action. We will show that if the moment map image of $M$ is a GKM-graph and if the graph is index-increasing, then $(M,ω)$ satisfies the hard Lefschetz property.

math.SG↗
arXiv2016

Embedded surfaces for symplectic circle actions

The purpose of this article is to characterize symplectic and Hamiltonian circle actions on symplectic manifolds in terms of symplectic embeddings of Riemann surfaces. More precisely, we will show that (1) if $(M,ω)$ admits a Hamiltonian $S^1$-action, then there exists an $S^1$-invariant symplectic $2$-sphere $S$ in $(M,ω)$ such that $\langle c_1(M), [S] \rangle > 0$, and (2) if the action is non-Hamiltonian, then there exists an $S^1$-invariant symplectic $2$-torus $T$ in $(M,ω)$ such that $\langle c_1(M), [T] \rangle = 0$. As applications, we will give a very simple proof of the following well-known theorem which was proved by Atiyah-Bott \cite{AB}, Lupton-Oprea \cite{LO}, and Ono \cite{O2} : suppose that $(M,ω)$ is a smooth closed symplectic manifold satisfying $c_1(TM)=λ\cdot [ω]$ for some $λ\in \R$ and let $G$ be a compact connected Lie group acting effectively on $M$ preserving $ω$. Then (1) if $λ< 0$, then $G$ must be trivial, (2) if $λ=0$, then the $G$-action is non-Hamiltonian, and (3) if $λ> 0$, then the $G$-action is Hamiltonian.

math.SG↗
arXiv2015

Equivariant pointwise clutching maps

In the paper, we introduce the terminology equivariant pointwise clutching map. By using this, we give details on how to glue an equivariant vector bundle over a finite set so as to obtain a new Lie group representation such that the quotient map from the bundle to the representation is equivariant. Then, we investigate the topology of the set of all equivariant pointwise clutching maps with respect to an equivariant vector bundle over a finite set. Results of the paper play a key role in classifying equivariant vector bundles over two-surfaces in other papers.

math.GR↗
arXiv2013

Hamiltonian circle action with self-indexing moment map

Let $(M,ω)$ be a $2n$-dimensional smooth compact symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points and let $μ: M \rightarrow \R$ be a corresponding moment map. Let $Λ_{2k}$ be the set of all fixed points of index $2k$. In this paper, we will show that if $μ$ is constant on $Λ_{2k}$ for each $k$, then $(M,ω)$ satisfies the hard Lefschetz property. In particular, if $(M,ω)$ admits a self-indexing moment map, i.e. $μ(p) = 2k$ for every $p \in Λ_{2k}$ and $k=0,1,\cdots,n,$ then $(M,ω)$ satisfies the hard Lefschetz property.

math.SG↗
arXiv2013

Crystal $\Bla$ in $B(\infty)$ for $G_2$ type Lie Algebra

A previous work gave a combinatorial description of the crystal $B(\infty)$, in terms of certain simple Young tableaux referred to as the marginally large tableaux, for finite dimensional simple Lie algebras. Using this result, we present an explicit description of the crystal $B(λ)$, in terms of the marginally large tableaux, for the $G_2$ Lie algebra type. We also provide a new description of $B(\la)$, in terms of Nakajima monomials, that is in natural correspondence with our tableau description.

math.RT↗
arXiv2013

An example of circle actions on symplectic Calabi-Yau manifolds with non-empty fixed points

Let $(X,σ,J)$ be a compact Kähler Calabi-Yau manifold equipped with a symplectic circle action. By Frankel's theorem \cite{F}, the action on $X$ is non-Hamiltonian and $X$ does not have any fixed point. In this paper, we will show that a symplectic circle action on a compact non-Kähler symplectic Calabi-Yau manifold may have a fixed point. More precisely, we will show that the symplectic $S^1$-manifold constructed by D. McDuff \cite{McD} has the vanishing first Chern class. This manifold has the Betti numbers $b_1 = 3$, $b_2 = 8$, and $b_3 = 12$. In particular, it does not admit any Kähler structure.

