Feichtner-Kozlov, Dmitry
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Feichtner-Kozlov, Dmitry
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Feichtner-Kozlov, Dmitry
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Kozlov, Dimitry Feichtner
Feichtner-Kozlov, Dimitry N.
Feichtner-Kozlov, Dmitrij Nikolaevič
Feichtner-Kozlov, Dmitrij
Kozlov, Dmitrij N.
Kozlov, Dmitrii Nikolaevich
Kozlov, D. N.
Feichtner-Kozlov, D.
Kozlov, Dmitrij
Kozlov, Dmitry
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Item-typ:Veröffentlichung, New invariants in topological data analysis and their applications in material and biological sciences(2025-10-29); ; Topological data analysis has proven to be a versatile and powerful enrichment for data analysis of diverse selections of data in all types of quantitative analyses. Strategies include incorporating descriptors and characterizing topological properties of datasets ranging from scalar fields, point clouds, graphs, or any other scientific dataset. Furthermore, more data-oriented combinatorial constructions can be used from which the topological properties are taken. In the following thesis, the focus is on three applications: A new combinatorial structure for describing phylogenetic networks as well as general filtered spaces, surface classification via roughness computations and its comparison to topological invariants on its scalar field, and an application for the spatial analysis of microscopic images. More specifically, the cliquegram and facegram models are established for phylogenetic models and their theoretical properties investigated, as well as their computational complexity, and algorithms for their efficient computation proposed. In addition, an approach to classify surfaces based on their roughness is presented and persistent homology is employed to extract multiscale topological features from surface data and integrate these features into machine learning models. Finally, embryonic and neural stem cells are distinguished after protein staining results in distinct spatial structures of their 3D microscopic images which are captured using cubical persistent homology and analyzed. These different approaches demonstrate the benefits of combining new topological and combinatorial methods with existing ones, providing a more comprehensive toolkit for complex network analysis. Code for reproducing and reusing the techniques is provided.Dissertation40 44 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Cosystoles and cheeger constants of the simplex(2021-06-25); ; ; The central interest of this thesis is to develop tools to get hands on the cosystolic norm and the coboundary expansion of a cochain, values which are important to determine the Cheeger constants of a simplicial complex. We develop some structural theory about the cosystolic norm of a cochain, in which we, among other small results, study an interesting connection between that norm and the hitting number of a certain set system (see Chapter 2). In Chapter 3 we restrict our research to 1- dimensional cosystoles of a simplex which are slightly easier to understand, so we can provide more explicit results for that case, including the explicit determination of the largest 1-dimensional cosystoles of a simplex and a rough insight, how all 1-cosystoles of a simplex in a certain dimension can be arranged in the so-called cosystolic complex. Furthermore, we prove the strict inequality h1(D[n]) > n3 for the case n = 16, which strengthens our conjecture that this strict inequality holds in general if n is a power of 2. In Chapter 4 we study some alternative ways to generalize the classical Cheeger constant which might be easier to access and prove that these different constants equal for a large family of simplicial complexes. In Chapter 5 we solve a beautiful combinatorial ordering problem, which is not directly related to the main subject of this thesis but arose during considerations about that and should be worth to be provided to the reader as well. In Appendix A we give an algorithm for the exact calculation of the solutions of the ordering problem from Chapter 5.Dissertation291 263 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Abstract Homotopy Theory and the Thomason Model structureThere is a closed model structure on the category of small categories, called Thomason model structure, that is Quillen equivalent to the standard model structure on the category of topological spaces. We will give an introduction to the concepts necessary to understand the definition, as well as the purpose of the Thomason model structure. These concepts include category theory, classical homotopy theory on topological spaces, simplicial homotopy theory on simplicial sets and abstract homotopy theory via the use of model categories. We will show, that there is a model structure on the category of small acyclic categories, that is Quillen equivalent to the Thomason model structure. Both of these model structures share the same cofibrant objects, and we will show that these include finite semilattices, countable trees, finite zigzags and posets with five or less elements.Dissertation534 541 