(click titles for the slides) Geometry of closed kinematic chain
 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 degreeoffreedom, 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 antisymmetry 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 ChataurMenichi’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.
 Representations on Real Toric Manifolds
 PrincetonRider workshop on the Homotopy Theory of Polyhedral Products, Princeton 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 Gmodule 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 typeA 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 typeA 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 + Asrigidaspossible Shape Deformation
 Poster at MEIS2016, Nishijinplaza, Fukuoka, 11 Nov. 2016
We demonstrate that two wellknown deformation techniques, Movingleast square and Asrigidas 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 nonlinear 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 wellknown, 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 nonlinear transformation with applications to shape deformation and animation. The codes used are available at github with the MIT license.
 Equivariant string products
 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 momentangle complexes. On the other hand, Bahri, Bendersky, Cohen, and Gitler gave homotopical decompositions of the suspensions of real momentangle complexes. We combine these results to give plocal 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う表現に大きな自由度を与える、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 LeibnizHopf 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 subHopf algebra of the LeibnizHopf 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 NavierStokes 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 (PLmap, in short) provide a natural framework for shape manipulation. In this talk, I will discuss an algorithm to blend/deform shapes based on PLmap and Lie theory. A shape is represented by a simplicial complex and its deformation by a PLmap. Then, creating an animation boils down to finding a nice path in the space of 3dimensional PLmaps.
 "Products in equivariant homology"
 The 6th (non)Commutative algebra and Topology, Shinshu Univ., 2022 Feb 2016.
 The 3rd Korea Toric Topology Winter Workshop, TheK 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; ChasSullivan's string product for the free loop space $LM$ over a manifold and ChataurMenichi'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 Gmanifold. 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; ChasSullivan's string product for the free loop space LM over a manifold and ChataurMenichi'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, 1213 Jan. 2015.
Let $G$ be a simplyconnected, 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, Ktheory, 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 quasiregular (that is, not too small) prime of $G$. In particular, we give a bound for the pprimary 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 simplyconnected, 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 subvarieties called the Schubert varieties. (2) Borel computed it with the rational coefficients as the “double coinvariant ring of the Weyl group." (3) Using GoreskyKottwitzMacPherson’s theory, it is given a combinatorial presentation called the GKMpresentation. 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. 1014 2014.
 "A new algorithm of Polar decomposition"
 Poster at Symposium MEIS2014: Mathematical Progress in Expressive Image Synthesis 2014, Nishijin Plaza Kyushu University, Nov. 1214 2014.
 "Nway morphing plugin for Maya" [video]
 Poster at Symposium MEIS2014: Mathematical Progress in Expressive Image Synthesis 2014, Nishijin Plaza Kyushu University, Nov. 1214 2014.
 "Modp 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. GoreskyKottwitzMacPherson (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 NavieStokes equations for generating clouds and fires, reproducing kernel Hilbert space for mixing different facial expressions, and piecewise linear map (PLmap, in short) for morphing shapes. I will discuss an algorithm to blend/deform shapes based on PLmap and an elementary Lie theory. A shape is represented by a polyhedron and its deformation by a PLmap. The idea is to find a suitable PLmap which minimizes a certain energy functional defined on the space of $3$dimensional PLmaps.
 "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.
 数学がつなぐカタチ  幾何学的な形状補間法 
 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 MSJSI2012 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 Gaction"
 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 Shortterm Joint Research Project "New Animation Interpolation and Proposal of its Evaluation Indicators", Institute of MathforIndustry, 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 socalled "Asrigidaspossible 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 Gmanifolds"
1. International Conference Toric Topology and Automorphic Functions, Pacific National University, Khabarovsk, Russia, Sep. 9th. 2011. 2. 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 Gmanifolds, by using socalled ``localization to fixed points'' developed in [GKM].  "Intersection of topology and combinatorics in Schubert calculus"
1. StudioPhones Seminar, Studio Phones, Osaka, Aug. 23rd. 2011. 2. 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" ,
1. 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" ,
1. JST CREST Groebner basis young researchers meeting, Yamaguchi Univ. Feb. 17th. 2011.
 "Equivariant cohomology of flag manifolds" ,
1. トポロジーの多様性, Kinosaki, Dec. 2nd, 2010. 2. 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)",
1. The International Conference "GEOMETRY, TOPOLOGY, ALGEBRA and NUMBER THEORY, APPLICATIONS" dedicated to the 120th anniversary of Boris Delone, Moscow, Aug. 19th, 2010. 2. International Conference JapanMexico on Topology and its Applications, Colima, Mexico, September 27  October 1, 2010. 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 subvarieties. 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 noncrystallographic 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 noncrystallographic 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",
1. 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 torusequivariant 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",
1. Fukuoka Seminar on Homotopy Theory, Jan. 9th. 2010. 2. The Third East Asia Conference on Algebraic Topology, Vietnam National University, Hanoi, Dec. 17th, 2009. 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]
1. 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",
1. Shinshu Topology Seminar, Shinshu University, Mar. 18th, 2009.
 "BottSamelson cycle in Schubert calculus",
1. Very informal seminar on Schubert calculus, Okayama University, Dec. 25th, 2008.
 "An algebraic topological approach toward concrete Schubert calculus",
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 Zmodule 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 BersteinGelfandGelfand, 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.  The 2nd East Asia Conference on Algebraic Topology, National University of Singapore, Dec. 19th, 2008.
 Topology Friday Seminar, Kyushu University, Jun. 20th, 2008. Reference list
 "Computers work exceptionally on Lie groups",
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.  First Global COE seminar on Mathematical Research Using Computers, Kyoto University, Oct 24, 2008.
 "Mod 2 cohomology of some low rank 2local finite groups",
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.  International Conference on Algebraic Topology, Korea University, Oct. 5th, 2008.
 "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",
This work should be viewed as a completion of a series of computations of Chow rings for simplyconnected 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 BernsteinGelfandGelfand 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. (based on [6])  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.
 "Homotopy Nilpotency in $p$compact groups",
Homotopy nilpotency is a homotopical analogue of the ordinary nilpotency for a group. This is an invariant for a Hspace 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 pcompact groups, which is a homotopy theoretical generalization of compact Lie groups. This is a joint work with D.Kishimoto. (based on [1])  Geometry & Topology Seminar, McMaster University, Nov. 22, 2007.
 "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 3manifolds in low dimensions", Homotopy Symposium 2004, Okinawa Senin 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.
