# Talks (slides)

## (click titles for the slides)

Tutorial on CubicalRipser and other TDA software using Python

POSTECH MINDS & PIAI Workshop on Topological Data Analysis and Machine Learning, Online, 7 July, 2021.

Techniques in Topological Data Analysis are vast, and there are many software packages that provide various functionalities. Oftentimes, it is not easy to find the right tool for the data and the task at hand. We will give hands-on tutorials on several software packages for use with Python on Google Colab (a Jupyter notebook is available here). Various tasks, including classification, regression, clustering, and visualisation, on multiple data types, including point cloud, image, volumetric data, time series, and graph, are covered. In particular, we will introduce our Cubical Ripser, a fast program for computing persistent homology of images and volumes (cubical complexes).

Fast computation of persistent homology of volumetric data and its application in medical image analysis

The 2nd Ajou-Kyushu joint workshop on industrial Mathematics: "Biomedical Mathematics"

(The 2nd Edition of Asia Pacific Online Seminars on Mathematics for Industry), 7 May 2021 (online)

Medical images such as CT and MRI are acquired in the form of volumetric data. We introduce our easy-to-use software, CubicalRipser, which is capable of fast persistent homology computation of volumetric data. As an application, we discuss how persistent homology can be enhanced to capture both local and global topological features of images to enable interpretable medical image analysis.

数学と諸分野の連携に向けた若手数学者交流会2020, 13 Mar. 2021 (online)

Kyushu-Illinois Strategic Partnership Colloquia Series #2: Mathematics Without Borders−Applied and Applicable, 11 Mar. 2021 (online)

Kaleidocycles are Origami models of flexible polyhedra which exhibit an intriguing turning-around motion (have a look at the pictures at https://github.com/shizuo-kaji/Kaleidocycle). The study of Kaleidocycles involves kaleidoscopic aspects and lies at the intersection of geometry, topology, and integrable systems (and mechanics). In this talk, we discuss two "incarnations" of them. (1) The states of a Kaleidocyce form a real-algebraic variety defined by a system of quadratic equations. In particular, the degree-of-freedom of its motion corresponds to the dimension of the variety. Using this formulation, we introduce a special family of Kaleidocycles, which we call the Mobius Kaleidocycles, having a single-degree-of-freedom (joint work with J. Schoenke at OIST). (2) A Kaleidocycle can be viewed as a discrete space curve with a constant torsion. Its motion corresponds to a deformation of the curve. Through this correspondence, we describe particular motions of Kaleidocycles using semi-discrete integrable systems (joint work with K. Kajiwara and H. Park at Kyushu University).

バイオフィジオロジー研究会特別企画Webカンファレンス2021, 19 Feb. 2021 (online)

A simple path tracer for 4D scene rendering (シンプルな４次元パストレーサーによるレンダリング)

VRを用いたインタラクティブな高次元認識, 12 Feb. 2021 (online)

パストレーシングは光輸送をモンテカルロ法で解くことで３次元シーンのレンダリング画像を得る方法である．この講演では，わずか99行のコードでパストレーシングを実現した smallpt を紹介し，それが安直に４次元シーンのレンダリングに拡張できる (smallpt4d) ことを述べる．

Nested Subspace Arrangement によるグラフの連続表現

日本オペレーションズ・リサーチ学会「超スマート社会のシステムデザインのための理論と応用」研究部会 第7回研究会, 19 Nov. 2020 (online)

グラフ埋め込みは，SNSなどの巨大なグラフをユークリッド空間に埋め込むことで，解析の助けにする手法の総称である．埋め込まれた各頂点は座標を与えられ，離散構造が連続化されるため，機械学習の前処理として用いられる．これまでには，広範かつ効率的な埋め込みを探すために，ポアンカレ円盤など一般の距離空間への埋め込みを考えたり，有向グラフを表現するために各頂点を点ではなく円盤として埋め込むなどの拡張が考えられてきた．後者の方法では，有向辺を円盤の包含で表現することで，非対称性を巧妙に実現しているのだが，サイクルが表現できないという弱点がある．この講演では，頂点を部分集合族によって表現する Nested Subspace Arrangement を導入することでこの弱点を克服する．さらに，WordNet や Twitter network などの巨大なグラフが先行研究に比べてより効率的に埋め込めることを紹介する．この研究は，九州大学の秦希望氏，吉田明広氏，藤澤克樹氏との共同研究である．

形状デザインの数理的方法の深化・発展とその社会実装

第9回 藤原洋数理科学賞 奨励賞 受賞講演, Keio University, 17 Oct. 2020

分子のようにミクロな世界から建築物や宇宙まで，この世界はさまざまな形で成り立っています．私は広い意味での形を，解析したりデザインしたりする研究を行っています．目に見える形だけではなく，数学的にはビッグデータも高次元の形とみなすことができます．数字や文字で表しにくい対象・データを，その形を通して見ることで，コンピューターによる記述や処理が可能になります．私の研究ではこれまでに，CTやレーダーといった医療・土木で用いられるイメージング技術，コンピューター上で形状をデザインする技術，データの形に着目することで少量のデータで深層学習を補うことのできる解析技術などを，共同研究者の方たちと一緒に開発してきました．また，工学的に有用で不思議な性質を持つ「メビウス・カライドサイクル」という形状(リンク機構)は，折り紙からスタートして背後に現代数学が絡み合う，工学・数学・折り紙が邂逅する学術上も応用上も面白い発見になりました．折り紙の展開図や3Dプリンタモデルを公開しているので，ぜひ触れて見てください．

深層学習とパーシステントホモロジーによるハイブリッド画像解析

日本応用数理学会 2020年度年会, 正会員OS 位相的データ解析, 9 Sep. 2020 (online)

畳み込みニューラルネットワーク(CNN)は画像解析において非常に強力である一方で、近視眼的でありテクスチャーなど局所的な情報に捉われるすぎる傾向があることが知られている。これは、人間がものを見る時により大域的な手がかりを重視するのと対照的である。本講演では、パーシステントホモロジーによって画像の大域構造を、CNNが学習しやすい形にエンコードする手法を紹介する。 ２通りのものの見方を融合することで、実際に画像認識精度が向上することを例示する。

Geometry of the configuration space of Kaleidocycles

African Mathematics Seminar, 5 Aug. 2020 (online)

Have you heard of a Kaleidocycle, which is an origami art consisting of tetrahedra joined by their edges to form a ring? (if not, have a look at https://github.com/shizuo-kaji/Kaleidocycle). It exhibits an intriguing turning-around motion. Mathematically, the set of states (the configuration space) of a Kaleidocycle is identified by a certain real-algebraic subvariety of the product of the real Grassmannians: each connecting edges define affine lines in 3-space, and they satisfy geometric conditions specified by a system of quadratic equations. Each real solution to the system corresponds to a state of the Kaleidocycle. Each one-dimensional subspace of the configuration space corresponds to a motion of the Kaleidocycle. We define a flow on the configuration space using discrete versions of mKdV and sine-Gordon equations. The one-dimensional orbits generated by the flow corresponds to the characteristic turning-around motion. We discuss some interesting open problems related to the configuration space of the Kaleidocycle, which lie at the intersection of geometry, topology, and integrable systems. This is joint work with K. Kajiwara and H. Park.

