Refinement of Interval Approximations for Fully Commutative Quivers

A fundamental challenge in multiparameter persistent homology is the absence of a complete and discrete invariant. To address this issue, we propose an enhanced framework that realizes a holistic understanding of a fully commutative quiver's representation via synthesizing interpretations obtained from intervals. Additionally, it provides a mechanism to tune the balance between approximation resolution and computational complexity. This framework is evaluated on commutative ladders of both finite-type and infinite-type. For the former, we discover an efficient method for the indecomposable decomposition leveraging solely one-parameter persistent homology. For the latter, we introduce a new invariant that reveals persistence in the second parameter by connecting two standard persistence diagrams using interval approximations. We subsequently present several models for constructing commutative ladder filtrations, offering fresh insights into random filtrations and demonstrating our toolkit's effectiveness in analyzing the topology of materials

完全可換クイバーの区間近似とその応用

京都大学新制・課程博士博士(理学)甲第25087号理博第4994号京都大学大学院理学研究科数学・数理解析専攻(主査)教授 平岡 裕章, 教授 COLLINSBenoit Vincent Pierre, 教授 坂上 貴之学位規則第4条第1項該当Doctor of ScienceKyoto UniversityDFA

Derived binomial rings I: integral Betti cohomology of log schemes

We introduce and study a derived version $\mathbf L\mathrm{Bin}$ of the binomial monad on the unbounded derived category $\mathscr D(\mathbb Z)$ of $\mathbb Z$-modules. This monad acts naturally on singular cohomology of any topological space, and does so more efficiently than the more classical monad $\mathbf L\mathrm{Sym}_{\mathbb Z}$. We compute all free derived binomial rings on abelian groups concentrated in a single degree, in particular identifying $C_*^{\mathrm{sing}}(K(\mathbb Z,n),\mathbb Z)$ with $\mathbf L\mathrm{Bin}(\mathbb Z[-n])$ via a different argument than in works of To\"en and Horel. Using this we show that the singular cohomology functor $C_*^{\mathrm{sing}}(-,\mathbb Z)$ induces a fully faithful embedding of the category of connected nilpotent spaces of finite type to the category of derived binomial rings. We then also define a version $\mathbf L \mathcal Bin_X$ of the derived binomial monad on the $\infty$-category of $\mathscr D(\mathbb Z)$-valued sheaves on a sufficiently nice topological space $X$. As an application we give a closed formula for the singular cohomology of an fs log complex analytic space $(X,\mathcal M)$: namely we identify the pushforward $R\pi_*\underline{\mathbb Z}$ for the corresponding Kato-Nakayama space $\pi\colon X^{\mathrm{log}}\rightarrow X$ with the free coaugmented derived binomial ring on the 2-term exponential complex $\mathcal O_X\rightarrow \mathcal M^{\mathrm{gr}}$. This gives an extension of Steenbrink's formula and its generalization by the second author to $\mathbb Z$-coefficients.Comment: 61 pages, comments welcom

Approximation methods in geometry and topology: learning, coarsening, and sampling

