設計の新パラダイムを拓く新しい離散的な曲面の幾何学
キーワード：幾何学的形状生成，離散微分幾何，可積分系，曲面論，曲線論，建築，工業意匠設計，計算幾何学，折紙
2019.10～2025.03.

離散可積分系による離散微分幾何の展開
キーワード：可積分系，離散系，ソリトン方程式，離散微分幾何
2008.04.

パンルヴェ系の研究
キーワード：可積分系，離散系，パンルヴェ系，特殊函数
1993.04.

可解カオス系の超離散化とトロピカル幾何
キーワード：可解カオス，超離散化，トロピカル幾何
2007.12～2011.03.

設計の新パラダイムを拓く新しい離散的な曲面の幾何学
2019.10～2025.03, 代表者：梶原健司, 九州大学, JST CREST
可展面など性質のよい曲面を形状要素にもつ新しい離散曲面の幾何学を創始し、美的形状の理論を取り入れ、その上に構造解析・最適化手法を構築する。その枠組みで美とアート性を備え、安全・安心を担保する構造物設計を効率的かつ低コストで可能にする革新的ソフトウェア基盤を開発する。研究費総額246,700千円（直接経費）.

Application of Discrete Integrable Systems to Numerical Analysis of Boundary Value Problems
2016.07～2016.08, 代表者：Philip Broadbridge, La Trobe University, La Trobe University (Australia)
Work on integrable discrete dynamical systems; discrete integrable systems to boundary value problems of practical interest in heat and mass flow; the ability to construct new solutions of the differential equations for scale-invariant curvature-driven evolving curves and scrolls.

Reflection groups and discrete dynamical systems
2016.01～2020.12, 代表者：Nalini Joshi, The University of Sydney, Australia
This project aims to solve long-standing problems in discrete dynamical systems that are of particular interest to physics, by using reflection groups to reveal unexpected geometric insights. Mathematics has the power to abstract crucial patterns from complex observations. Symmetries familiar in the real world, like the hexagonal patterns of honeycombs, arise inside convoluted structures in high-dimensional systems. By revealing the structure of space-filling polytopes in integrable systems, the project seeks to find sought-after reductions of high-dimensional discrete models to two dimensions. Expected outputs include new reductions to discrete Painlevé equations, new circle patterns useful for computer graphics and discrete holomorphic functions..

グローバルCOEプログラム「マス・フォア・インダストリ教育研究拠点」
2008.04～2012.03, 代表者：若山正人, 九州大学大学院数理学研究院, 九州大学大学院数理学府数理学専攻
マス･フォア･インダストリ(Math-for-Industry以下｢MI｣と称する)とは、純粋・応用数学を流動性・汎用性をもつ形に融合再編しつつ産業界からの要請に応えようとすることで生まれる、未来技術の創出基盤となる数学の新研究領域です。本プログラムでは、統計・確率、数値/数式計算、神戸大学と連携した可積分系をはじめとする優れた研究実績をもとに、数学と産業界のいまだ類を見ない協働に向けた先駆的、組織的取組をさらに押し進め、世界最高水準のMI教育研究拠点の形成を目指します。MI研究の展開には新しい研究人材が不可欠です。本プログラムでは、MIの教育研究を推進し、技術の未来をになう国際的な若手数学者を育成します。.

21世紀COEプログラム「機能数理学の構築と展開」計算数理プロジェクト
2003.04～2008.03, 代表者：中尾充宏, 九州大学, 九州大学大学院数理学府
本プロジェクトでは、「計算機援用証明」を２１世紀の高度情報化社会における、数学解析の方法論として定着させ、数値シミュレーションの信頼性向上と「計算機援用解析学」の構築を目ざす。また、解析学に限らず純粋数学分野の研究への計算機活用について、分野を超えた共同研究を企画・推進します。さらに流体の運動をはじめとする複雑現象の解明に向けて、偏微分方程式の数値解析の高性能な近似スキームを開発しその理論的誤差解析の研究を展開する。特に地球規模の環境問題と深く関わる現象の数値シミュレーションにおいて必要となる、大規模数値計算の高速・高信頼化手法の確立を図る。また、他の学問分野との連携によって、偏微分方程式を中心とした数学モデルによる自然・社会現象の数学的定式化を行うとともに、その理論的および数値解析的研究を進め複雑現象解明に貢献するよう努める。.

