Presentation

(click titles for the slides)

We consider a ring of tetrahedra interconnected by opposite edges as hinges, forming a structure known as a Kaleidocycle. This intriguing object, foldable from a sheet of paper, rotates like a bubble ring. We model the Kaleidocycle as a discrete space curve with constant torsion, and identify its configuration space as a certain point configuration on the two sphere. Then, we describe its motion by the flow generated by a semi-discrete integrable system. Through numerical experiments, we present open problems, highlighting disparities between discrete and continuous geometry. 

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. Persistent homology (PH), a tool from the emerging field of topological data analysis, has been successfully used to detect global characteristics of data that are overlooked by conventional methods. We present Cubical Ripser, an open-source software designed for high-efficiency computation of PH of cubical complexes. We will discuss how PH can be integrated into a standard image analysis pipeline with some practical applications.

Abstract: Persistent homology (PH) serves as a powerful tool for image feature extraction. We present Cubical Ripser, an open-source software designed for high-efficiency computation of persistent homology of cubical complexes. We will illustrate how Cubical Ripser can be integrated into a standard image analysis pipeline.
Hands-on Jupyter Notebook (Google Colab)

Medical imaging provides detailed visual representations of internal structures and functions of the human body and plays a pivotal role in diagnosing, monitoring, and treating various medical conditions.  Mathematical disciplines intersect with medical imaging in multifaceted ways, encompassing:
- Image reconstruction involves the transformation of raw measurements across diverse modalities such as computed tomography (CT), magnetic resonance imaging (MRI), and ultrasound into coherent, human-interpretable images.
- Image enhancement and information Extraction aim at refining image quality while extracting vital information embedded within.
- Quantitative analysis unveils deeper insight into the heterogeneity and progression of diseases in an objective and reproducible manner.
We will present some of our collaborative endeavours, bridging the expertise of medical doctors, medical physicists, and the realm of mathematics. Our work showcases applications of machine learning and topology that fortify and enrich the field of medical imaging.

We propose a new scheme for convolutional neural networks to learn visual representation with synthetic images and mathematically-defined labels that capture topological information. Our scheme can be viewed as a type of self-supervised learning, where the regression of vectorised persistent homology of an image is learned. We show that the acquired visual representation supplements the one obtained by the usual supervised learning with manually-defined labels by confirming an improved convergence in training for image classification. Our method provides a simple way to encourage the model to learn global features through a specifically designed task based on topology. It requires no real images nor manual labels and can be utilised at a minimal extra cost.

The interval decomposition theorem states that the persistent homology of a finite filtered complex decomposes uniquely into a direct sum of intervals. The decomposition enables the definition of persistence diagrams and barcodes and provides the foundation for the theory of persistent homology. There are different ways to prove (and generalise) the theorem, each conveying a unique facet of the subject. In this talk, I will present an elementary and concise proof of the theorem, which requires only basic linear algebra and is therefore suitable for the classroom.

Modelling preference data collected from many individuals with various tastes is a subject of preference learning. A person’s preference on a set of options, such as political parties and film genres, can be modelled by a (partial) order on the set. There are two major approaches to modelling preference data; based on the distance between orders and based on a utility function defined over the set of options. These approaches lack flexibility (or are biased) since too much structure is forced on the preference data to be modelled by the mathematical structure that the models utilise. Instead, we rely on a geometric entity, hyperplane arrangement, to model preference data. Given n points in the Euclidean ball, we have an arrangement of n(n-1)/2 equidistant planes. This defines a probability distribution on the symmetric group Sn, where the probability of a permutation(=ranking) is proportional to the volume of the compartment corresponding to it. For ranking data given in the form of the histogram over Sn, we construct an algorithm to find the coordinates of n points so that the resulting probability distribution fits well with the data. The geometric and combinatorial structure of hyperplane arrangement provides a good balance of flexibility and regularisation. This is joint work with T. Abe, A. Horiguchi, and Y. Watanabe.