math.SG↗
arXiv2011

Log-concavity of complexity one Hamiltonian torus actions

Let $(M,ω)$ be a closed $2n$-dimensional symplectic manifold equipped with a Hamiltonian $T^{n-1}$-action. Then Atiyah-Guillemin-Sternberg convexity theorem implies that the image of the moment map is an $(n-1)$-dimensional convex polytope. In this paper, we show that the density function of the Duistermaat-Heckman measure is log-concave on the image of the moment map.

math.SG↗
arXiv2011

Classification of equivariant vector bundles over real projective plane

We classify equivariant topological complex vector bundles over real projective plane under a compact Lie group (not necessarily effective) action. It is shown that nonequivariant Chern classes and isotropy representations at (at most) three points are sufficient to classify equivariant vector bundles over real projective plane except one case. To do it, we relate the problem to classification on two-sphere through the covering map because equivariant vector bundles over two-sphere have been already classified.

math.GR↗
arXiv2011

Classification of equivariant vector bundles over two-sphere

We exhaustively classify topological equivariant complex vector bundles over two-sphere under a compact Lie group (not necessarily effective) action. It is shown that inequivariant Chern classes and isotropy representations at (at most) three points are sufficient to classify equivariant vector bundles except a few cases. To do it, we calculate equivariant homotopy of the set of equivariant clutching maps. Holomorphic version of this will be treated in other paper. Classification on two-torus, real projective plane, Klein bottle will appear soon.

math.GR↗
arXiv2010

Classification of equivariant vector bundles over two-torus

We exhaustively classify topological equivariant complex vector bundles over two-torus under a compact Lie group (not necessarily effective) action. It is shown that inequivariant Chern classes and isotropy representations at (at most) six points are sufficient to classify equivariant vector bundles except a few cases. To do it, we calculate homotopy of the set of equivariant clutching maps. And, classification on real projective plane, Klein bottle will appear soon

math.GR↗
arXiv2009

On Homoclinic points, Recurrences and Chain recurrences of volume-preserving diffeomorphisms without genericity

Let $M$ be a manifold with a volume form $ω$ and $f : M \to M$ be a diffeomorphism of class $\mathcal{C}^1$ that preserves $ω$. In this paper, we do \textit{not} assume $f$ is $\mathcal{C}^1$-generic. We have two main themes in the paper: (1) the chain recurrence; (2) relations among recurrence points, homoclinic points, shadowability and hyperbolicity. For (1) (without assuming $M$ is compact), we have the theorem: if $f$ is Lagrange stable, then $M$ is a chain recurrent set. If $M$ is compact, then the Lagrange-stability is automatic. For (2) (assuming the compactness of $M$), we prove some various implications among notions, such as: (i) the $\mathcal{C}^1$-stable shadowability equals to the hyperbolicity of $M$; (ii) if a point $p\in M$ has a recurrence point in the unstable manifold $W^u (p, f)$ and there is no homoclinic point of $p,$ then $f$ is nonshadowable; (iii) if $f$ has the shadowing property and $p$ has a recurrence point in $W^u (p, f),$ then the recurrent point is in the limit set of homoclinic points of $p$.

math.DS↗
arXiv2004

Frankel's theorem in the symplectic category

We prove that if an (n-1)-dimensional torus acts symplectically on a 2n-dimensional manifold, then the action has a fixed point if and only if the action is Hamiltonian. One may regard it as a symplectic version of Frankel theorem. The case of n=2 is the well known theorem of McDuff. From the well known example of McDuff (a six dimensional symplectic non-Hamiltonian circle action with fixed tori) and its products with copies of a two dimensional sphere with the usual rotation, the condition on the dimension of the acting torus is optimal to obtain the result.

math.SG↗
arXiv2003

On extensions of representations for compact Lie groups

Let $H$ be a closed normal subgroup of a compact Lie group $G$ such that $G/H$ is connected. This paper provides a necessary and sufficient condition for every complex representation of $H$ to be extendible to $G$, and also for every complex $G$-vector bundle over the homogeneous space $G/H$ to be trivial. In particular, we show that the condition holds when the fundamental group of $G/H$ is torsion free.

math.RT↗