主要著書

1.	若山正人（編）, 可視化の技術と現代幾何学, 岩波書店, 2010.04.

主要原著論文

1.	Nalini Joshi, Kenji Kajiwara, Tetsu Masuda, Nobutaka Nakazono, Discrete power functions on a hexagonal lattice I: derivation of defining equations from the symmetry of the Garnier system in two variables, JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL, 10.1088/1751-8121/ac11bd, 54, 33, 2021.08.
2.	Sampei Hirose, Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta, Discrete local induction equation, Journal of Integrable Systems, 10.1093/integr/xyz003, 4, 1, xyz003, 2019.06, [URL].
3.	Jun ichi Inoguchi, Kenji Kajiwara, Kenjiro T. Miura, Masayuki Sato, Wolfgang K. Schief, Yasuhiro Shimizu, Log-aesthetic curves as similarity geometric analogue of Euler's elasticae, Computer Aided Geometric Design, 10.1016/j.cagd.2018.02.002, 61, 1-5, 2018.03, [URL], In this paper we consider the log-aesthetic curves and their generalization which are used in CAGD. We consider those curves under similarity geometry and characterize them as stationary integrable flow on plane curves which is governed by the Burgers equation. We propose a variational formulation of those curves whose Euler–Lagrange equation yields the stationary Burgers equation. Our result suggests that the log-aesthetic curves and their generalization can be regarded as the similarity geometric analogue of Euler's elasticae..
4.	Kenji Kajiwara, Masatoshi Noumi, Yasuhiko Yamada, Geometric aspects of Painlevé equations, Journal of Physics A: Mathematical and Theoretical, 10.1088/1751-8121/50/7/073001, 50, 7, 2017.01, [URL], In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlev equations, with a particular emphasis on the discrete Painlev� equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on P1xP1. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeometric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases..
5.	Hisashi Ando, Mike Hay, Kenji Kajiwara, Tetsu Masuda, An explicit formula for the discrete power function associated with circle patterns of Schramm type, Funkcialaj Ekvacioj, 57, 2014, 1-41, 2014.04.
6.	Jun-ichi Inoguchi, Kenji Kajiwara, Nozomu Matsuura, Yasuhiro Ohta, Motion and Bäcklund transformations of discrete plane curves, Kyushu Journal of Mathematics, doi:10.2206/kyushujm.66.303, 66, 2, 303-324, 2012.10, We construct explicit solutions to the discrete motion of discrete plane curves that has been introduced by one of the authors recently. Explicit formulas in terms of the τ function are presented. Transformation theory of the motions of both smooth and discrete curves is developed simultaneously..
7.	Mike Hay, Kenji Kajiwara and Tetsu Masuda, Bilinearization and special solutions to the discrete Schwarzian KdV equation, Journal of Math-for-Industry, 3, 2011A, 53-62, 2011.04.
8.	Kenji Kajiwara, Nobutaka Nakazono and Teruhisa Tsuda, Projective reduction of the discrete Painlevé system of type (A2+A1)(1), International Mathematical Research Notices, 10.1093/imrn/rnq089, Vol. 2010, article ID: rnq089, 2010.05.
9.	Kenji Kajiwara, Masanobu Kaneko, Atsushi Nobe and Teruhisa Tsuda, Ultradiscretization of a solvable two-dimensional chaotic map assciated with the Hesse cubic curve , Kyushu Journal of Mathematics, 63巻2号315-338ページ, 2009.09.