Geometric toolbox for data analysis

応用のためのトポロジカルデータ解析チュートリアル＆ワークショップ, 19 Jun. 2020 (online)

データというものは有限の計算機で扱うために本質的に離散であるが、連続な対象が背後にあり、そのサンプリングとして得られている場合など、その大局的な構造を位相や幾何を通して捉えるのは往々にして有効である。このチュートリアルでは、persistent homology やラプラシアンといった道具を機械学習と組み合わせつつデータの図形的な性質を調べる方法を、実問題への応用例をあげつつ紹介する。

Topological characterisation of Interstitial Lung Disease

"医学と数理”（第２回 京大―ハイデルベルク大―理研 ワークショップ ）, Kyoto University, 17 Sep. 2020 (postponed from 21 Mar. 2020 due to coronavirus outbreak)

Discrete surface deformation with a specified Gaussian curvature

Closed kinematic chains and discrete space curves

Closing Workshop of Joint Project between Austria (FWF) and Japan (JSPS): Geometric shape generation, Tokyo Institute of Technology, 20 Feb. 2020

Flood prediction by geographic data analysis with GANs and tailored loss functions

Forum-Math-for-Industry 2019, Massey University, New Zealand, 21 Nov. 2019

In recent years, Japan has suffered from severe damage caused by floods. There is an urgent need for accurate flood forecast. Flood prediction usually takes the form of time series analysis in which future water level is regressed from that of the observed past and additionally precipitation, geography, and so on. There are standard machine learning models such as ARIMAX models to perform this type of analysis. Here, we investigate the use of neural network-based models for the problem. The two key factors are (1) use of generative adversarial networks; this is to cope with the lack of data (2) a loss function which is tailored for the problem; in flood prediction, usual losses such as the mean squared error are not optimal, since, for example, undershooting in prediction is much more harmful than overshooting.

Time evolution of Kaleidocycles

International Workshop Mathematical Sciences and Applications, Yamaguchi, 7 Nov. 2019

Kaleidocycle is an origami toy which can be folded from a sheet of paper. Unlike usual origami, it has some mobility; in fact, it can be seen as an example of linkage mechanisms which consists of hinges. We model (a generalisation of) Kaleidocycles as discrete space curves with constant torsions. In particular, we see that the motion of a Kaleidocycle is governed by a semi-discrete (discrete space, continuous time) integrable system. The origami piece demonstrates visually how different fields of mathematics, topology, algebraic geometry, and analysis meet as well as art. This is joint work with K. Kajiwara and H. Park.

Geometry of hinged linkage systems

International Congress on Industrial and Applied Mathematics (ICIAM2019), Valencia, 17 Jul 2019.

We study a certain class of spacial linkage mechanisms consisting of hinges by identifying them as discrete space curves and discuss their motion in terms of isoperimetric and torsion preserving deformation of the curves. In particular, we construct explicit motions corresponding to closed hinged linkages consisting of congruent tetrahedra, governed by the semi-discrete mKdV and sine-Gordon equations. This is joint work with K. Kajiwara and H. Park.

Homology assisted neural networks for images [PowerPoint(heavy!)]

International Congress on Industrial and Applied Mathematics (ICIAM2019), Valencia, 16 Jul 2019.

Tools in topological data analysis (TDA) look at global features of the data while convolutional neural networks (CNNs) are good at detecting local features. The question is how we can combine the power of these two. We report our attempt to make CNNs utilise the global information obtained by TDA for a practical task on non-destructive inspection using sensor images. Then, we will investigate if CNNs can be trained to compute (persistent) homology. Finally, we describe how a CNN approximating persistent homology of the distance transformation of binary images may be used to "optimise" the topology of images.

かたちの線型代数・微積分

東京大学数理情報学談話会, 3 Jul 2019

画像処理では、画像を二次元の格子グラフ上の関数とみなし、その関数を線型代数や微積の道具立てを用いて料理する。同様に、３次元空間内の形状は、グラフやより一般に単体複体上の関数とみなすことで、やはり線型代数やベクトル解析を用いて処理することができる。本講演では、形状処理、特にかたちの変形について、この観点からの研究を紹介したい。

やわらかい幾何の拡がり

マス・フォア・イノベーション シンポジウム, 九州大学, 6 Jun 2019

データから見える液晶の転移温度と秩序 -- QSPRの先へ

高分子学会九州支部フォーラム, 九州大学, 14 Mar 2019

トポロジーに基づく形状設計法

芝浦工業大学 数理科学科 談話会, 19 Feb 2019

３次元形状をコンピューター上で扱う技術は、computer graphics (CG) や computer aided design (CAD) とともに発展してきた。また最近では 3D プリンタの発展により、digital fabrication も盛んである。これらの分野においては、トポロジーのアイデアが本質的な役割を果たす技術課題が散見される。計算機の上で"かたち”を扱うには、まず対象を有限の記号におとす必要があるが、代数トポロジーはまさにうってつけの道具である。この講演では、講演者がこれまで行ってきた共同研究の中から２つのトピックを取り上げ、単体複体で表された曲面の変形と、離散的な空間曲線としてモデル化されたリンク機構の設計についてお話ししたい。

多数のヒンジからなる、自由度１の閉リンク機構

日本応用数理学会：応用数理ものづくり研究会, 政策研究大学院大学, 18 Feb 2019

微分形式を用いたメッシュ形状処理の基本(Introduction to discrete differential forms for mesh processing)

IMI短期共同研究2018「造船工学における曲面幾何」, Kyushu univ., 25 Dec 2018

A linkage mechanism that follows a discrete sine-Gordon equation

SIDE13: Symmetries and Integrability of Difference Equations, Fukuoka, 15 Nov 2018

We consider a family of linkage mechanisms which consist of $n$-copies of a rigid body joined together by hinges to form a ring. Each hinge joint has its own axis of revolution and rigid bodies joined to it can be freely rotated around the axis. The family includes the famous 3-fold symmetric Bricard6R linkage which exhibits a ``turning over'' motion. We can model such a linkage as a discrete closed curve in R^3 with a constant torsion up to sign. Then, its motion is described as the deformation of the curve preserving torsion and arc length. In general, the degree of freedom of the motion, or equivalently, the dimension of the configuration space of such a linkage increases as n gets bigger. In this talk, we describe particular paths in the configuration space that are governed by semi-discrete sine-Gordor/mKdV equations. The infinitesimal motion is seen to be confined in the osculating plane. This is joint work with K. Kajiwara and H. Park at Kyushu university.