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Combinatorial Topology of Quotients of Posets(2020-08-04); ; ; In this thesis we study the topology of quotients of posets. By the topology of a poset we mean the topology of its order complex called nerve in this thesis. An action of some group on a poset induces an action on its nerve. The posets we consider are partition lattices of finite sets. It is well-known that the nerve of a partition lattice is homotopy equivalent to a wedge of spheres of equal dimension. The symmetric group acts on a partition lattice in a natural way. We consider quotients of such a nerve by subgroups of the symmetric group. Especially we consider subgroups which fix at least one element. It turns out that quotients by such subgroups are also homotopy equivalent to wedges of spheres of equal dimension. Furthermore we consider sublattices of the partition lattice where certain block sizes are forbidden. For the proofs we use Discrete Morse Theory as well as Equivariant Discrete Morse Theory. We use the notion of an acyclic matching. We also develop new methods for Equivariant Discrete Morse Theory by adapting the Patchwork Theorem and poset maps with small fibers from Discrete Morse Theory. There exists an adaption of Discrete Morse Theory to free chain complexes. In this thesis we develop an adaption for the equivariant case.Dissertation504 454 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, On different witness complex filtrations and their landmark choices(2025-06-16); ; ; The main aim of this thesis is to study different Witness Complex filtrations. By the notion of Witness Complex we mean filtrations of simplicial complexes characterized by the fact that the data set they are based on is not automatically included in the complex but rather landmark points are chosen and the rest of the points is used for additional information on the higher dimensional simplices between them. We consider two constructions that are commonly referred to in literature simply by the term Witness Complex, two constructions based on the Cech and Vietoris-Rips Complexes as well as a possible construction for a Witness Complex bifiltration. First, we compare them to one another and to other known filtrations, look at methods of choosing the landmark points and search for approximations to the Vietoris-Rips Complex before introducing multiparameter filtrations. One of the main parts of this work is after that to construct a different method of acquiring landmark points that is robust with respect to outliers in the data set. This method is derived from the DBSCAN algorithm and with it we can compare one of the Witness Complex constructions to the Degree-Rips filtration which is a density-sensitive bifiltration of the classical Vietoris-Rips Complex. Another goal of this work is then to expand on some existing interleaving results for Witness Complexes to all of the different introduced constructions and especially to expand them before finally stating some additional stability results for all of the constructions which allow both the landmark points as well as the data points to shift to a certain extent.Dissertation67 328 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Selected topics in combinatorial topology and geometry : Nested set complexes, equivariant trisp closure maps, rigid Rips complexes, and vector space partitionsThis cumulative dissertation consists of four papers: Nested set complexes for posets and the Bier construction, Equivariant closure operators and trisp closure maps, Rigid Rips complexes and topological data analysis (joint work with Eva-Maria Feichtner and Dmitry Kozlov), and Some necessary conditions for vector space partitions (joint work with Olof Heden). Those got partially enhanced with some additional expository material. In the first part, we generalize the framework of combinatorial nested set complexes to the setting of posets, and demonstrate that some older proofs seamlessly generalize along, on the example of the topology of the Bier poset and the relationship between the complex of k-trees and the order complex of a certain subposet of the partition lattice. A trisp closure map is a compact certificate for collapsibility of a trisp onto a subtrisp. We discuss the relationship to closure operators (on posets) and the interaction of trisp closure maps with group operations on the trisp. We introduce Rigid Rips complexes as another filtration obtained from a finite metric space, which is sufficiently easy to calculate and has persistent homology provably different from the Vietoris-Rips filtration, showing features that are missed using the latter filtration. In the final part, we introduce a family of necessary conditions for the existence of a partition of a finite vector space into subspaces. We exploit these in the situation of a partition of a 2t-dimensional space with spaces of dimension at most t: We give bounds on the number of t-dimensional spaces in terms of the number and dimension of lower-dimensional spaces, and we remark on the relationship between the t-dimensional spaces in the partition and the same t-dimensional spaces seen as a partial t-spread. Two