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 → 25日

Deadline for registration if joining the social gathering (conference dinner): 2026 January 23 → 25

26日以降の登録では、懇親会参加はできません。
Please note that participants who register on or after 26 January 2026 will not be able to attend the social gathering.

Invited speakers

Yasuhiko Asao (Fukuoka University)

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))

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
Title: Ladder modules from the perspective of chromatic topological data analysis
Abstract: A ladder module is a representation of an N by 2 grid with commuting squares. On one hand, we can view it as the simplest non-trivial multi-persistence module. On the other hand, it is a morphism of two one-parameter persistence modules. The latter perspective inspires studying the kernel, image and cokernel of this morphism, defining three more one-parameter persistence modules.

In chromatic TDA, we start with a colored point set, based on which we construct a pair of filtered topological spaces—an N by 2 grid of inclusions—and obtain a ladder module by applying the homology functor. In addition to the five persistence modules described above, we obtain one more as the quotient homology of the filtered pair. Passing to bar decompositions, we get the 6-pack of persistence diagrams.

In my talk, I will show how 6-packs can detect that the ladder module is not interval decomposable, show various options for the construction of the topological pairs from a chromatic point set, and describe a Delaunay-like construction allowing efficient computations that enable practical use of the concepts both in theory and in applications.

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
Title: The Knowledge Matrices of a Neural Network
Abstract: At any stage of training, a neural network can be viewed as a choice of A_n-representation for each input M(x). Taking the ordered product of the matrices in this representation yields the knowledge matrix M(x). The network’s output at x is obtained by summing the columns of M(x), and can thus be interpreted as a specific dimension reduction of this matrix.

In this talk, we investigate the properties and potential applications of knowledge matrices and examine alternative dimension reductions.

This is joint work with Thomas Brüstle, William Forget, and Samuel Leblanc.

Speaker: Yasuhiko Asao
Title: Magnitude homology and representation of metric spaces
Abstract: Magnitude is a numerical invariant of enriched categories with finitely many objects, in particular of finite metric spaces, introduced by T. Leinster. Magnitude homology is its categorification, introduced by Hepworth-Willerton and Leinster-Shulman. 

In this talk, I will explain an interpretation of magnitude homology as the homology of a representation of metric spaces following joint work of the speaker and S. O. Ivanov. If time permits, I would like to discuss blurred magnitude homology from this perspective.

Speaker: Michio Yoshiwaki
Title: On interval modules in the category of 2-dimensional persistence modules
Abstract: For 2-dimensional persistence modules, it is a crucial problem to define a descriptor, persistent diagram. Since there are infinitely many indecomposable 2-dimensional persistence modules up to isomorphism, it is difficult to handle all of them. Therefore, attempts have been made to focus solely on interval modules. I would like to talk about an interval approximation and attempts at characterization via the Bridgeland stability condition.

Speaker: Bjørnar Gullikstad Hem
Title: Decomposing multipersistence modules using functor calculus
Abstract: Multiparameter persistent homology has attracted growing interest in the topological data analysis community, in part due to its ability to handle noisy data. However, unlike the single-parameter case, multipersistence modules do not generally admit an interval decomposition, which makes the multiparameter setting considerably more complicated. Nevertheless, there exist certain sufficient conditions that guarantee interval decomposability, such as a locally defined condition called middle exactness.

In this talk, I introduce poset cocalculus, which is a variant of functor (co)calculus that is defined for functors from a poset to a model category. The motivation for this framework lies in the relevance of functors from posets to the model category of chain complexes over a field, as any multipersistence module is the homology of such a functor. Poset cocalculus provides tools for relating local conditions on these functors to their global structure. I apply this to give a novel, more synthetic proof of the fact that middle exactness implies interval decomposability.

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.