Equivariant loop product (同変ループ積について)

日本数学会2018年度秋季総合分科会 トポロジー分科会 特別講演, 岡山大学, 26 Sep 2018

Image and Shape Manipulation (slides available upon request)

2018 NCTS Summer Course, 30 Jul - 17 Aug, 2018(with Chun-Chi Lin and Mei-Heng Yueh (NTNU))

We will primarily be interested in altering shapes and images based on topological and geometrical techniques. Images are vector-valued functions on a rectangular domain, whose values are the color intensity of pixels. A mesh is an imbedding of a polygon into the three-space. Thus, we can manipulate images and shapes by defining operations on these maps. However, these maps are inevitably discrete to be dealt with on computer. Discrete differential geometry is a relatively new and active area in geometry to handle “smooth” shapes represented in a discrete way. We will see how basic notions in calculus are translated in a discrete language and can be applied to manipulate shapes and images. We will also discuss concrete algorithms and their implementations.

Representing the diagonal as the zero locus in a flag manifold

Friday Topology Seminar, Kyushu University, 29 June 2018.

The zero locus of a generic section of a vector bundle over a manifold defines a sub-manifold. A classical problem in geometry asks to realise the fundamental class of a specific sub-manifold in this way. We study the class of a point in a generalised flag manifold and of the diagonal in the direct product of two copies of a generalised flag manifold. These classes are important since they in a sense generate the cohomology of the flag manifold and the torus equivariant cohomology of the flag manifold, respectively.

A closed linkage mechanism having a degenerate configuration space

La Trobe-Kyushu Joint Seminar on Mathematics for Industry 34, 8 May 2018.

Linkage is a mechanism consisting of rigid bars connected by revolute joints (hinges).Each hinge joins two adjacent bars and can rotate them around its axis. A linkage mechanism is said to be closed if it has the topology of a circle. We consider a closed linkage with n-hinges (hence, n-bars). Possible states of such a linkage form the configuration space, which can be thought of as a subspace of (S^1)^n. By dimension counting, the configuration space is generically n-6 dimensional. We discovered a closed linkage mechanism, which seems to have several interesting properties; most notably, the 1-dimensional configuration space regardless of n. Most of the results are only numerically confirmed and yet to be proved.

Geometry seminar, Tokyo Institute of Technology, 24 May 2018.

曲線の幾何学から生まれた閉リンク機構 (in Japanese)

精密工学会2018年春季大会 AIMaP数学応用シンポジウム: 精密工学と幾何学の新たな出会い, 中央大学理工学部, 17 Mar. 2018

CGと数学 (in Japanese)

情報処理学会第８０回全国大会 イベント企画 「工芸から科学へ - CG技術の新たな挑戦 -」, 早稲田大学理工学部, 15 Mar. 2018

線形代数や流体力学を始めとして，CGには様々な数学が用いられていますが，数学を研究する立場からCG研究はどう映るのでしょうか．海外では意外にも古くから，また日本でも最近は，離散微分幾何学といった新しい数学の源泉としてCGが認識されてきています．数学には「役に立たないほど高等だ」という考え方と同時に，歴史的に物理や工学にアイデアの起源を見つけてきたという二面性があります．CGと数学の関わりについて，トポロジーの研究者である私個人の経験を中心にお話したいと思います．

柔らかい幾何の拡がり -トポロジーの応用- (in Japanese)

AIMaP公開シンポジウム「数学と産業の協働ケーススタディ」, 日本橋ライフサイエンスビルディング, 20 Jan 2018.

Interpolating Shapes

Bilateral Mini-Workshop of NTNU and Yamaguchi University on Mathematics and its Applications, NTNU, Taiwan, 26 Dec. 2017

Interpolation is one of the major principles in science and engineering. Curve fitting to observed data in experiments and image and audio up-scaling involve interpolation. Soap film naturally interpolates the boundary wire to a surface. In topology, we identify two shapes, such as a doughnut and a mag, which can be “interpolated" continuously. I will discuss a method to interpolate shapes based on topology to demonstrate possibilities of applying abstract mathematics to something “visible".

３次元形状モデリングへのトポロジーの応用 (in Japanese)

数学と諸分野の連携を通した知の創造, 東北大学, 9 Dec. 2017.

PolyArc Fitter —An approximation of a curve using line segments and arcs —(with A. Hirakawa, Y. Onitsuka, C. Matsufuji, D. Yamaguchi)

Poster at MEIS2017, Fukuoka, 16 Nov. 2017

Given a sequence of points on the plane, we propose an algorithm to approximate them by a curve consisting of line segments and circular arcs. It has a practical use In computer aided manufacturing, and our algorithm has already been used in ship building. In a typical pipeline, ship parts are designed by a CAD software and numerical control machines (NCM) are used to actually cut steel plates. Many NCMs are capable of cutting only line segments and circular arcs, so the designed curves have to be converted in such forms. Moreover, it is desirable to have as few segments as possible due to efficiency and physical limitations of the machine. Given a sequence of points on the plane, our algorithm produces a curve consisting of a small number of line segments and circular arcs which passes within a user specified neighbour from every point.

Image deformer on iPad [codes]

Poster at MEIS2017, Fukuoka, 16 Nov. 2017

We developed grid (2D-mesh) based image deformers which work on iOS devices. A loaded image or real-time input from the camera is converted to a texture placed on a triangulated grid, and the grid is deformed in real-time according to user’s touch gestures. A few different deformation algorithms are implemented. The source codes are available at my github under MIT license.

Point and diagonal classes in flag varieties

The 7th East Asian Conference on Algebraic Topology (EACAT 2017), IISER Mohali, India, 1 Dec. 2017

The zero locus of a generic section of a vector bundle over a manifold defines a sub-manifold. A classical problem in geometry asks to realise the fundamental class of a specified sub-manifold in this way. We study the classes of a point in a generalised flag manifold and of the diagonal in the direct product of two copies of a generalised flag manifold. These classes are important since they are related to ordinary and equivariant Schubert polynomials.

Topology Seminar, University of Southampton, 18 Dec. 2017

画像関係の機械学習 Hands-on (Hands-on Machine Learning for Image analysis)

IMI短期共同研究 三次元幾何モデリング評価手法の提案とソフトウェア開発, Kyushu university, Fukuoka, 5 Sep. 2017.

