| Applied Geometry & Topology 2018 Project of Karlsruhe, Kyoto, and Southampton |
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| Date : 24 - 25 July | ||
| Venue : Room 108, Dept of Math, Kyoto University (Map) | ||
| Aim : This workshop will bring together researchers and students interested in applied geometry & topology, in a broad sense, to discuss and share their ideas. | ||
| Speakers | Jacek Brodzki | David Croydon |
| Emerson G. Escolar | Yasuaki Hiraoka | |
| Shu Kanazawa | Ippei Obayashi | |
| Tomoo Yokoyama | ||
| Organizers | Tsuyoshi Kato | Daisuke Kishimoto |
| Schedule | ||
| 24 July | 25 July | |
| 9:30-10:30 | T. Yokoyama | J. Brodzki |
| 10:30-11:00 | Coffee | Coffee |
| 11:00-12:00 | Y. Hiraoka | S. Kanazawa |
| 12:00-13:30 | Lunch | |
| 13:30-14:30 | D. Croydon | |
| 14:30-15:00 | Coffee | |
| 15:00-16:00 | I. Obayashi | |
| 16:15-17:15 | E.G. Escolar | |
| Title & Abstract | ||
| Jacek Brodzki (University of Southampton) | ||
| Geometry, Topology, and the structure of data | ||
| Modern data is astonishing in its variety, and is far removed from anything that could be held in spreadsheet or analysed using traditional methods. Through intense research effort, topology has emerged as a source of novel methodology to provide insight into the structure of very complex, high dimensional data. It provided us with tools like persistent homology, which are used to compute numerical topological characteristics of the data. More recently, these methods have been augmented by geometric insights, which are valuable in capturing the structure of and relationships between complex shapes.
In this talk, I will provide an overview of new techniques from topology and geometry and illustrate them on particular examples. One set of data was created through the study of CT scans of human lungs, and another addresses the problem of classification of three-dimensional shapes. |
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| David Croydon (Kyoto University) | ||
| Dynamics of the box-ball system with random initial conditions via Pitman's transformation | ||
| The box-ball system (BBS), introduced by Takahashi and Satsuma in 1990, is a cellular automaton that exhibits solitonic behaviour. It is related to the Korteweg–de Vries equation, which is used for modelling shallow water waves. In a joint work with Tsuyoshi Kato (Kyoto University), Makiko Sasada (University of Tokyo) and Satoshi Tsujimoto (Kyoto University), we explore the dynamics of the BBS started from random initial conditions. In particular, we show that the model can be described using the transformation of a nearest neighbour path encoding of the particle configuration given by `reflection in the past maximum', which was shown by Pitman to connect Brownian motion and a three-dimensional Bessel process. We use this to characterise the set of configurations for which the dynamics are well-defined and reversible for all times. We give simple sufficient conditions for random initial conditions to be invariant in distribution under the BBS dynamics, which we check in several natural examples, and also investigate the ergodicity of the relevant transformation. Furthermore, we analyse various probabilistic properties of the BBS that are commonly studied for interacting particle systems, such as the asymptotic behavior of the integrated current of particles and of a tagged particle. | ||
| Emerson G. Escolar (RIKEN AIP) | ||
| Realization of Indecomposable Persistence Modules of Large Dimension | ||
| We start with a quick review of the persistent homology of filtrations, which compactly summarizes the scale and robustness of topological features using a so-called persistence diagram. We then discuss persistence modules over commutative ladders (2 by n commutative grids), where we provide a generalization of persistence diagrams in the representation finite setting (n < 5). This setting can be seen as a partial generalization towards multidimensional persistence over commutative grids, which naturally appears when considering data filtered by more than one parameter, for example. The general case has proven more difficult, as it has the serious drawback of no longer having a concise and complete descriptor analogous to the persistence diagrams. We illustrate the existence of infinite families of indecomposable persistence modules over commutative grids of sufficient size, which provides a constructive proof of representation infinite type. We also give realizations by topological spaces (and Vietoris-Rips filtrations for the 2 by 5 case). We aim by this example to gain insight into one possible topological source of the algebraic difficulties, and to show that they can actually appear in real data and are not the product of degeneracies. | ||
| Yasuaki Hiraoka (Kyoto University) | ||
| Topological data analysis on materials science and several problems in random topology | ||
| Topological data analysis (TDA) is an emerging concept in applied mathematics in which we characterize “shape of data” using topological methods. In particular, the persistent homology and its persistence diagrams are nowadays applied to a wide variety of scientific and engineering problems. In my talk, I will explain our recent activity of TDA on materials science, e.g. glass, granular systems, iron ore sinters etc. By developing several new mathematical tools based on quiver representations, inverse analysis, and machine learnings, we can explicitly characterize significant geometric and topological (hierarchical) features embedded in those materials, which are practically important for materials properties. I will also present several interesting mathematical problems in random topology (limit theorems of persistence diagrams and higher dimensional percolations) which are motivated from those applications. | ||
| Shu Kanazawa (Tohoku University) | ||
| Asymptotic behavior of lifetime sums for random simplicial complex processes | ||
| Persistent homologies can describe the topological features of an increasing family of simplicial complexes (simplicial complex processes). In particular, these provide rigorous definitions for the concepts of birth and death times of higher-dimensional holes (cycles and cavities). The lifetimes, which are defined as the difference between the birth time and death time, measure the persistence of each hole in the process. In this work, we consider the asymptotic behavior of lifetime sums for a class of random simplicial complex processes, which is a higher-dimensional counterpart of Frieze's zeta function theorem for the Erdős-Rényi graph process. Main results include solutions to the questions on the Linial-Meshulam complex process and the clique complex process that were posed in the preceding study by Hiraoka and Shirai. One of the key ingredients of the arguments is a new upper bound of Betti numbers of general simplicial complexes in terms of the number of small eigenvalues of Laplacians on links. This bound can be regarded as a quantitative version of the cohomology vanishing theorem. | ||
| Ippei Obayashi (Tohoku University) | ||
| Analysis of the shape of data by persistent homology and machine learning | ||
| The main topic of this presentation is the combination of persistent homology (PH) and machine learning (ML). PH enables us to characterize the shape of data efficiently and quantitatively. ML enables us to clarify the characteristic features behind a lot of data. By the combination of these two concepts, we can find the geometric characteristic features from data. Many methods for the combinations are already proposed, and many ML methods are now available. This study uses simple methods, persistence images for the combination of PH and ML, and linear machine learning models for ML. Both ideas are simple, and they have some disadvantage about generalization ability (prediction ability), but the combination of two simple concepts allows us to concretely identify the important geometric structures for prediction by using another idea of inverse analysis from persistence diagram to original data. | ||
| Tomoo Yokoyama (Kyoto University of Education) | ||
| Topological methods for analyzing flows on surfaces | ||
| We introduce topological methods, called a word representation and a tree representation, for analyzing 2D flows. Applying the topological methods to a plate in a time-dependent uniform flow under mild conditions, we can estimate when the lift-to-drag ratios of the plate are maximal and can determine the intermediate topologies of the uniform flow. Our talk consists of three parts. First we quickly review relations between topology and dynamical systems. Second, we present applications of our methods. Finally we introduce the theoretical background and an implementation of representations of 2D flows, and discuss the relative works and the relations between topological structures and data structures (e.g. implementation of representation algorithms, contour extraction of streamlines from image data, generating all word representation by an automaton, generating all tree representation by "regular tree grammar + cyclic order", improvement of industrial machines, possibilities of analyzing reversal phenomena, ocean phenomena and medical phenomena). | ||