Title: Interval Multiplicities of Persistence Modules (one lecture)

Abstract:For any persistence module $M$ over a finite poset $\mathbf{P}$, and any interval $I$ of $\mathbf{P}$, we give a formula for the multiplicity $d_M(V_I)$ of the interval module $V_I$ in the indecomposable decomposition of $M$ in terms of the ranks of matrices consisting of structure linear maps of $M$. This generalizes the corresponding formula for 1-dimensional persistence modules.

The formula tells us which morphisms of $\mathbf{P}$ are essential to compute $d_M(V_I)$. This leads to the notion of an order-preserving map $\ze \colon Z \to \mathbf{P}$ essentially covering $I$, for which the multiplicity is preserved under the induced restriction functor $R \colon \mod \mathbf{P} \to \operatorname{mod} Z$. When $Z$ is of Dynkin type $\mathbb{A}$, also known as a zigzag poset, this allows the multiplicity to be computed more efficiently from the filtration level of topological spaces, without computing all structure linear maps of $M$.

Finally, we give a formula for $d_M(V_I)$ in terms of a projective (or injective) (co)presentation of $M$. In the 2D-grid case, this is more practical since such resolutions can be computed from the filtration level of topological spaces. This is a joint work with Enhao Liu.

Title: Complex Matching Distance and Stability for Minimal Projective Resolutions, with Applications to Persistence (two lectures)

Abstract: We develop a stability theory for minimal projective resolutions of $\mathbf{P}$-modules, where $\mathbf{P}$ is a finite metric poset. We use the G\"ulen--McCleary distance on $\mathbf{P}$-modules together with a new complex matching distance on bounded complexes of finitely generated projective $\mathbf{P}$-modules. The latter yields an extended metric on homotopy classes of such complexes and restricts to minimal projective resolutions. Our main theorem shows that this induced distance on minimal projective resolutions is bounded above by the G\"ulen--McCleary distance.

As an application, we pass to the interval poset and kernel construction, interpreting persistence diagrams as the homotopy class of minimal projective resolutions of kernel modules. This gives a corresponding stability theorem, which in the one-parameter case recovers classical bottleneck stability and in the multiparameter case addresses existing notions of signed diagrams. This is a joint work with Amit Patel.

Talk 1 Title: Density-Sensitive Bifiltrations in Topological Data Analysis.

Abstract: The standard filtrations of TDA (Rips, Čech, Delaunay) are stable to small perturbations of the input data, but are notoriously unstable to outliers, and can be insensitive to variations in density. One of the main motivations of 2-parameter persistence is to address these limitations. There are several natural ways one can define a density-sensitive bifiltration of point cloud or metric data, offering different tradeoffs between generality, computability, and robustness to outliers. Such bifiltrations have been actively studied in recent years, leading to substantial advances in our understanding of them, as well as new computational tools. In this talk, I will survey this progress.

Talk 2 Title: $\ell^p$-metrics on Multiparameter Persistence Modules and Merge Trees.

Abstract: The standard stability and inference theory for persistent homology is an $\ell^\infty$ theory.  This talk concerns foundational work towards developing an $\ell^p$ extension of this theory.  Bjerkevik and I have introduced a generalization of the $p$-Wasserstein distance on barcodes to multiparameter persistence modules, called the $p$-presentation distance and denoted $d_I^p$.  We have shown that $d_I^{\infty}$ equals the interleaving distance and that on 1- or 2-parameter persistence modules over prime fields, $d_I^p$ is the universal (i.e., largest) metric satisfying a natural stability property.  This universality result extends a stability theorem of Skraba and Turner for the $p$-Wasserstein distance on barcodes in the 1-parameter case, and is also a close analogue of a universality property for the interleaving distance.  Cardona, Curry, Lam and I have subsequently introduced an analogous $p$-presentation distance on merge trees, with a closely analogous theory.
  In spite of these good theoretical properties, fundamental aspects of the $p$-presentation distances are poorly understood.  Most critically, the distances are defined in terms of two infima, and the question of whether these infima are attained has remained open. Recently, Riley Decker and I have proven that for both modules and merge trees, the infima are attained in the case $p=1$. We have also established bounds on the size of the objects attaining these infima. These bounds depend polynomially both on the size of minimal presentations for the input and on a measure of input fineness called the \emph{algebraic spread}.  We conjecture that these results extend to arbitrary $p$ with polynomial bounds that do not depend on the algebraic spread.

