Persistent homology is one of the main methods in topological data analysis, and can analyze the shape of data at multiple scales. It has been applied in many fields, for example in materials science, and also has been used as a preprocessing of data for machine learning. The theory of persistent homology is supported by algebra. In particular, understanding the category of persistence modules is important for multiparameter persistence and exploring issues of stability with respect noise. For further developments, we decided to organize this workshop to explore various interactions between topological data analysis, representation theory, geometry, and machine learning.

パーシステントホモロジーは位相的データ解析における主たる手法のうちの一つであり、データのかたちをマルチスケールで解析できる。材料科学に直接適用されるだけでなく機械学習の前処理にも用いられるパーシステントホモロジーの、その理論は代数によって支えられている。その中でもノイズ安定性についてはパーシステントホモロジーのなす圏を理解することが重要である。時空間同時解析などパーシステントホモロジーの理論の高次元化が望まれているが、その圏はより複雑になって行くため、代数的により多様な理論を必要とすることは間違いない。位相的データ解析、表現論、幾何学と機械学習のさまざまなインタラクションを模索するため、この研究集会を企画することとした。

Dates:
      12 & 13 February 2026: talks and discussions (open)
      11 & 14 February 2026: discussions (by invitation only)

Hybrid event

• In-person (registration required):
  TOKYO ELECTRON House of Creativity
  Tohoku University Katahira Campus, Sendai, Japan
  
  現地開催場所（要参加登録）：
  知の館（TOKYO ELECTRON House of Creativity）講義室

• Online (registration required):
  Zoom We will send the link to your registered email address.

• 懇親会に参加する場合の参加締め切り：2026年1月23日

Deadline for registration if joining the social gathering (conference dinner): 2026 January 23

Invited speakers

Justin Desrochers (Université de Sherbrooke)

Ondřej Draganov (Institut National de Recherche en Informatique et en Automatique(Inria))

Bjørnar Gullikstad Hem (École Polytechnique Fédérale de Lausanne(EPFL))

Woojin Kim (Korea Advanced Institute of Science and Technology(KAIST)

Yuya Mizuno (Osaka Metropolitan University)

Kaveh Mousavand (Okinawa Institute of Science and Technology(OIST))

Asao Yasuhiko (Fukuoka University)

11 February 2026

10:00-17:00 Discussions (by invite only)

14 February 2026

10:00-17:00 Discussions (by invite only)

Speaker: Woojin Kim
Title: Interleaving Distance as an Edit Distance
Abstract: The concept of edit distance, which dates back to the 1960s in the context of comparing word strings, has since found numerous applications with various adaptations in computer science, computational biology, and applied topology. By contrast, the interleaving distance, introduced in the 2000s within the study of persistent homology, has become a foundational metric in topological data analysis. In this talk, we show that the interleaving distance on finitely presented single- and multi-parameter persistence modules can be formulated as an edit distance. The key lies in clarifying a connection between the Galois connection and the interleaving distance, via the established relation between the interleaving distance and free presentations of persistence modules.  In addition to offering new perspectives on the interleaving distance, we expect that our findings facilitate the study of stability properties of invariants of multi-parameter persistence modules. As an application of the edit formulation of the interleaving distance, we present an alternative proof of the well-known bottleneck stability theorem.
This is joint work with Won Seong, and a preprint is available at https://arxiv.org/abs/2509.24233

Speaker: Ondřej Draganov
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Speaker: Emerson Escolar
Title: "On angle-optimization and simplification of homology representatives"
Abstract: In applications of persistent homology, extracting "optimal" representatives for homology classes is crucial for identifying geometric regions of interest detected as holes by persistent homology. In prior work, "optimal" is defined in terms of minimizing length or volume. In this work, we restrict our attention to a single homology class, and introduce a cost function based on angles that generalizes the total curvature. The show that our cost function penalizes departures from planarity, convexity, and simple-ness of the cycle representative.  We formulate the optimization problem as a binary quadratic problem, and show results of experiments on artificial toy data. This talk is based on joint work with Yuta Shimada.

Speaker: Justin Desrochers
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Speaker: Asao Yasuhiko
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Speaker: Michio Yoshiwaki
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Speaker: Bjørnar Gullikstad Hem
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Speaker: Yuya Mizuno
Title: Tilting theory of preprojective algebras
Abstract: Gabriel's theorem, which establishes a correspondence between indecomposable representations of Dynkin quivers and positive roots, is a cornerstone of the representation theory of quivers. In this talk, I will discuss preprojective algebras and tilting theory, both of which originated from this fundamental theorem.

First, I will overview the development of tilting theory, starting from its origins as a generalization of classical Morita theory, passing through tilting complexes in derived categories, and leading to the recent concepts of silting complexes and τ-tilting theory.
Finally, building on this general framework, I will present recent results concerning the tilting theory of preprojective algebras.

Speaker: Kaveh Mousavand
Title: Directedness of indecomposables, bricks, and spread (interval) modules
Abstract: The study of cycles of indecomposable modules has long played a central role in the representation theory of algebras. In particular, directing modules (i.e., indecomposables that lie on no cycles) exhibit many important structural properties. As a result, representation-directed algebras have many remarkable features; however, they form a rather restrictive subclass of representation-finite algebras.

In a series of recent works, we have investigated and introduced new generalizations of representation-directed algebras. In this talk, I first present some new realizations of locally-representation-directed algebras, based on joint work with C. Paquette. I then introduce the novel notion of brick-directed algebras, developed in joint work with S. Asai, O. Iyama, and C. Paquette, and explain how this brick-analogue significantly generalizes the classical concept of representation-directedness. If time permits, I will also discuss some aspects of ongoing joint work with E. Escolar, where we study directedness in the context of spread modules.