Data materialize in many different forms and formats. These can be continuous or discrete, from algebraic expressions to unstructured pointclouds and highly structured graphs and simplicial complexes. Their sheer volume and plethora of different modalities used to manipulate and understand them highlight the need for expressive abstractions and approximations, enabling novel insights and efficiency. Geometry and topology provide powerful and intuitive frameworks for modelling structure, form, and connectivity. Acting as a multi-focal lens, they enable inspection and manipulation at different levels of detail, from global discriminant features to local intricate characteristics. However, these fundamentally algebraic theories do not scale well in the digital world. Adjusting topology and geometry to the computational setting is a non-trivial task, adhering to the “no free lunch” adage. The necessary discretizations can be inaccurate, the underlying combinatorial structures can grow unmanageably in size, and computing salient topological and geometric features can become computationally taxing. Approximations are a necessity when theory cannot accommodate for efficient algorithms. This thesis explores different approaches to simplifying computations pertaining to geometry and topology via approximations. Our methods contribute to the approximation of topological features on discrete domains, and employ geometry and topology to efficiently guide discretizations and approximations. This line of work fits un der the umbrella of Topological Data Analysis (TDA) and Discrete Geometry, which aim to bridge the continuous algebraic mindset with the discrete. We construct topological and geometric approximation methods operating on three different levels. We approximate topological features on discrete combinatorial spaces; we approximate the combinatorial spaces themselves; and we guide processes that allow us to discretize domains via sampling. With our Dist2Cycle model we learn geometric manifestations of topological features, the “optimal” homology generating cycles. This is achieved by a novel simplicial complex neural network that exploits the kernel of Hodge Laplacian operators to localize concise homology generators. Compression of meshes and arbitrary simplicial complexes is made possible by our general spectral coarsening strategy. Functional and structural properties are preserved by optimizing for important eigenspaces of general differential operators, the Hodge Laplacians, at multiple dimensions. Finally, we offer a geometry-driven sampling strategy for data accumulation and stochastic integration. By employing the kd-tree geometric partitioning algorithm we construct a sample set with provable equidistribution guarantees. Our findings are contextualized within prior and recent work, and our methods are thoroughly discussed and evaluated on diverse settings. Ultimately, we are making a claim towards the usefulness of examining the ever-present topological and geometric properties of data, not only in terms of feature discovery, but also as informed generation, manipulation, and simplification tools

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

On the Categorical Theory of Persistence Modules

¿Cómo se infieren formas a partir de datos? Se pueden emplear técnicas del álgebra y la topología computacional para responder a esta pregunta, dando lugar al análisis topológico de datos (TDA), un campo que experimenta actualmente un gran crecimiento. La homología persistente es una herramienta clave en TDA que computa características topológicas de un espacio en diferentes resoluciones espaciales a partir de la construcción de complejos simpliciales y su caracterización. Esta herramienta tiene aplicaciones en varias áreas de la matemáticas aplicada y una base teórica sólida formalizada en el marco de la teoría de categorías. En esta tesis hacemos una revisión de la teoría en torno a uno de los pilares centrales de la homología persistente, los módulos de persistencia. Examinamos la descomposición y comparación de estos y teoremas de estabilidad relacionados. En este proceso, nos centramos en buscar el enfoque matemático más útil para expresar estas ideas, que quedan cohesionados por la teoría de categorías. Además de la literatura revisada, también aportamos una serie de contribuciones propias en materia de una construcción particular de módulos de persistencia denominados módulos escalera (ladder modules). Asimismo, presentamos este trabajo de forma lo más autocontenida posible, para que sirva de introducción fluida tanto a esta vertiente de la teoría de categorías como a los conceptos de homología persistente aquí empleados.How is shape infered from data? Algebraic and topological techniques can be employed computationally to answer this question, giving birth to the fast growing field of topological data analysis (TDA). Persistent homology is a key tool in TDA that computes topological features of a space at different spatial resolutions from the construction of simplicial complexes and their characterization. It has both a broad applicability in various areas of applied mathematics and a strong theoretical base that has been formalized in the framework of category theory. In this thesis we review the theory around one of persistent homology’s key elements: persistence modules. We examine their decomposition and comparison and related stability theorems. Along the way, we focus on finding the most useful mathematical language to convey these ideas, which end up being glued by category theory. In addition to the reviewed literature on persistent homology, we also add some contributions of our own regarding ladder modules, a particular construction of persistence modules. Furthermore, we present this work in a self-contained way, to serve as a painless introduction both to this side of category theory and to persistent homology.Universidad de Sevilla. Máster Universitario en Matemática

Topological Data Analysis

International audienceIt has been observed since a long time that data are often carrying interesting topological and geometric structures. Characterizing such structures and providing efficient tools to infer and exploit them is a challenging problem that asks for new mathematics and that is motivated by a real need from applications. This paper is an introduction to Topological Data Analysis (), a new field that emerged during the last two decades with the objective of understanding and exploiting the topological structure of modern and complex data. The paper surveys some important mathematical and algorithmic developments in as well as software solutions that are currently used to address various applied and industrial problems