# Presentation

(click titles for the slides)

Configuration space of Moebius Kaleidocycle

HSE International Laboratory of Algebraic Topology and its Applications seminar, 8 Apr. 2022 (online)

The configuration of points in the Euclidean space has long been a research topic in geometry and topology, sometimes in relation to the analysis of mechanical linkages. In this talk, we consider the configuration of lines in the Euclidean space, which provides a model for a certain type of mechanical linkage. The linkage is also popular as an origami toy and is called Kaleidocycle. We see how geometry and topology help to analyse Kaleidocycles; in particular, we construct

- a rare example of a single-degree-of-freedom underconstrained linkage, which we have named the Moebius Kaleidocycle

- a motion of a Kaleidocycle governed by some integrable systems, and show it preserves a discretised total elastic energy.

We will discuss a wide range of open problems.

Geometry and Topology of Mobius Kaleidocycles

Asia Pacific Seminar on Applied Topology and Geometry, 4 Mar. 2022 (online)

A set of finite points, as trivial as it may seem, turns out to possess rich geometric and topological structures. For example, a theorem of McCord states that the weak homotopy types of finite simplicial complexes are realised by finite spaces. The prime power conjecture asks what number of points can form a finite projective plane. Finite metric spaces are the primary source of applications of persistent homology. In this talk, we look at the configuration of finite points in $R^3$ satisfying certain geometric constraints. The object gives a model for a flexible origami toy called Kaleidocycle. We see how the geometrical and topological notions are used to formulate and analyse the properties of Kaleidocycle, and in particular, how they have led to the discovery of a rare example (in fact, the only example to my best knowledge) of 1-DoF underconstrained linkage, which we name the Mobius Kaleidocycle. Also, we construct a motion of a general Kaleidocycle that preserves a discretised total elastic energy through the flow generated by some integrable systems.

POSTECH MINDS Seminar, online, 26 Oct. 2021

For information processing, the data should be given a representation that is easy to manipulate. Often, geometric objects are used to give a representation of a discrete data type to endow the data with rich structures such as differentiability and metric. For instance, graph embedding techniques associate vertices of a graph with points in a Riemannian manifold so that the adjacency relation is modelled by vicinity. I will talk about the following two representations: (1) a directed graph by subspaces of the Euclidean space (2) a probability distribution on a permutation group by hyperplanes in the Euclidean space. The latter data type is relevant to recommendation. I will discuss both mathematical backgrounds and their applicability to real-world data.

Homological image analysis

The 5th ZIB-RIKEN-IMI-ISM MODAL Workshop on Optimization, Data Analysis and HPC in AI, online, 29 Sep. 2021

Deep convolutional networks have proved to be extremely powerful in image analysis. However, they tend to be biased toward local features such as texture and often fail to capture the global structure of image and volume data. On the other hand, persistent homology, a tool from an emerging field of topological data analysis, have been successfully used to detect global characteristics of data that are overlooked by conventional methods. I will discuss how they can be combined to extract both local and global features of image and volume data.

Topological Methods for Causal Inference from Time-series Data

The Society of Instrument and Control Engineers (SICE) Annual Conference 2021, SICE-JSAE-AIMaP session "Advanced Automotive Control and Mathematics", online, 8 Sep. 2021.

The real world consists of a lot of inter-related systems that evolve over time. Understanding and modelling such relations is the main topic of (data) science. Given two systems, detecting causality between them is an important task. It is particularly difficult when we cannot intervene in the systems but can only observe their behaviours. In this talk, we give an overview of causal inference from observed data. There are mainly two approaches in causal inference that differ in the fundamental assumption for the system; deterministic or probabilistic. We mainly focus on the former case. We discuss the idea of topological methods, including the widely-used “convergent cross mapping (CCM)” and its variants, that do not require the underlying model identification and are applicable to complex non-linear systems.

Analysis of a closed kinematic chain using discrete differential geometry

RIMS Workshop Mathematical methods for the studies of flow, shape, and dynamics, online, 30 Aug. 2021.

A closed kinematic chain is a linkage system consisting of rigid bars connected by joints to form a cycle. We show a geometric approach to model a closed kinematic chain. In particular, we analyse a family of closed kinematic chains consisting of copies of an identical part connected by hinge joints, which can be seen as a Mobius band constructed from twisted panels (see here for images). We describe their motion by integrable systems, define topological invariants of the shape, and consider the degree-of-freedoms.

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.

Geometry of Kaleidocycles

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).

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.

Discrete surface deformation with a specified Gaussian curvature

JST CREST Research Area [Mathematical Information Platform] Project Kickoff International Conference “Evolving Design and Discrete Differential Geometry — towards Mathematics Aided Geometric Design” , Kagoshima University, 6 Mar. 2020

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.

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.

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.

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".

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

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.

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.

A C++ library for 3D affine transformation and shape deformation

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.

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.

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.

"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.

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.

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.

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.

"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.

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.