Geometry of closed kinematic chain(** p. 21 the conjecture about a constant curvature is obviously wrong. I will fix it later. ** The slide with Possible Applications was not included in the talk due to patent application reasons.)

IMI Workshop Mathematics in Interface, Dislocation and Structure of Crystals, Nishijin plaza, Fukuoka, 29 Aug. 2017

Consider a system consisting of rigid bodies connected to each other. Such a system can be modelled by a graph with edges labelled by elements of the Euclidean group SE(3), where each cycle satisfies a certain closedness condition. We are particularly interested in a system consisting of hinges. To each vertex is assigned one degree-of-freedom, namely the rotation angle, and the configuration space of the system is described by the real solution to a system of polynomial equations. We found an interesting family of systems on cycle graphs, whose configuration spaces form positive dimensional real algebraic varieties. They are a type of so called Kaleidocycle, but exhibit intriguing properties such as anti-symmetry and constant bending energy. This is joint work with Eliot Fried, Michael Grunwald, and Johannes Schoenke at OIST.

A topological view on shape deformation

Applied Algebraic Topology 2017, Hokkaido University, 11 Aug. 2017.

Manipulating shapes with computer has now become ubiquitous; Computer graphics (CG) is essential to film making, and so is Computer Aided Design (CAD) to industrial design. From a topological view point, a shape is a certain space (typically, a simplicial complex or a surface) and its deformation (animation) is a family of imbeddings into the ambient space (typically, $\R^3$). In this talk, I will review some of my recent work to demonstrate how this topological view can actually be made into algorithms of shape deformation.

A secondary and equivariant string product

Young Researchers in Homotopy theory and Toric topology 2017, Kyoto University, 4 Aug. 2017.

A product on homology of the space of free loops LM over a closed manifold M is first defined by Chas and Sullivan and various similar constructions have been discovered since then. Among them is Chataur-Menichi’s product on homology of LBG where BG is the classifying space of a finite group G. We give a common generalisation to both by defining a product on homology of L(M_G), where M_G is the Borel construction of a compact (not necessarily connected) Lie group action on M. We also discuss a secondary product and show it is related to the cup product in negative Tate cohomology of G. This is joint work with Haggai Tene.

Topology of brain-wide dynamics in consciousness (Satohiro Tajima and Shizuo Kaji)

Poster at

*ASSC 21*, Beijing, China, 13-17 Jun. 2017

Representations on Real Toric Manifolds

Princeton-Rider workshop on the Homotopy Theory of Polyhedral Products, Princeton Univ./Rider Univ., 29 May 2017

When a group G acts on a manifold M, the (co)homology of M is equipped with a G–module structure. One of the central questions in representation theory asks to realise a given G-module geometrically in this manner. In this talk, we consider finite group actions on real toric manifolds combinatorially through the correspondence between real toric manifolds and simplicial complexes with characteristic matrices. In particular, we see interesting representations of (signed) permutations appear on the homology of certain real toric manifolds. This is joint work with Soojin Cho and Suyoung Choi.

A type-A Weyl group action on the associated real toric manifold

The 4th Korea Toric Topology Workshop, Jeju unipark, 27 Dec. 2016

A simplicial complex $K$ with vertex labelling in an F_2 vector space gives rise to a real toric space. When a finite group $G$ acts on $K$ satisfying a certain condition, it induces an action on the toric space. We investigate the case when $(K, \lambda)$ is associated to a root system, and $G$ is its Weyl group. In particular, we give a concrete description of the Weyl group representation on the cohomology of the associated real toric space for type-A root systems. This is joint work with Soojin Cho and Suyoung Choi.

Polar Decomposition of Square matrices

Mathematical methods and practice in cryptography, security and bigdata, Hokkaido Univ. 21 Dec. 2016

Polar decomposition is in the standard toolbox to analyse and visualise point cloud data in the Euclidean space. In this talk, I will discuss how it is used, in particular, in computer graphics. I will also talk about an efficient algorithm to compute the polar decomposition of a regular square matrix. This is joint work with Hiroyuki Ochiai at Kyushu university.

かたちの平均 ( some gif animations do not work with the online version of Power Point...)

信州大学数理科学談話会, 22 Nov. 2016

３次元空間内に複数の形状が与えられた時、 その”平均”とは何であろうか。この講演では、コンピューターで計算・出力するという観点から、 この問に一つの答えを与える。扱う数学の道具自体は高度ではないが、リー群論や離散幾何の アイデアが応用されることで、ペンギンたちが平均される様子をお見せしたい。

Moving Least Square + As-rigid-as-possible Shape Deformation

Poster at MEIS2016, Nishijin-plaza, Fukuoka, 11 Nov. 2016

We demonstrate that two well-known deformation techniques, Moving-least square and As-rigid-as possible schemes, can be combined straightforwardly and produce nice results.

Designing transformations with simple ingredients

GEMS2016: Geometry and Materials Sciences 2016, OIST, 17 Oct. 2016

When we deign curves and surfaces with computer, it is enough to specify a small number of control points and parameters. Can we design transformations just as simply? I will propose a method to generate non-linear transformations from small input data which can be easily specified by the user. I will demonstrate this technique by applying it to character animation and shape modeling. Most of the codes used in the demonstration are available at my github repository.

トポロジーを応用した計算機による形状処理

京都大学応用数学セミナー, 11 Oct. 2016

３次元空間内の形状をコンピューター内であらわす際，一般的には，点群の座標+単体複体といった構造情報を用いる．この構造情報を保ったまま，点の座標のみを動かす変換を作用させることで，基本形状を変形してデザインしたり，キャラクターをアニメーションさせたりすることができる．この様な視点から，様々な場面で，それぞれの入力に応じて実際に変換を構成する手法を紹介する．扱う数学の道具自体は高度ではないが，リー群論や離散幾何のアイデアが応用され，ペンギン計算が行われる様子をお見せしたい．

３次元アフィン写像と形状変形ライブラリ (A C++ library for 3D affine transformation and shape deformation)

数学ソフトウェアとフリードキュメント XXIII, 関西大学, 14 Sep. 2016

３次元空間のアファイン写像は，計算機で形状変形やアニメーションを扱う上で基本構成要素となる．アフィン写像は行列で表すことができるが，これは効率は良い反面，例えば正則行列の平均が必ずしも正則にはならないなど，用途によっては使い勝手が良くない．講演者は落合啓之氏と共同で，リー環のカルタン分解を用いてアフィン写像をパラメトライズする方法を考案した．また，アフィン変換をベースに，より複雑な非線形変換を構成することができるが，これを形状変形とアニメーション生成に利用する例を示す．全ての実装はC++のソースコードも含めて MIT License で公開している．

Affine transformation serves as one of the fundamental tools in computer graphics. As is well-known, affine transforms are represented by matrices. This is computationally efficient, but not always convenient. H. Ochiai and I developed a parametrisation method of affine transforms using the Cartan decomposition of their Lie algebra. Based on this method, we see how one can construct more complex non-linear transformation with applications to shape deformation and animation. The codes used are available at github with the MIT license.