Talk 3 Title: Limit Computation Over Posets via Minimal Initial Functors. (Joint work with Tamal Dey)

Abstract: It is well known that limits can be computed by restricting along an initial functor, and that this often simplifies limit computation.  We systematically study the algorithmic implications of this idea for diagrams indexed by a poset.  We say an initial functor $F\colon C\to D$ with $C$ small is \emph{minimal} if the sets of objects and morphisms of $C$ each have minimum cardinality among the sources of all initial functors with target $D$.  For $Q$ a finite poset or $Q$ an interval in $\mathbb{N}^d$ (i.e., a convex, connected subposet), we describe all minimal initial functors $F\colon P\to Q$ and in particular, show that $F$ is always a subposet inclusion.  We give efficient algorithms to compute a choice of minimal initial functor. In the case that $Q$ is an interval in $\mathbb{N}^d$, we give asymptotically optimal bounds on $|\hom P|$, the number of relations in $P$, in terms of the number $n$ of minima of $Q$: We show that $|\hom P|=\Theta(n)$ for $d\leq 3$, and $|\hom P|=\Theta(n^2)$ for $d>3$.  We apply these results to give new bounds on the cost of computing $\lim G$ for a functor $G\colon Q\to \mathbf{Vec}$ valued in vector spaces.  For $Q$ connected, we also give new bounds on the cost of computing the generalized rank of $G$  (i.e., the rank of the induced map $\lim G \to \operatorname*{colim} G$), which is of interest in topological data analysis.

Title: The TF equivalences on the real Grothendieck groups

Abstract: Let $\mathcal{A}$ be an abelian length category with finitely many isoclasses of simple objects. Then the dual real Grothendieck group $K_0(\mathcal{A})_\mathbb{R}^*$ is identified with an Euclidean space $\mathbb{R}^n$. The theme of my talk is the TF equivalence on $K_0(\mathcal{A})_\mathbb{R}^*$, which I introduced as a way to visualize important numerical data of $\mathcal{A}$ by using semistable torsion pairs of Baumann-Kamnitzer-Tingley. I will discuss how the TF equivalence is related to stability conditions of King and the wall-chamber structure of Brüstle-Smith-Treffinger. Then I will explain that the TF equivalence recovers the silting fan of 2-term silting complexes in the case $\mathcal{A}$ is the category of finitely generated modules over a finite dimensional algebra.

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Title: On Angle-optimization and Simplification of Degree-1 Homology Representatives

Abstract: In topological data analysis, in particular persistent homology analysis, extracting "optimal" representatives for homology classes is crucial for identifying geometric regions of interest. In prior work, optimality is defined in terms of minimizing length or volume. In this work, we restrict our attention to a single homology class in degree 1 and introduce the total absolute curvature of cycles as the cost function. We show that this cost function, based on angles between edges of cycles, penalizes departures from planarity, convexity, and simple-ness of the cycle representative. We formulate the "angle-optimal homologous cycle problem", recast it as a binary quadratic optimization problem, and show the results of experiments on artificial toy data. This talk is based on joint work with Yuta Shimada (arXiv:2608.23949). 

Title: A Topological Fingerprint of the Frenkel Line

Abstract: Supercritical liquid copper at 10 GPa exhibits an unusual viscosity rise above ~8000 K, accompanied by a Stokes-Einstein breakdown. This is coincident with the Frenkel line (FL), the boundary separating liquid-like oscillatory dynamics from gas-like ballistic motion. Pair distribution functions show only smooth broadening across this range. We compute persistent homology on Alpha complex filtrations of 500,000-atom molecular dynamics configurations (2000-12000 K) and find dimension-resolved reorganization between 5000 and 10000 K: occupied diagram bins increase 125% for 1D loops (H1) and 119% for 2D voids (H2), with total pair frequency declining 15% and 23% respectively. Shannon entropy of the bin-multiplicity distribution rises 14-15% for both dimensions, tracking the logarithm of the number of occupied bins. H1 entropy and viscosity are rank-correlated on either side of the FL with opposite signs (Spearman rho = -1.00 below, +0.80 above), reflecting their shared temperature dependence. These results indicate that medium-range cages dissolve progressively into a more diverse population of transient, longer-persistence features. Comparison with density-matched null point processes and diagram-level distance metrics shows this evolution to be a smooth crossover, and HomCloud's inverse analysis is used to examine the atoms behind short- and medium-range order.