new constructions for vector space partitions are also given.Dissertation340 125 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Chain Complexes over Principal Ideal DomainsChain complexes are studied here as an abstract algebraic generalisation of geometric simplicial complexes. From this point of view, we extend the notions of shellability and of a cone, which are both defined for simplicial complexes, to chain complexes. We define a cone for chain complexes in an abstract way abandoning the geometrical idea of an apex and compare it with mapping cones. Indeed, there are cones which cannot be regarded as mapping cones, in contrast to the simplicial case. And conversely, we name certain conditions on which a mapping cone is a cone matching our definition. Our notion of shellability given here for chain complexes is a generalisation of this well-known term which is defined for simplicial complexes as well as for regular finite CW-complexes. But in contrast to shellable simplicial complexes, there is no information about the homology of shellable chain complexes, so we claim additional conditions on them which imitate other properties of simplicial complexes. This leads to our notions of regular and totally regular chain complexes. We obtain complete homological information for totally regular chain complexes which have a specific augmentation map. In the end, we consider mapping cones over shellable or regular chain complexes and show that they also are shellable or regular, respectively.Dissertation308 116 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, Combinatorial aspects of spatial frameworksThe rigidity or flexibilty of a skeletal structure might be investigated by asking questions about its underlying graph. While combinatorial criteria are known to determine whether a framework is rigid by knowledge about its underlying graph, provided the graph is embedded on a line or in the plane, a combinatorial criteria to determine the rigidity of spatial frameworks is not at hand. Isostatic graphs (the underlying graphs of rigid frameworks which become flexible if an arbitrary bar is removed from the framework) have several interesting properties and the problem of finding combinatorial criteria for a framework to be rigid reduces to the problem of finding combinatorial criteria for graphs to be isostatic. So called abstract rigidity matroids present a matroid approach to rigidity theory. Alternative characterizations of this family of matroids in terms of bases, circuits and hyperplanes of these matroids are presented. Afterwards, framework decompositions are introduced, which generalize 3T2-decompositions of graphs of 2-isostatic frameworks to higher dimensions. We show that it is sufficient for a graph to admit a proper spatial framework decomposition to be 3-isostatic. After discussing some properties of graphs with such decompositions, we present an algorithm that finds such proper decompositions for planar 3-isostatic graphs.Dissertation317 311 - Some of the metrics are blocked by yourconsent settings
Item-typ:Veröffentlichung, The persistence transformation; a new methodology of topological data analysis(2024-01-29); ; ; The field of Topological Data Analysis (TDA) continues to evolve as a powerful tool for the analysis of complex data. The motivation behind this research lies in the need to extend existing TDA tools to provide more accurate, efficient, and comprehensive analyses of intricate datasets. The primary research problem addressed herein pertains to the limitations of the Persistence Diagram, a fundamental TDA tool that does not inherently incorporate positional information of topological features. The absence of this crucial spatial context can lead to inaccurate results, especially when analyzing low-dimensional data. To tackle this issue, this dissertation introduces the Persistence Transformation, an innovative extension of the Persistence Diagram. It is designed to capture the positional information of topological peaks, enhancing the robustness and depth of TDA analyses. Key findings of this research include a comprehensive analysis of the properties and stability of the Persistence Tansformation. Furthermore, a real-world application of this method demonstrates its effectiveness in the classification of MALDI data, highlighting the practical utility of the extension. The originality and contribution of this work are underscored by the extension of the traditional Persistence Diagram. The introduction of the Persistence Transformation empowers mathematicians and data analysts to tackle a broader spectrum of complex problems, fostering more accurate results across diverse application domains. However, it is important to note that the Persistence Transformation generates results of a higher dimensionality when compared to the Persistence Diagram. While this enables a richer analysis of complex data, it may necessitate additional computational resources. Nevertheless, this research not only advances the domain of TDA but also opens the door to a wider array of analytical possibilities for complex datasets, offering valuable insights across various fields.Dissertation409 383