The 5th GeToPhyMa: Summer school "Rational homotopy theory and its interactions" celebrating Jim Stasheff and Dennis Sullivan for their 80th and 75 anniversary, UIR, Rabat, Morocco, 20 Jul 2016.

A product in the homology of the free loop space LM of a manifold was discovered by Chas and Sullivan together with other structures. Chataur and Menichi defined among other things a similar product in the homology of the free loop space LBG of the classifying space of a topological group G. We define a product in the homology of the free loop space L(M_G) of the Borel construction of a manifold with a Lie group action. We show it vanishes under a certain degree condition, and in that case, we define a secondary version of the product. This is joint work with Haggai Tene.

Homotopy decomposition of a suspended real toric space

Toric Topology 2016 in Kagoshima, Kagoshima University, 20 Apr 2016.

Choi and Park gave a formula for the homology of real topological toric manifolds with coefficients in a ring in which two is invertible, based on the work by Cai on the homology of real moment-angle complexes. On the other hand, Bahri, Bendersky, Cohen, and Gitler gave homotopical decompositions of the suspensions of real moment-angle complexes. We combine these results to give p-local homotopy decompositions of the suspensions of real topological toric manifolds for odd primes p. This is joint work with S. Choi and S. Theriault.

トポロジーのCGへの応用

サロン時間学, 時間学研究所, 山口大学, 29 Jan 2016

コンピューターグラフィックス(CG)と伝統的な絵画や映像表現との違いの一つは、前者は時間変化を伴い表現に大きな自由度がある、と言えるのではないか。絵画は自由に視覚効果を創造できるが、動きを表現することは難しい。一方で、実写映像で可能な表現は、物理的な制約を受ける。宇宙空間での大爆発や、恐竜が闊歩する世界は、コンピューター無しでは実現が難しいだろう。CGは、お金と時間のかかる実写の代替として、シミュレーションを用いて映像を作るという方向とは別に、全く物理的な制約を離れて、自由に仮想的な表現を生み出すという可能性も拓いた。その一つの例として、モーフィングと呼ばれる特殊効果がある。少年がカレーを食べるとサッカー選手に変化するCM(古いですが)のあれである。今回は、トポロジーという数学を用いたモーフィングの実現方法について紹介したい。 モノの形状の時間変化を数学モデルを用いて定式化するのだが、実験結果をいかによく説明するかという物理モデルの話と違い、人間の目に心地よければ良いという、正解のない世界で自由にモデルを設計できる。そんなところにどこか純粋数学に通じるものがありおもしろい。

Steenrod algebra and Leibniz-Hopf algebra, and their duals

Topology Seminar, Fukuoka University, Japan, 18 Jan 2016.

A famous theorem by Milnor tells a lot about the structure of the dual Steenrod algebra; it is a polynomial algebra. He also gave a formula for the antipode on the generators. In this talk, we regard the dual Steenrod algebra as a sub-Hopf algebra of the Leibniz-Hopf algebra and provide another proof and an extension to Milnor's result purely combinatorially. This is joint work with N.D. Turgay.

A topological algorithm for shape deformation in computer graphics

Topology and Combinatorics seminar, Ajou University, Korea, 31 Dec 2015.

In computer graphics, mathematics has been a fundamental toolbox. The Navier-Stokes equations is indispensable for generating clouds and fires, reproducing kernel Hilbert space is used for mixing different facial expressions, and simplicial complex and piecewise linear map (PL-map, in short) provide a natural framework for shape manipulation. In this talk, I will discuss an algorithm to blend/deform shapes based on PL-map and Lie theory. A shape is represented by a simplicial complex and its deformation by a PL-map. Then, creating an animation boils down to finding a nice path in the space of 3-dimensional PL-maps.

Products in equivariant homology

The 6th (non)Commutative algebra and Topology, Shinshu Univ., 20-22 Feb 2016.

The 3rd Korea Toric Topology Winter Workshop, The-K Hotel, Gyeongju, Korea, 29 Dec 2015.

We discuss an external product associated with the homology of a compact manifold with a Lie group action. It is then used to define a product on $H_G^*(LM)$ the homology of the free loop space over the Borel construction. This product unifies two known constructions in string topology; Chas-Sullivan's string product for the free loop space $LM$ over a manifold and Chataur-Menichi's string product for the free loop space $LBG$ over the classifying space of a compact Lie group. We will also introduce its secondary product, which generalises Tate's cup product in the homology of $BG$. This is joint work with Haggai Tene.

Tetrisation of triangular meshes and its application in shape blending

MEIS2015, Nishijin plaza, Kyushu university, 26 Sep 2015.

Products in Equivariant Homology

Topology Seminar, CRM, Barcelona, 12 June 2015.

Let G be a compact Lie group and M be a compact G-manifold. Denote its Borel construction by M_G. Given a homotopy pullback square with the base space M_G, we will introduce a generalised external product in the homology such that it reduces to the ordinary external product when M is a point and G is trivial. ( In this case, the homotopy pullback is just the direct product. ) It unifies two constructions in string topology; Chas-Sullivan's string product for the free loop space LM over a manifold and Chataur-Menichi's string product for the free loop space LBG over the classifying space of a compact Lie group. We will also introduce its secondary product, which generalises Tate's cup product in the homology of BG. This is joint work with Haggai Tene.

The dual Steenrod algebra and the overlapping shuffle product

Antalya Algebra Days XVII, Nesin Maths Village, Turkey, 23 May 2015.

"Homotopy decomposition of a suspended flag manifold"

Poster at IMPANGA15, Banach center at Bedlewo, Poland, 13 Apr 2015.

External products in equivariant homology

Torus Actions in Geometry, Topology, and Applications, Skoltech, Moscow, 17 Feb, 2015.

Mod p decompositions of the loop spaces of flag manifolds

International Mathematics Conference, in honour of the 70th Birthday of Professor S. A. Ilori, Ibadan, 12-13 Jan. 2015.