Title: Characterisation of Cluster Graphs via Persistent Homology

Abstract: Ordinary homology captures topological features of data, while persistent homology provides a stable, continuous relaxation that is suitable for optimisation. From this perspective, it is natural to construct persistence-based scores that quantify desirable structural properties of data. A basic and widely used example is connectivity, particularly in image analysis. In this talk, we consider edge-weighted graphs and introduce a persistence-based descriptor that measures their connectivity structure. We characterise the local minima of this descriptor and show, in particular, that its strict local minima are precisely the cluster graphs, that is, disjoint unions of cliques. We also present examples in which this score is incorporated as a regulariser for basis learning. Beyond its applications in machine learning, the result illustrates a different use of persistent homology: not merely as a descriptor of data, but as a tool for characterising a combinatorially defined class of graphs through an optimisation principle.

Title: Interleaving Distance as a Galois-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 work, we show that the interleaving distance on finitely presented single- and multi-parameter persistence modules can be formulated as a so-called Galois-edit distance. The key lies in clarifying a connection between the Galois connection and the interleaving distance by exploiting an established relation between the interleaving distance and free presentations of persistence modules. In addition to offering new perspectives on the interleaving distance, this result enables us to exploit the maximality property of the edit distance in the search for invariants of persistence modules that are stable under the interleaving distance. Consequently, we obtain a new characterization of such invariants. As an application of this characterization, we present an alternative proof of the well-known bottleneck stability theorem. 

This is joint work with Won Seong.

Title: Computing relative Betti numbers for interval resolutions

Abstract: Interval resolutions, introduced by Asashiba, Escolar, Nakashima, and Yoshiwaki, provide a homological approach to approximate finite-poset-indexed persistence modules by interval modules. The relative Betti numbers record multiplicities of the interval modules in each degree of the interval resolution.

I present our recent progress on computing these numbers using injective coresolutions of persistence modules, which can, in particular, be efficiently computed from bifiltrations. I also provide criteria for filtering out intervals before applying our proposed formula. If time permits, I will introduce MPIntInv, a Python software implementing a practical workflow for computing interval-based invariants of multiparameter persistence modules.

Title: Persistent Homology in Materials Science: Toward Understanding Structure–Property Relationships

Abstract: Topological data analysis provides powerful tools for describing complex structures hidden in data. In particular, persistent homology has found growing applications in materials science. One representative application is the analysis of structure–property relationships in disordered materials such as amorphous solids and glasses.

Although the atomic structures of amorphous materials and glasses appear random at first glance, the constituent atoms interact through physical forces, giving rise to structural correlations that extend beyond the nearest-neighbor length scale and involve many atoms. Such correlations are commonly referred to as medium-range order and have long been considered important for understanding the properties of disordered materials. However, how to describe these complex structural features and, more importantly, how to relate them quantitatively to material properties remain long-standing challenges.

By regarding atomic coordinates as a point cloud and applying persistent homology, one can extract geometric features such as rings and voids formed by atoms over multiple length scales. These topological features reflect aspects of medium-range order in disordered structures. By using persistence diagrams to predict material properties, identifying the regions of the diagrams that contribute most strongly to the prediction, and tracing these contributions back to the corresponding real-space atomic structures, it becomes possible to investigate the relationship between medium-range order and material properties in a more physically interpretable manner.

In this talk, I will present examples of our studies using persistent homology to clarify the role of medium-range order in thermal transport and mechanical responses, mainly in amorphous silicon. I will also discuss our recent efforts to establish a systematic workflow that connects topological descriptors, property prediction, identification of important features, and inverse analysis back to real-space structures, with the broader goal of using topological data analysis to understand structure–property relationships in materials.