10.	Kenji Kajiwara, Atsushi Nobe and Teruhisa Tsuda, Ultradiscretization of solvable one-dimensional chaotic maps, Journal of Physics A: Mathematical and Theoretical, 41巻，395202, 2008.09.
11.	Kenji Kajiwara and Yasuhiro Ohta, Bilinearization and Casorati determinant solution to the non-autonomous discrete KdV equation, Journal of the Physical Society of Japan, 77巻, 054004, 2008.05.
12.	Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta and Yasuhiko Yamada, Point configurations, Cremona transformations and the elliptic difference Painleve equation, Séminaires et Congrès , 14巻，175-204., 2007.08.
13.	Taro Hamamoto, Kenji Kajiwara and Nicholas S. Witte, Hypergeometric solutions to the q-Painleve equation of type $(A_1+A_1')^{(1)}$, Interenational Mathematics Research Notices, 2006, Article ID 84619, 2006.10.
14.	Nalini Joshi, Kenji Kajiwara and Marta Mazzocco, Generating Function Associated with the Determinant Formula for the Solutions of the Painlev\'e II Equation, Ast\'erisque, 297, 67-78, Vol.274(2004) 67-78, 2005.06.
15.	Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta and Yasuhiko Yamada, Cubic Pencils and Painlev\'e Hamiltonians, Funkcialaj Ekvacioj, Vol.48(1) (2005) 147-160, 2005.04.
16.	Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta and Yasuhiko Yamada, Construction of Hypergeometric Solutions to the q-Painlev\'e Equations, International Mathematical Research Notices, 24, 1439-1463, Vol.2005(24) (2005) 1439-1463, 2005.01.
17.	Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta and Yasuhiko Yamada, Hypergeometric solutions to the q-Painleve equations, International Mathematical Research Notices, 47, 2497-2521, 2004:47 (2004) 2497-2521, 2004.08.
18.	Kenji Kajiwara, On a q-Painleve III equation.II: rational solutions, Journal of Nonlinear Mathematical Physics, Vol.22 282-303, 2003.08.
19.	K. Kajiwara, T. Masuda, M. Noumi, Y. Ohta and Y. Yamada, 10E9 solution to elliptic Painleve equation, Journal of Physics A: Mathematica and General, Vol.36 L263-L272, 2003.05.
20.	Kenji Kajiwara and Kinji Kimura, On a $q$-Painlev\'e III Equation. I: Derivations, Symmetry and Riccati Type Soutions, Journal of Nonlinear Mathematical Physics, Vol.10, 86-102, 2003.02.
21.	Kenji Kajiwara, Masatoshi Noumi and Yasuhiko Yamada, q-Painlev\'e Systems Arising from q-KP Hierarchy, Letters in Mathematical Physics, 10.1023/A:1022216308475, 62, 3, 259-268, Vol.62, 259-268., 2002.12.
22.	Katsunori Iwasaki, Kenji Kajiwara and Toshiya Nakamura, Generating Function Associated with the Rational Solutions of the Painlev\'e II Equation, Journal of Physics A: Mathematical and General, 10.1088/0305-4470/35/16/101, 35, 16, L207-L211, Vol.35, L207-L211, 2002.04.
23.	Kenji Kajiwara, Masatoshi Noumi and Yasuhiko Yamada, A Study on the Fourth q-Painlev\'e Equation, Journal of Physics A: Mathematical and General, Vol.34,8563-8581, 2001.10.
24.	Kenji Kajiwara, Tetsu Masuda, Masatoshi Noumi, Yasuhiro Ohta and Yasuhiko Yamada, Determinant Formulas for the Toda and Discrete Toda Equations, Funkcialaj Ekvacioj, Vol. 44,291-307, 2001.08.
25.	Kenji Kajiwara and Yasuhiro Ohta, Determinant Structure of the Rational Solutions for the Painlev\'e IV Equation, Journal of Physics A: Mathematical and General, Vol. 31, 2431--2446, 1998.03.