Let $G$ be a simply-connected, simple, complex Lie group and $P$ be its parabolic subgroup. The homogeneous space $G/P$ is called the flag manifold. Topological invariants of $G/P$ such as the cohomology ring, K-theory, and its equivariant variants have been extensively studied under the name of Schubert calculus. However, much less is known about the homotopy groups. In this talk, we outline an attempt to give a better understanding of the homotopy groups of $G/P$ using standard techniques from algebraic topology; localisation at a prime and homotopy decomposition. We give a homotopy decomposition of the based loop space of $G/P$ into a product of irreducible spaces ( which are fairy simple ) when localised at a quasi-regular (that is, not too small) prime of $G$. In particular, we give a bound for the p-primary exponent of the homotopy groups of $G/P$. This is a joint work with A. Ohsita and S. Theriault.

Torus equivariant cohomology of flag manifolds

Topology Seminar at University of Ibadan 15. Jan, 2015

Let $G$ be a simply-connected, simple, complex Lie group and $P$ be its parabolic subgroup. The homogeneous space $G/P$ is called the flag manifold. Fix a maximal torus $T$ of $G$ which is contained in $P$. It acts on $G/P$ by the group multiplication. One of the main goals of the "equivariant Schubert calculus" is to study the $T$-equivariant (Borel) cohomology $H^*_T(G/P)$. There are three presentations known for $H^*_T(G/P)$: (1) Chevalley gave a free $H^*(BT)$-module basis of it consisting of sub-varieties called the Schubert varieties. (2) Borel computed it with the rational coefficients as the “double coinvariant ring of the Weyl group." (3) Using Goresky-Kottwitz-MacPherson’s theory, it is given a combinatorial presentation called the GKM-presentation. Each presentation has both advantages and disadvantages; for example, in (1) geometric meaning of a class is clear but it is hard to compute the product of two classes while in (2) elements are just polynomials and one can easily multiply them. Therefore, it is preferable to know how to convert an element in one presentation to another. In this talk, I will review those three presentations and investigate how they are related to each other.

Shape deformation in Computer graphics

Poster at Discrete, Computational and Algebraic Topology, University of Copenhagen, Nov. 10-14 2014.

A new algorithm of Polar decomposition

Poster at Symposium MEIS2014: Mathematical Progress in Expressive Image Synthesis 2014, Nishijin Plaza Kyushu University, Nov. 12-14 2014.

N-way morphing plugin for Maya [video]

Poster at Symposium MEIS2014: Mathematical Progress in Expressive Image Synthesis 2014, Nishijin Plaza Kyushu University, Nov. 12-14 2014.

Mod-p decompositions of the loop spaces of compact symmetric spaces

29th British Topology Meeting, Southampton, UK, Sep. 8. 2014.

We give homotopy decompositions of the based loop spaces of compact symmetric spaces after they are localised at large primes. The factors are fairly simple; namely spheres, sphere bundles over spheres, and their loop spaces. As an application, upper bounds for the homotopy exponents are determined. This is a joint work with A. Ohsita and S. Theriault.

A product in equivariant homology for compact Lie group actions

Topology Seminar, CRM, Barcelona, Spain, July 18. 2014.

The Tate cohomology for a finite group integrates group homology and cohomology into one theory. It is equipped with a cup product, which coincides with the usual one on cohomology and gives a ring structure on homology. A few attempts have been made to generalise this product structure on homology. We follow the line of Kreck and Tene. Kreck defined a product on H_*(BG;Z) for a compact Lie group G based on his geometric homology theory and Tene showed it coincides with the cup product on the Tate cohomology when G is finite. We will generalise this product to one on the equivariant homology of a manifold with a nice action of a Lie group. Our construction is simple and purely homotopy theoretical. This is a joint work with Haggai Tene.

A topological algorithm for Blending Shapes

Directed Algebraic Topology and Concurrency, Université Lyon 1, France, Jan 30th 2014.

Lie group action on a GKM manifold

Manifold Atlas Writing Seminar, Max Planck Institute for Mathematics, Bonn, Apr 28th 2014.

Topology Seminar, University of Southampton, UK, Dec 9th 2013.

A GKM manifold is a manifold with a "nice" torus action. Many interesting spaces are examples of GKM manifold including toric manifolds and flag manifolds. Goresky-Kottwitz-MacPherson (for which GKM is an abbreviation) showed that a certain kind of graph, called the GKM graph, is associated to a GKM manifold, from which we can read off the torus equivariant cohomology of the manifold purely combinatorially. I will discuss the case when the torus action extends to a Lie group action, and show how the action is reflected on the combinatorial structure of the GKM graph.

An Application of Lie theory to Computer Graphics

Applied Topology, Banach center in Bedlewo, Poland, July 25th 2013.

In computer graphics, various mathematics is used such as the Navie-Stokes equations for generating clouds and fires, reproducing kernel Hilbert space for mixing different facial expressions, and piecewise linear map (PL-map, in short) for morphing shapes. I will discuss an algorithm to blend/deform shapes based on PL-map and an elementary Lie theory. A shape is represented by a polyhedron and its deformation by a PL-map. The idea is to find a suitable PL-map which minimizes a certain energy functional defined on the space of $3$-dimensional PL-maps.

Cohomology of a GKM graph with symmetry

Toric Topology 2012 in Osaka, Osaka city university, Nov. 18th 2012.

「代数トポロジーと組み合わせ論の相互作用」, 兵庫教育大学神戸サテライト, Feb. 16th 2013.

Haifa university colloquium, Israel, Mar. 19th 2013.

Ordinary and Equivariant Schubert classes

Characteristic classes and intersection theory seminar, Higher School of Economics, Moscow, Sep. 20th 2012.

ワークショップ「数理科学と情報科学の周辺」, Shinshu University, Feb. 14th 2013.

CEDEC2012, Pacifico Yokohama, Aug. 20th 2012.

少数のキーフレーム(物体の形状データ)から、それらを補間する連続的なフレームデータを生成する技術は、フレーム補間とよばれ、アニメーション作成やインタラクティブな物体変形に使われています。ここでは、特に局所的な形状をなるべく保ったままフレーム補間を行うアルゴリズムを、背後にある数学に焦点を当てながら紹介し ます。技術自体の即効性よりも、アルゴリズムの発見法や数学者の物の見方について主にお話ししたいと思います。

"Ordinary vs Double Schubert polynomials"

Poster session at MSJ-SI2012 Schubert calculus, Osaka city university, July 26th 2012.

"An invitation to Schubert calculus"

Combinatorics Seminar in Kyushu University, Nishijin plaza, Kyushu university, July 14th 2012.

Postnikov Seminar, Moscow State University, Sep. 18th 2012.

"The equivariant cohomology of a manifold with a G-action"

Topology of Mapping space and around, Okinawa Senin Kaikan May 14th 2012.

"Frame interpolation for character animation - Overview of the problem and our progress"

IMI Short-term Joint Research Project "New Animation Interpolation and Proposal of its Evaluation Indicators", Institute of Math-for-Industry, Kyushu University, Mar. 5th. 2012.