Title: Characteristic classes of data manifolds from learned atlases

Abstract: Persistent homology identifies the space of natural image patches as a Klein bottle [1], but the Betti numbers alone do not yield coordinates on the patches that reflect this structure. However, the cohomology class can be interpreted, by representability, as a map to a classifying space, and any cocycle representing it as the gluing data of a bundle over the data. This idea underlies circular coordinates, Perea's projective coordinates, and the more recent work on vector bundles and persistent Stiefel–Whitney classes. Common to all of them is a system of transition functions, which determines both the coordinates and the characteristic classes.

In joint work with Yuichi Ike [2], we obtain the transition functions from the other end. Rather than deriving them from a cocycle, we learn the charts themselves, as an atlas of local autoencoders, and read off the transition maps, the Z/2 cocycle they define, and w1​ of the tangent bundle. A stability theorem and a certificate computable from the sample say when that reading can be trusted. The talk covers the mathematics of the passage from classes to bundles and the atlas construction, accompanied by experiments on synthetic manifolds and on real data, including the image patches and the conformation space of cyclo-octane.

[1] G. Carlsson, T. Ishkhanov, V. de Silva, A. Zomorodian. On the local behavior of spaces of natural images. Int. J. Comput. Vis. 76 (2008), 1–12.

[2] E. Paluzo-Hidalgo, Y. Ike. Learning tangent bundles and characteristic classes with autoencoder atlases. arXiv:2602.22873 (2026).

Title: Rank invariants and chain-modularity

Abstract: Persistence modules indexed by arbitrary posets are challenging objects to study. Unlike the totally ordered case, these modules do not decompose nicely into barcodes. One popular and successful approach to these modules is to instead study their rank invariant, that is, the function that to each pair of comparable poset elements associates the rank of the corresponding transition map.

These rank invariants satisfy certain inequalities related to supermodularity, a notion that initially appeared in lattice theory and combinatorial optimization. In this talk, we introduce the notion of chain-supermodularity, as well as the stricter notion of chain-modularity, which is the case where the defining inequalities are equalities. We explore the space of chain-modular functions, and determine which persistence modules' rank invariants are chain-modular.

This is the first step in an ongoing project to understand the space of rank invariants, done in collaboration with Wojciech Chachólski, Andrea Guidolin, Christina Kapatsori, Woojin Kim, and Francesca Tombari.

Title: Central limit theorems for persistent Betti numbers

Abstract: Various persistent homological invariants have been developed in recent years, raising the question of whether their probabilistic behavior can be understood in a unified way. In this talk, I will introduce a method based on homological algebra for deriving central limit theorems for persistent Betti numbers associated with \mathbb{R}-indexed chain complexes constructed from a homogeneous Poisson point process. The key point is that the translation invariance, weak stabilization, and bounded moment assumptions in the Penrose–Yukich framework can be verified by studying the kernels and cokernels of add-one maps between chain complexes. As applications, we recover the known central limit theorem for persistent Betti numbers arising from simplicial complex filtrations and establish central limit theorems for blurred magnitude homology and magnitude Betti numbers of random geometric graphs.

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Title: Computing Anisotropic Čech Filtrations: Geometry, Algorithms, and AI-Assisted Verification

Abstract: Topological data analysis is usually built from isotropic neighbourhoods, such as Euclidean balls. In anisotropic data, however, local geometry may have strongly preferred directions, suggesting that balls should be replaced by ellipsoids whose shapes reflect the local structure of the data. This leads naturally to anisotropic Čech filtrations, but also makes their computation substantially more involved.

In this talk, we discuss the computation of Čech filtrations generated by growing ellipsoids. For a simplex, its filtration value can be formulated as a convex minimax problem describing the first common intersection of the corresponding ellipsoids. We explain numerical methods for solving this problem, its relation to pairwise ellipsoid tangency, and techniques for reducing the computational cost through certified pruning and active-set information. Numerical examples illustrate how these ideas can be used to construct anisotropic persistent homology in practice.

We also discuss how AI agents and formal verification can support mathematical research and software development for TDA. Parts of the theory have been formalised in Lean 4, providing a setting in which AI-assisted mathematical development can be checked against precise formal statements. We conclude with a brief outlook on other formalisation efforts in TDA, including work on Reeb graphs.