Abstract:
The purpose of this study is to consider crime prevention measures in ethnic areas focusing on Crime Prevention through Environmental Design (CPTED) by an analysis of crime data and field survey. In this study, it was found that the main type of foreign crime that occurred in the research area was violence, and crimes committed by Koreans, which were mainly violence and crimes such as burglary, theft, robbery, and sexual offences, occurred steadily. Because it was found that crimes were related to the urban planning elements comprised of land use such as traditional market, inn, pub, and complicated space structure and the architectural design for natural surveillance and security facilities such as CCTV, lighting, alarm, and target hardening device, a new strategy for crime prevention design should include street environmental management, improvement of commercial facilities, and reinforcement security device of each buildings has to be spread through support of policy. In conclusion it was thought that CPTED would be a valuable measure to prevention crime and support community activities in ethnic area as expecting an improvement of physical environment and resident participatory for safer community.

Abstract:
In this article, we explain the main philosophy of 2-representation theory and quantum affine Schur-Weyl duality. The Khovanov-Lauda-Rouquier algebras play a central role in both themes.

Abstract:
In this paper, we give a realization of crystal bases for quantum affine algebras using some new combinatorial objects which we call the Young walls. The Young walls consist of colored blocks with various shapes that are built on the given ground-state wall and can be viewed as generalizations of Young diagrams. The rules for building Young walls and the action of Kashiwara operators are given explicitly in terms of combinatorics of Young walls. The crystal graphs for basic representations are characterized as the set of all reduced proper Young walls. The characters of basic representations can be computed easily by counting the number of colored blocks that have been added to the ground-state wall.

Abstract:
We give a realization of crystal graphs for basic representations of the quantum affine algebra $U_q(C_2^{(1)})$ in terms of new combinatorial objects called the Young walls.

Abstract:
Let S be the direct sum of algebra of symmetric groups C S_n for a non-negative integer n. We show that the Grothendieck group K_0(S) of the category of finite dimensional modules of S is isomorphic to the differential algebra of polynomials Z[D^n x]. Moreover, for a non-negative integer m, we define m-th products on K_0(S) which make the algebra K_0(S) isomorphic to an integral form of the Virasoro-Magri Poisson vertex algebra. Also, we investigate relations between K_0(S) and K_0(N) where K_0(N) is the direct sum of Grothendieck groups K_0(N_n) of finitely generated projective N_n-modules. Here N_n is the nil-Coxeter algebra generated by n-1 elements.

Abstract:
We construct a canonical basis for quantum generalized Kac-Moody algebra via semisimple perverse sheaves on varieties of representations of quivers. We compare this basis with the one recently defined purely algebraically by Jeong, Kang and Kashiwara.

Abstract:
Let $B(\Lambda_0)$ be the level 1 highest weight crystal of the quantum affine algebra $U_q(A_n^{(1)})$. We construct an explicit crystal isomorphism between the geometric realization $\mathbb{B}(\Lambda_0)$ of $B(\Lambda_0)$ via quiver varieties and the path realization ${\mathcal P}^{\rm ad}(\Lambda_0)$ of $B(\Lambda_0)$ arising from the adjoint crystal $\adjoint$.

Abstract:
We introduce a new family of superalgebras $\overrightarrow{B}_{r,s}$ for $r, s \ge 0$ such that $r+s>0$, which we call the walled Brauer superalgebras, and prove the mixed Scur-Weyl-Sergeev duality for queer Lie superalgebras. More precisely, let $\mathfrak{q}(n)$ be the queer Lie superalgebra, ${\mathbf V} =\mathbb{C}^{n|n}$ the natural representation of $\mathfrak{q}(n)$ and ${\mathbf W}$ the dual of ${\mathbf V}$. We prove that, if $n \ge r+s$, the superalgebra $\overrightarrow{B}_{r,s}$ is isomorphic to the supercentralizer algebra $_{\mathfrak{q}(n)}({\mathbf V}^{\otimes r} \otimes {\mathbf W}^{\otimes s})^{\op}$ of the $\mathfrak{q}(n)$-action on the mixed tensor space ${\mathbf V}^{\otimes r} \otimes {\mathbf W}^{\otimes s}$. As an ingredient for the proof of our main result, we construct a new diagrammatic realization $\overrightarrow{D}_{k}$ of the Sergeev superalgebra $Ser_{k}$. Finally, we give a presentation of $\overrightarrow{B}_{r,s}$ in terms of generators and relations.

Abstract:
Using combinatorics of Young walls, we give a new realization of arbitrary level irreducible highest weight crystals $\mathcal{B}(\lambda)$ for quantum affine algebras of type $A_n^{(1)}$, $B_n^{(1)}$, $C_n^{(1)}$, $A_{2n-1}^{(2)}$, $A_{2n}^{(2)}$, and $D_{n+1}^{(2)}$. The irreducible highest weight crystals are realized as the affine crystals consisting of reduced proper Young walls. The notion of slices and splitting of blocks plays a crucial role in the constructions of crystal graphs.

Abstract:
Using combinatorics of Young tableaux, we give an explicit construction of irreducible graded modules over Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^{\lambda}$ of type $A_{n}$. Our construction is compatible with crystal structure. Let ${\mathbf B}(\infty)$ and ${\mathbf B}(\lambda)$ be the $U_q(\slm_{n+1})$-crystal consisting of marginally large tableaux and semistandard tableaux of shape $\lambda$, respectively. On the other hand, let ${\mathfrak B}(\infty)$ and ${\mathfrak B}(\lambda)$ be the $U_q(\slm_{n+1})$-crystals consisting of isomorphism classes of irreducible graded $R$-modules and $R^{\lambda}$-modules, respectively. We show that there exist explicit crystal isomorphisms $\Phi_{\infty}: {\mathbf B}(\infty) \overset{\sim} \longrightarrow {\mathfrak B}(\infty)$ and $\Phi_{\lambda}: {\mathbf B}(\lambda) \overset{\sim} \longrightarrow {\mathfrak B}(\lambda)$.