The interpolation technique, which produces continuous images from two or more keyframes, is important in the video production fields since it drastically reduces tedious work of creating animations. Among many suggested methods, Alexa et. al. introduced so-called "As-rigid-as-possible interpolation" (ARAP, in short) in 2000. Since then, improvements and extensions have been studied mostly from CG side, and the way of comparison and evaluation for different techniques is aesthetic. Mathematically, ARAP can be formulated as the problem of finding an appropriate path connecting given polyhedral shapes. Our aim is to establish rigorous framework to study it. In this talk, I will: (1) give a brief survey of the original method, (2) introduce mathematical evaluation indicators for "rigidity", (3) provide possible improvements and extensions.

**"Schubert calculus for G-manifolds"**International Conference Toric Topology and Automorphic Functions, Pacific National University, Khabarovsk, Russia, Sep. 9th. 2011.

Toric Topology in Osaka 2011, Osaka city university, Osaka, Nov. 28th. 2011.

Let G be a Lie group and T be its maximal torus. The homogeneous spaces G/T is known to be a smooth variety and called the flag variety. It admits the action of T by the group multiplication, and the equivariant cohomology H^*_T(G/T) brings up an interesting subject of study called equivariant Schubert calculus. Among remarkable properties of H^*_T(G/T) is the fact that there is a distinguished geometric basis consisting of the Schubert classes, which fits into a hierarchy posed by the divided difference operators. We try to extend this to a litter wider class of G-manifolds, by using so-called ``localization to fixed points'' developed in [GKM].

**"Intersection of topology and combinatorics in Schubert calculus"**StudioPhones Seminar, Studio Phones, Osaka, Aug. 23rd. 2011.

Ajou university Colloquim, Suwon, Korea, Nov. 25th. 2011.

"How many lines are there in the three space which meet all the four given lines ?" In 19th century, H.Schubert considered this problem in an insightful but not rigorous way. He invented a symbolic ``calculation'' for the conditions on lines as follows: [intersecting a given line]^{¥cap 4} = [lying on a given line] ¥cup [lying on a given line],and obtained the answer two. In fact, the ``algebra'' of the conditions on lines is isomorphic to the ring of the symmetric polynomials called Schur polynomials. D.Hilbert asked for a rigorous foundation for the above calculus as the 15th problem in his 1900 lecture and now Schubert’s quiz can be rephrased in terms of cohomology, or equivalently, intersection theory of a Grassmaniann manifold. In this talk, I will briefly review the basics of Schubert calculus with a focus on the correspondence of several algebras occurring in this subject including the above one.

**"A rational homotopy model of quasitoric manifolds"**,Toric Topology in Himeji 2011, Egret Himeji, Apr. 3rd. 2011.

**"グレブナー基底を用いると、解けそうで解けない少し難しい代数トポロジーの問題"**,JST CREST Groebner basis young researchers meeting, Yamaguchi Univ. Feb. 17th. 2011.

**"Equivariant cohomology of flag manifolds"**,トポロジーの多様性, Kinosaki, Dec. 2nd, 2010.

京大微分トポロジーセミナー, Kyoto Univ. Dec. 7th.

**"Equivariant Schubert calculus of Coxeter group I_2(m)"**,Let $G$ be a Lie group and $T$ be its maximal torus. The homogeneous spaces $G/T$ is known to be a smooth variety and called the

*flag variety*of type $G$. Its cohomology group has a distinguished basis consisting of Schubert classes, which arise from a certain family of sub-varieties. The ring structure of $H^*(G/T)$ with respect to this basis reveals interesting interactions between topology, algebraic geometry, representation theory, and combinatorics, and has been studied under the name of Schubert calculus. One way to study $H^*(G/T)$ is to identify it with the coinvariant ring of the Weyl group $W$ of $G$, i.e. the polynomial ring divided by the ideal generated by the invariant polynomials of $W$. From this point of view, the problem can be rephrased purely in terms of $W$ and extended to any Coxeter group including non-crystallographic ones. In fact, H. Hiller pursued this way in his book ``The geometry of Coxeter groups'' and gave a characterization of a ``Schubert class'' in the coinvariant ring. On the other hand, $G/T$ has the canonical action of $T$ and we can consider the equivariant topology with respect to this action. A similar story goes for the equivariant cohomology $H^*_T(G/T)$ and we can consider equivariant Schubert calculus for Coxeter groups. This time we consider a double version of a coinvariant ring. Along this line, the first difficulty is how to find polynomials in it representing Schubert classes. By several people, such polynomial representatives have been found for type $A_n, B_n, C_n, D_n$. Here we give polynomial representatives for the non-crystallographic group of type $I_2(m)$. The main ingredients is the localization technique, a powerful machinery of equivariant topology.

**"Schubert calculus, seen from torus equivariant topology"**,KAIST Toric Topology Workshop 2010, Feb. 25th. 2010.

I will give two talks on Schubert calculus, with emphasis on the torus action. Schubert calculus is a study of the geometry of a flag variety, which is defined as the homogeneous space of a compact Lie group $G$ divided by its subgroup $P$ containing a maximal torus $T$. Since flag varieties are so basic objects in various areas of mathematics, there are a lot of ways to explore this fertile land. Here we'll take an inclination for a view from torus equivariant topology. A flag variety has an ideal action of the maximal torus $T$, which is hamiltonian with isolated fixed points corresponding to the elements of the Weyl group. Hence, the localization technique, widely known as "GKM theory," offers a powerful machinery to deal with topological invariants such as the equivariant cohomology of flag varieties combinatorially. Fortunately enough for those who have a liking for computation, this is actually applicable to calculations, through which we can see a concrete aspect of the subject. I will start with brief history of the subject and then review basic notions, playing with the most fundamental example of Grassmannian manifolds. Then I will introduce two descriptions of the torus-equivariant cohomology of flag varieties, one by the GKM graph, the other by the polynomial ring with two series of indeterminants, and discuss the interaction between them. With these preparations, we'll set out for a somewhat outskirts region of the study, namely a concrete calculation of the torus equivariant cohomology of the flag varieties associated to exceptional Lie groups.

