## Professor KASUE Atsushi

### Faculty, Affiliation

Faculty of Mathematics and Physics, Institute of Science and Engineering

### College and School Educational Field

Division of Mathematical and Physical Science, Graduate School of Natural Science and Technology

School of Mathematics and Physics, College of Science and Engineering

### Laboratory

Department of Mathematics TEL:076-264-5643 FAX:076-264-5738

### Academic Background

**【Academic background(Doctoral/Master's Degree)】**

Osaka University Master 1980**【Academic background(Bachelor's Degree)】**

Osaka University 1978

### Career

### Year & Month of Birth

1954/07

### Academic Society

### Award

### Specialities

Laplace operator Spectral convergence、differential geometry

### Speciality Keywords

Riemannian manifold, network, energy form, Laplace operator, potential theory

### Research Themes

#### Compactifications of Riemannian manifolds and embeddings of graphs

We study p-harmonic functions of finite p-Dirichlet sum

on networks in Hadamard manifolds in relation to their

geometric compactifications and embeddings of finite

graphs into the maifolds.

#### Dirichlet finite maps and asymptotic geometry of infinite graphs

We investigate the spaces of functions of finite Dirichlet sums in indinite graphs or

infinite networks in relation to geometric structure of graphs or networks,via quasimono-morphisms or Dirichlet finite maps. Our study is focuced on behavior of such maps near Royden boundary, Kuramochi boundary, Floyd boundary, and Gromov hyperbolic boun-

dary of graphs or networks. It is important to treat with general exponents other than 2

in Dirichlet finite functions or maps.

#### Convergence theory of metric measure spaces

I study sets of finite networks (weighted graphs). By using effective resistance (combined resistance), Hausdorff convergence in the sense of Gromov and variation convergence based on energy form can be defined. A variety of things appear in the diagrams (spaces) that emerge in the limit of the sets based in these phases, including so-called fractal sets and infinite networks. The present research analyzes the Markov-type family that appears in the limit and Kuramochi compactization associated with each, primarily focusing on infinite networks (reversible Markov chains).

#### Random walks and Dirichlet finite maps

I study the relationship between random walk on an infinite network, and Dirichlet energy finite function. More generally, I research Dirichlet energy finite mapping onto complete metric space. Dirichlet energy finite mapping, when followed along random walk, approaches a finite definite value. I investigate the relationship between this fact and the space called Kuramochi compactization of infinite network.

### Books

- Riemmannian manifolds and their limits Mathematical Society of Japan 2004/08

### Papers

- Random walks and Kuramochi boundaries of infinite networks OSAKA JOURNAL OF MATHEMATICS 50 1 31-51 2013/03
- Functions with finite Dirichlet sum of order p and quasi-monomorphisms of infinite graphs Tae Hattori NAGOYA MATHEMATICAL JOURNAL 207 95-138 2012/08
- Covergence of metric graphs and energy forms 26 2 367-448 2010/04
- Convergence of Riemannian manifolds and Laplace operators, II POTENTIAL ANALYSIS 24 137-194 2006/01
- Variational Convergence of Finite Networks 12 1 57-70 2006

- A Thmson's principle and a Rayleigh's monotonicity law for nonlinear networks Atsushi Kasue Potential Analysis 2016/06/06
- Dirichlet finite harmonic functions and points at infinity of graphs and manifolds, Tae Hattori PROCEEDINGS OF THE JAPAN ACADEMY SERIES A-MATHEMATICAL SCIENCES 83 129-134 2007
- EXPANSION CONSTANTS AND HYPERBOLIC EMBEDDINGS OF FINITE GRAPHS Hattori, Tae; Kasue, Atsushi Mathematika 2014/11/19

### Conference Presentations

### Arts and Fieldwork

### Patent

### Theme to the desired joint research

### Grant-in-Aid for Scientific Research

○「倉持境界の幾何解析の展開」(2016-)

○「測度距離空間の収束理論とその展開」(2007-)

○「測度距離空間の収束とエネルギー形式」(2003-)

○「リーマン多様体の収束とラプラス作用素」(2001-)

○「リーマン多様体のコンパクト化とグラフの埋め込み」(2012-2015)

### Classes (Bachelors)

○Mathematical Thinking(2017)

○Analysis 2C(2017)

○Advanced Calculus 2A(2017)

○Advanced Calculus 2B(2017)

○Geometry 2B(2017)

○Geometry 2A(2017)

○Mathematical Thinking(2017)

○Geometry 2A(2016)

○Geometry 2B(2016)

○Introduction to Current Mathematics(2016)

○Differential and Integral Calculus 1(2016)

○Mathematical Thinking(2016)

### Classes (Graduate Schools)

○Exercise A(2017)

○Seminar A(2017)

○Research Work A(2017)

○Geometric Analysis(2017)

○Seminar A(2016)

○Mathematics Education b(2016)

○Exercise A(2016)

○Research Work A(2016)

○Mathematics Education a(2016)

○Geometric Analysis(2016)

### International Project

### International Students

### Lecture themes

○Lines, circles and elementary geometry