Modern medical imaging techniques have enabled access to the interior of the human body in the form of not only 2D images but also 3D volumes. It is, however, not easy to utilise the 3D information and analysis is often limited to a slice-by-slice investigation. We need a set of features for volumetric data to take full advantage of the 3D measurements. On the one hand, radiomic features have been proposed to capture the textural characteristics of a volume. They are computed from small patches of a volume and encode only local properties. On the other hand, persistent homology (PH) provides computational machinery to extract the global structure of a volume. In this talk, we present our software, Cubical Ripser [1], for efficient computation of persistent homology of volumetric data. Then, we define a few types of invariants of a volumetric image based on PH and demonstrate their clinical relevance to abnormality quantification and detection in lung CT [2].
[1] S. Kaji, T. Sudo, and K. Ahara, Cubical Ripser: Software for computing persistent homology of image and volume data, arXiv:2005.12692
[2] N. Tanabe, S. Kaji, et al., A homological approach to a mathematical definition of pulmonary fibrosis and emphysema on computed tomography, J Appl Physiol, vol 131-2, 2021 

A person's preference on a set of options, such as political parties and film genres, can be modelled by a (partial) order on the set. Modelling preference data collected from many individuals with various tastes is a subject of preference learning. There are two major approaches to modelling preference data; based on the distance on orders and based on a utility function defined over the set of options. These approaches lack flexibility (or are biased) since too much structure is forced on the preference data to be modelled by the mathematical structure that the models utilise. Instead, we rely on a geometric entity, hyperplane arrangement, to model preference data. The geometric and combinatorial structure of hyperplane arrangement provides a good balance of flexibility and regularisation.

Given a finite set X of n items, a complete order (permutation) of the items is called a ranking of X. A ranking distribution over X is a collection of rankings of X. We will discuss a high fidelity geometric model of a ranking distribution together with efficient learning and sampling algorithms.

Modern medical imaging techniques have enabled access to the interior of the human body in the form of not only 2D images but also 3D volumes. It is, however, not easy to utilise the 3D information and analysis is often limited to a slice-by-slice investigation. We need a set of features for volumetric data to take full advantage of the 3D measurements. On the one hand, radiomic features have been proposed to capture textural characteristics of a volume. They are computed from small patches of a volume and encode only local properties. On the other hand, persistent homology provides computational machinery to extract the global structure of a volume. In this talk, we present our software, Cubical Ripser [1], for efficiently computing persistent homology of volumetric data. Then, we demonstrate its clinical relevance to abnormality quantification and detection in lung CT [2].

Topological data analysis (TDA) is an emerging field in the intersection of mathematics and data science that utilises the power of algebraic topology to analyse data given in the form of point clouds, time-series, images, and graphs. TDA focuses on the shape of the data by looking at the local-global structures, quantifying the characteristics of data complementary to the ones obtained by conventional methods. Persistent homology (PH) is one of the main tools of TDA, and it provides quantification of holes and cliques together with their scales in a mathematically rigorous and computable way. We discuss the basic idea of PH and demonstrate its usability through examples of simple graph analysis. In particular, we see how similarity metrics and features of graphs are defined by PH and used for downstream tasks such as classification and regression.

The configuration of points in the Euclidean space satisfying certain geometric constraints has long been a research topic in geometry and topology, sometimes concerning 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:
1) We construct a flow on the configuration space by a semi-discretisation of the classical sine-Gordon and mKdV equations, which generates the characteristic "everting motion" of a Kaleidocycle.
2) We show the motion preserves a discretised version of the elastic energy.
3) We construct a special family of Kaleidocycles, which we have named the Mobius Kaleidocycle, by a variational calculus. 

We will also discuss a wide range of conjectures and open problems:
a) There are discrete Mobius strips with a three pi twist, but there does not seem to exist a pi twist one.
b) The Mobius Kaleidocycle may be a rare example of a single-degree-of-freedom underconstrained linkage; its configuration space is identified as a singular manifold of a function, and it is conjectured to be homeomorphic to a circle.
c) Two variational problems, one on the twist rate and the other on the discrete elastic energy, seem to coincide to provide a characterisation of the Mobius Kaleidocycles.

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.

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.

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.

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.

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.

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).

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.

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).

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.

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.

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.

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.

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. 

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.

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. 

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. 

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.

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.

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. 

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.