In this talk, we will survey the book "Arithmeticity in the theory of automorphic forms - G.Shimura (2000)".

In view of scheme language, we start from the beginning of the theory of elliptic curves and geometric modular forms, and further cover the topics about Jacobians and Galois representation as well as modularity problems.

In this talk, we will survey the article "Modular forms and projective invariants - J.Igusa(1967)".

In this talk, we will survey the article "Class fields over real quadratic fields and Hecke operators - G.Shimura(1972)".

In this talk, we will review arithemetic properties of Siegel-Ramachandra invariants. And, we further construct ray class fields over imaginary quadratic fields by means of the Siegel-Ramachandra invariant.

In view of scheme language, we start from the beginning of the theory of elliptic curves and geometric modular forms, and further cover the topics about Jacobians and Galois representation as well as modularity problems.

In view of scheme language, we start from the beginning of the theory of elliptic curves and geometric modular forms, and further cover the topics about Jacobians and Galois representation as well as modularity problems.

In this talk, we will survey the book "Arithmeticity in the theory of automorphic forms - G.Shimura (2000)".

In this talk, we will survey the article "Modular forms and projective invariants - J.Igusa(1967)".

In this talk, we will survey the article "Class fields over real quadratic fields and Hecke operators - G.Shimura(1972)".

A braid is a structure formed by intertwining a number of strands, such as textiles or human hairs. As a mathematical object, a set of braids forms a group, called a braid group which was firstly introduced by E. Artin in 1920’s, and generalized to any topological space via configuration spaces. Nevertheless, the braid theory has been researched only on manifolds until the late 1990’s when Ghrist published some results about the braid group on graphs. After Ghrist, many people studied braid groups on graphs. However for general CW (or simplicial) complexes of dimension greater than 1, the braid theory is still an unexplored field.

In this talk, we focus on the braid group on a finite regular CW complex of dimension 2 and we explain how a decomposition of given space is related to a decomposition of its braid group and how to build up the braid group from the simple ones. As an application, we figure out the hierarchy structure that the braid group admits and the relations between group theoretical properties of the braid group and geometrical properties of a given CW complex, such as, embeddability into a manifold or planarity.

A braid is a structure formed by intertwining a number of strands, such as textiles or human hairs. As a mathematical object, a set of braids forms a group, called a braid group which was firstly introduced by E. Artin in 1920’s, and generalized to any topological space via configuration spaces. Nevertheless, the braid theory has been researched only on manifolds until the late 1990’s when Ghrist published some results about the braid group on graphs. After Ghrist, many people studied braid groups on graphs. However for general CW (or simplicial) complexes of dimension greater than 1, the braid theory is still an unexplored field.

In this talk, we focus on the braid group on a finite regular CW complex of dimension 2 and we explain how a decomposition of given space is related to a decomposition of its braid group and how to build up the braid group from the simple ones. As an application, we figure out the hierarchy structure that the braid group admits and the relations between group theoretical properties of the braid group and geometrical properties of a given CW complex, such as, embeddability into a manifold or planarity.

The $¥Gamma$-polynomial is an invariant of an oriented link in the 3-sphere, which is contained in both the HOMFLYPT and Kauffman polynomials as their common zeroth coefficient polynomial. As applications of the $¥Gamma$-polynomial, I will talk about the following

three topics:

(1) On the arc index of cable knots (joint with Hwa Jeong Lee, KAIST)

(2) On the braid index of Kanenobu knots

(3) On the arc index of Kanenobu knots (joint with Hwa Jeong Lee, KAIST)

In view of scheme language, we start from the beginning of the theory of elliptic curves and geometric modular forms, and further cover the topics about Jacobians and Galois representation as well as modularity problems.

In this talk, we will survey the book "Arithmeticity in the theory of automorphic forms - G.Shimura (2000)".

In this talk, we will survey the article "Modular forms and projective invariants - J.Igusa(1967)".

In this talk, we will survey the article "Class fields over real quadratic fields and Hecke operators - G.Shimura(1972)".

In view of scheme language, we start from the beginning of the theory of elliptic curves and geometric modular forms, and further cover the topics about Jacobians and Galois representation as well as modularity problems.

In this talk, we will investigate the algebraic construction of the Jacobian of a hyperelliptic curves.

In this talk, we will survey the article "Modular forms and projective invariants - J.Igusa(1967)".

In this talk, we will survey the article "Class fields over real quadratic fields and Hecke operators - G.Shimura(1972)".

Modeling of debris flow is of great importance in various research areas, and there have been several debris flow models have been proposed in the last couple of decades. However, most of them do not consider the erosional effect. Here, we discuss a mathematical approach to model a debris flow with erosional effect. The (energetic) variational approach is applied to derive the resulting system of partial differential equation (PDEs) with the erosional effect. Since the erosional effect plays a key role on the interface between flow and the base, it is crucial to find kinematic boundary condition on the interface to elucidate erodible debris or granular flows. We first model an erosional potential energy with power law inspired by the frictional potential. In order to find proper kinematic boundary conditions on the surface and interface, the modeled erosional potential is incorporated with Bernoulli's equation of velocity potential, and Luke's variational principle is used. Then we employ a shallow-water assumption to derive the system of PDEs describing debris or granular flow with erodible base. In order to ensure stable finite volume discretization for shallow water type equation, the hydrostatic reconstruction method and implicit treatment of additional source terms is implemented. In order to verify the resulting mathematical system with the erodible effects, we simulated three reference experiments. The derived mathematical system properly describes erosional effect of granular flows and the simulated results are agreed well with experimental data. One of 2011 Umyeon Mt. debris flows is also simulated by the derived model.