主要総説, 論評, 解説, 書評, 報告書等

1.	梶原健司, 大学新入生の数学の学力--九州大学新入生数学基礎学力調査の結果より, 科学，岩波書店, 2010.11.

主要学会発表等

1.	梶原 健司, 可積分幾何による美的形状の生成, 日本数学会2024年度年会, 2024.03, [URL].
2.	Kenji Kajiwara, Yoshiki Jikumaru, Wolfgang K. Schief, Generation of Aesthetic Shape by Integrable Geometry, 10th International Congress of Industrial and Applied Mathematics (ICIAM2023), 2023.08, [URL].
3.	Kenji Kajiwara, Development of Mathematics for Industry in Japan: New Research Area, Education and Platform, 10th International Congress of Industrial and Applied Mathematics (ICIAM2023), 2023.08, [URL].
4.	梶原健司, 可積分系と幾何学的形状生成 — 「よい」方程式は「よい」形状を生成するか? —, 日本応用数理学会 2021 年研究部会連合発表会, 2021.03.
5.	Shota Shigetomi and Kenji Kajiwara, Explicit formulas for motions of smooth/discrete elasticae, Australia New Zealand Industrial and Applied Mathematics Conference 2021, 2021.02, Elastica is known as the shape of a thin elastic rod on the plane. Mathematically, it is characterised by the differential equation for its curvature. It is also known that traveling wave solutions of the modified KdV equation satisfy the equation for Elastica. Thanks to its integrability, it is possible to construct a discrete version of Elastica characterised by the integrable discrete analog of the modified KdV equation. We construct explicit solutions to the motion of Elastica described by the modified KdV equation. We also construct the explicit solutions to continuous and discrete motions of the discrete Elastica described by the semi-discrete potential modified KdV equation and the discrete potential modified KdV equation, respectively. Explicit formulas in terms of the tau function are presented..
6.	Kenji Kajiwara, Integrable discrete deformations of discrete curves: geometry and solitons, old and new, Representation theory, special function and Painlevé equation, 2015.03.
7.	Kenji Kajiwara, Torsion-preserving isoperimetric deformation of discrete space curves and its exact solutions, Symmetries and Integrability of Difference Equation X!, 2014.06.
8.	Kenji Kajiwara, Some explicit formulas in discrete differential geometry, Nonlinear Dynamical Systems, 2012.09.
9.	梶原 健司, 可積分系入門, 離散可積分系・離散微分幾何チュートリアル 2012, 2012.02.
10.	梶原 健司, Discretization of planar curve motions and discrete integrable systems, Winter School for Young Researchers on Mathematical Aspects of Image Processing and Computer Vision, 2011.11.
11.	Kenji Kajiwara, Discretization of Integrable Systems - From Nonlinear Waves to Geometry, Distinguished Lecture, Colloquium, 2011.02.
12.	井ノ口順一，梶原健司，松浦望，太田泰広, ユークリッド平面上の離散曲線の運動とBäcklund変換, 日本応用数理学会年会, 2010.09.
13.	梶原健司, 可積分系入門, 離散可積分系・離散微分幾何チュートリアル, 2010.02.
14.	Kenji Kajiwara, Hypergeometric solutions to the symmetric discrete Painlevé equations , Discrete systems and special functions, 2009.07.
15.	Kenji Kajiwara, Atsushi Nobe and Teruhisa Tsuda, Ultradiscretization of solvable chaotic maps and the tropical geometry, Geometric aspects of discrete and ultra-discrete integrable systems, 2009.04.
16.	Kenji Kajiwara, Hypergeometric solutions to the Painlevé systems , Workshop on Symmetries and Integrabiity of Difference Equations, 2008.06.

所属学会名

学協会役員等への就任

学会大会・会議・シンポジウム等における役割

学会誌・雑誌・著書の編集への参加状況

学術論文等の審査

海外渡航状況, 海外での教育研究歴

外国人研究者等の受入れ状況