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- Interdisciplinary Seminar on Topology and Its Applications

Information: Toshitake Kohno, Masaaki Suzuki

** Organizer:**

MIMS (Meiji Institute for Advanced Study of Mathematical Sciences)

** Venue：**On-line via Zoom webinar

※ The seminar will be lectured in Japanese.

The 6th Interdisciplinary Seminar on Topology and Its Applications

Date：June 2 (Thu.), 2022 17:30--18:30

Venue：On-line via Zoom webinar

Lecturer:

OBAYASHI Ippei

(Okayama University)

Title:

"Stable volumes for persistent homology"Abstract:

Persistent homology (PH) allows us to quantify information about the shape of data using homology.

Mathematically, we consider homology on an increasing sequence of topological space called a filtration.

The output of PH is a two-dimensional scatter plot called a Persistence Diagram (PD), in which each point corresponds to a homological structure such as connected components, loops, voids, and so on.

It would be very useful for data analysis using PD if it were possible to extract the structure of the input data corresponding to each point of the PD.

However, there are multiple candidates for the solution to this problem, and we need to select an appropriate one.

Various methods have been proposed to solve this problem using mathematical optimization techniques.

Optimal volume (volume optimal cycle)[1] proposed by Obayashi is one of them.

These methods have already been effectively used, and some problems have emerged.

Recently, Obayashi proposed stable volumes[2] to solve the problems of(1)instability against noise and(2)missing the smallest component probabilistically.

In this talk, I will introduce the background, mathematical definition, computational method, and application examples of stable volumes.[1] Ippei Obayashi. Volume Optimal Cycle: Tightest representative cycle of a generator in persistent homology. SIAM Journal on Applied Algebra and Geometry 2(4), 508–534, (2018).

https://epubs.siam.org/doi/abs/10.1137/17M1159439[2] Ippei Obayashi. Stable Volumes for Persistent Homology.

https://arxiv.org/abs/2109.11711

The 5th Interdisciplinary Seminar on Topology and Its Applications

Date：January 27 (Thu.), 2022 17:30--18:30

Venue：On-line via Zoom webinar

Lecturer:

KIN Eiko

(Osaka University)

Title:

"Braids, metallic ratios and periodic solutions of the 2n-body problem"Abstract:

A collision-free periodic motion of n points in the plane give rise to a braid with n strands through the trajectory of the n points.We use ideas of braids for a classification of periodic solutions in the planar N-body problem. According to the Nielsen-Thurston theory, braids fall into three types: periodic, reducible and pseudo-Anosov. If a braid is of pseudo-Anosov type, there is an associated stretch factor which is a conjugacy invariant of pseudo-Anosov braids.Which pseudo-Anosov stretch factor is realized by a periodic solution? It is a pseudo-Anosov type for the figure-8 solution of the 3-body problem by Chenciner-Montgomery, and its stretch factor is expressed in the golden ratio, i.e., the 1st metallic ratio.In this talk, I introduce a family of periodic solutions of the 2n-body problem found by M. Shibayama in 2006.We prove that braids coming from the solutions in the family are of pseudo-Anosov type.Intriguingly, stretch factors of the solutions in the family are expressed in metallic ratios.This is a joint work with Yuka Kajihara (Kyoto University) and Mitsuru Shibayama (Kyoto University).

The 4th Interdisciplinary Seminar on Topology and Its Applications

Date：November 18 (Thu.), 2021 17:30--18:30

Venue：On-line via Zoom webinar

Lecturer:

KAJI Shizuo

(Institute of Mathematics for Industry, Kyushu University)

Title:

"Presenting discrete structures by geometric entities"Abstract:

Discrete objects such as graphs and orders are sometimes tricky to handle on a computer, as suggested by the fact that combinatorial optimization is usually much more difficult than its continuous counterpart.Therefore, as the first step to information processing, discrete objects are often replaced with continuous ones.For example, a graph is replaced by a point cloud (vectors) via an isometric embedding of the graph into a Euclidean space so that the combinatorial data of the graph is translated into the metric and algebraic structures.

In this talk, we will look at two examples where discrete objects are given geometric entities.Namely, we will see how a directed graph is modeled by nested subspace sequences in a metric space, and how a probability distribution over the set of the total orders on a fixed set is modeled by a hyperplane arrangement.

The 3rd Interdisciplinary Seminar on Topology and Its Applications

Date：July 29 (Thu.), 2021 17:30--18:30

Venue：On-line via Zoom webinar

Lecturer:

HIRAOKA Yasuaki

(Kyoto University Institute for Advanced Study, Kyoto University)

Title:

"Towards multi-parameter persistent homology"Abstract:

Persistent homology is an emerging concept in applied mathematics which characterizes “shape of data” by using topological methods, and its mathematical theory is being actively developed, inspired from applied fields such as materials science, life science, etc.

In particular, the multi-parameter persistent homology is one of the most important generalizations required for practical data analysis. In this talk, I will explain several topics about multi-parameter persistent homology, including motivations from applications, approaches based on representation theory, and probabilistic decomposition theory.

The 2nd Interdisciplinary Seminar on Topology and Its Applications

Date：June 17 (Thu.), 2021 17:30--18:30

Venue：On-line via Zoom webinar

Lecturer:

TSUBOI Takashi (Musashino University)

Title:

"Flat tori and Origami"Abstract:

We consider origami embeddings of flat tori. For piecewise-linearly embedded surfaces, one can define lengths of piecewise smooth paths, and hence one can consider this question. Intuitively, the geodesics are lines on the surface locally developed to the plane. My motivation to look at this question was that an explanation of the Nash-Kuiper embedding theorem for piecewise smooth embedding cases looked much easier. In particular, since flat manifolds are easily (semi-)locally origami embedded, it looked possible to give explicit global origami embeddings. It has already been known that flat tori can be origami embedded, however, it is interesting for me to show them explicitly.※ The seminar is lectured in Japanese.

The 1st Interdisciplinary Seminar on Topology and Its Application

Date：May 27 (Thu.), 2021 17:30--18:30

Venue：On-line via Zoom webinar

Lecturer:

SHIMOKAWA Koya (Saitama University)

Title:

"Knot theoretical analysis of site-specific recombination of DNA"Abstract:

We explain the application of knot theory to the study of site-specific recombination of DNA. We characterize the unlinking pathway of a site-specific recombination system and discuss related topics.※ The seminar is lectured in Japanese.