**"Torus equivariant cohomology of flag varieties"****,**Let $G$ be a simple complex Lie group and $B$ its Borel subgroup. Then the (right) homogeneous space $G/B$ is called the flag variety. One of its rich structures is that $G/B$ has a stratification given by the

*Schubert varieties*, which is a class of subvarieties indexed by the Weyl group of $G$, and form a module basis of $H^*(G/B;Z)$. Determining the ring structure of $H^*(G/B;Z)$ with respect to this particular basis is a classical but still active theme in*Schubert calculus*. There are a lot of variant of this problem and here we consider the equivariant version. Let $T$ be the maximal torus of $G$. Then $T$ acts on $G/B$ from the left. Since the Schubert varieties are $T$-invariant, the story immediately fits into the $T$-equivariant setting. In this talk we observe a strategy for the concrete calculation of the torus equivariant cohomology $H^*_T(G/B;Z)$, and apply it to the specific case of $G=G_2$, the rank two exceptional group.

**"Schubert calculus of exceptional types"****,****[Abstract]**The 56th Topology Symposium, Hokkaido University, Aug. 11th, 2009.

"How many lines in the space meet four general lines ?" Starting with the inevitable introduction of

*Schubert calculus*, we make a whirlwind trip through this actively studied area. After introducing the three main characters, the*flag variety*, the*Weyl group*, and the*Schubert variety*, brought together on the*Hasse(GKM) diagram*, we focus on the cases of exceptional types. We explain how one can obtain the explicit description of integral cohomology of flag varieties of exceptional types as quotient rings of polynomial rings, and how to find representatives of Schubert classes there.

**"Divided difference operator and equivariant cohomology"**,Shinshu Topology Seminar, Shinshu University, Mar. 18th, 2009.

**"Bott-Samelson cycle in Schubert calculus"**,**"An algebraic topological approach toward concrete Schubert calculus"**,Topology Friday Seminar, Kyushu University, Jun. 20th, 2008. Reference list

Let G be a connected simple complex Lie group and P be a parabolic subgroup. The homogeneous space G/P is known to be a projective variety called the generalized flag variety. The Chow ring A(G/P) of G/P ( ,which is isomorphic to the ordinary cohomology with integral coefficients) is known to have a good Z-module basis consisting of so called Schubert varieties. Schubert calculus in a narrow sense is a study of the ring structure of A(G/P), or more precisely, a study of the structure constants for the intersection products of two arbitrary Schubert varieties. In this talk, we give an algebraic topological method to present A(G/P) explicitly in the form of the quotient of a polynomial algebra, especially when G is an exceptional group. Main tools and ingredients are the divided difference operators introduced by Berstein-Gelfand-Gelfand, and the Borel presentations of the cohomology rings of homogeneous spaces, which are laboriously calculated by Japanese algebraic topologists. This is a joint work with M.Nakagawa.

**"Computers work exceptionally on Lie groups"**,First Global COE seminar on Mathematical Research Using Computers, Kyoto University, Oct 24, 2008.

The well known classification theorem states that there are only nine types of simple compact Lie groups; the four classical infinite families and the five exceptional ones. Although topological invariants of classical Lie groups are often given in simple unified forms, those of exceptional groups are usually of ugly looks, which insist on handling by a case by case analysis; this is where computers would do themselves justice. By presenting an example of the computation of the Chow rings of complex Lie groups, I will demonstrate the possibility of using a computer in such a kind of research where tedious and technical labor is required.

**"Mod 2 cohomology of some low rank 2-local finite groups"**,International Conference on Algebraic Topology, Korea University, Oct. 5th, 2008.

The cohomology of finite groups have been studied successfully by homotopy theoretical approaches. The mod p cohomology of some finite groups of Lie type can be obtained from that of the homotopy fixed points of unstable Adams operations, which have similar properties as the free loop spaces over the classifying spaces of Lie groups. By inspecting this similarity, recently Kishimoto and Kono developed a method to compute the mod p cohomology ring over the Steenrod algebra of the finite groups. We will discuss the actual computation of some low rank cases for p=2, and show that the mod 2 cohomology of the groups and the free loop spaces are isomorphic as rings over the Steenrod algebra in those cases.

**"Chow rings of Complex Algebraic Groups"**, (in Japanese, hand written)1. COE tea time, Kyoto University, Jan. 17th, 2008.

**"Chow rings of Complex Algebraic Groups"**,Workshop on Schubert calculus 2008, Kansai Seminar House, Mar. 20, 2008. [Slide]

The 4th COE Conference for Young Researchers, Hokkaido Univ, Feb. 13, 2008. [Slide]

Topology Seminar, Kyoto University, Jan. 29, 2008.

International Conference on Topology and its Applications, Kyoto University, Dec. 4, 2007.

Symplectic Geometry Seminar, University of Toronto, Nov. 26, 2007.

Topology Seminar, Johns Hopkins University, Nov. 13, 2007.

This work should be viewed as a completion of a series of computations of Chow rings for simply-connected Lie groups by Chevalley, Grothendieck, and most recently by Marlin (1970s), who computed the Chow rings for G = Spin(n), G2, and F4. His method did not cover the cases of the exceptional Lie types G=E6, E7, and E8. On the other hand, these Chow rings A(G) can be determined from the cohomology of the corresponding flag varieties, the latter of which were computed by Borel, Toda, Watanabe, and Nakagawa. We use the divided difference operators of Bernstein-Gelfand-Gelfand and Demazure to obtain efficient presentations of the Chow rings of G=E6, E7, and E8 via geometric generators coming from Schubert classes on G/B. This is a joint work with M.Nakagawa.

**"Homotopy Nilpotency in $p$-compact groups"**,Geometry & Topology Seminar, McMaster University, Nov. 22, 2007.

Homotopy nilpotency is a homotopical analogue of the ordinary nilpotency for a group. This is an invariant for a H-space which measures how far its product is from being homotopy commutative. We determine the homotopy nilpotency of compact Lie groups when localized at large primes. This work is a continuation of that of McGibbon (1984) which determined the homotopy commutativity of localized compact Lie groups. By the same methods, we also obtained a similar result for all the p-compact groups, which is a homotopy theoretical generalization of compact Lie groups. This is a joint work with D.Kishimoto.

**"Certain $p$-local $H$-space structure on the classifying spaces of gauge groups"**, Takamatsu seminar on homotopy theory, Takamatsu National College of Technology, March 2007.**"Homotopy nilpotency in localized Lie groups"**, The 3rd COE Conference for Young Researchers, Hokkaido University, Feb. 13th. 2007.**"Homotopy nilpotency in localized groups"**, Homotopy Symposium 2006, Ehime University, Nov. 28th. 2006.**"Cohomology of the classifying spaces of gauge groups over 3-manifolds in low dimensions"**, Homotopy Symposium 2004, Okinawa Sen-in Kaikan Hotel, November. 2004.**"Introduction to rational homotopy theory"**, The 1st COE Kinosaki Young Researchers' Seminar, Kinosaki Town Hall, February 2004.**"On the rational $H$-spaces"**, Algebraic and Geometric Models for Spaces and Around, Okayama University, September 2003.