# Seminars:

The Centre is running a virtual seminar on Wednesdays, 12:15 am via Zoom. The Meeting-ID is 390 769 7241. Please contact Paweł Dłotko to obtain the passcode.

If you would like to give a talk as part of our seminar series please contact Paweł Dłotko.

If you are interested in research talks by members of the center, see below.

### Upcoming seminars

- 14/06/2023, 12:15

Dominika Jedynak (UJ): Hodge Laplacian on simplicial complexes and its application in trajectory clustering.

**Abstract:**In my presentation, I will introduce the Laplacian operator for graphs and the Hodge-Laplacian operator for complexes. I will discuss what valuable information these operators provide about signal on corresponding structures. Furthermore, I will explore Hodge Decomposition of signals on complexes and demonstrate how it can be used to obtain embeddings of trajectories on a complex. The presentation will be followed by some illustrative examples of results of above and other methods for trajectory clustering on complexes. The talk will be based on “Signal processing on higher order networks: Living on the edge… and beyond” written by Michael T. Schaub, Yu Zhu, Jean-Baptiste Seby, T. Mitchell Roddenberry and Santiago Segarra. - 21/06/2023, 12:15

tba

**Abstract:**tba

### Past seminars

- 07/06/2023, 12:15

Soumik Dutta (MIMUW): On Strong Collapse of Random Simplicial Complexes

**Abstract:**The strong collapse of a simplicial complex, proposed by Barmak and Minian, is a combinatorial collapse of a complex onto its sub-complex. Recently, it has received attention from computational topology researchers, owing to its empirically observed usefulness in simplification and size-reduction of the size of simplicial complexes while preserving the homotopy class. We consider the strong collapse process on random simplicial complexes. For the Erd ̋os-R ́enyi random clique complex X(n,c/n) on n vertices with edge probability c/n with c > 1, we show that after t many iterations of maximal strong collapsing phases the remaining subcomplex, or t-core, has (1 − γ_{t+1} − cγ_t + cγ_t^2)n non-isolated vertices asymptotically almost surely (a.a.s.), where γ_t is recursively determined by γ_{m+1} = e^{−c(1−γ_m)} and γ_0 = 0. - 31/05/2023, 12:15

Paweł Pilarczyk (Gdansk University of Technology): Computation of the Conley index using cubical grids

**Abstract:**Conley index is a topological invariant that is a generalization of the Morse index and provides qualitative information about an isolated invariant set in a (semi)dynamical system. Effective computation of its homological version for specific dynamical systems (with both discrete and continuous time) is possible with the use of the Computational Homology Project (CHomP) C++ software library https://www.pawelpilarczyk.com/chomp/. Actual applications of this approach prove its practical usefulness. In this talk, I am going to introduce some algorithms behind the software, and I am going to explain selected theoretical foundations. In particular, I am going to discuss two stages of discretization aimed at converting a continuous-time dynamical system into a finite object represented by means of a directed graph on a cubical grid in R^n. Additionally, I am going to show how one can effectively construct index pairs in this context. - 17/05/2023, 12:15

Tomasz Kaczynski: Morse theory for multi-filtrations: smooth and discrete.

**Abstract:**In this talk, I will present joint efforts to develop an analogy of Morse Theory for functions with values in R^k, k>1, in the context of multi-filtered persistent homology. In [1], a Forman-like multidimensional discrete Morse function is defined with the purpose of the matching algorithm for reduction of the underlying complex. The extension of the theory was partial and missing some geometric insight. It was pointed out in [1] that an appropriate application-driven extension of the Morse theory to multi-filtrations for smooth functions was not much investigated yet, and it would help in understanding the discrete analogy. My joint work [2] is a step in that direction and the main part of my talk. We describe the evolution of bi-filtrations in terms of cellular attachments. A concept of persistence path is introduced, analogies of Morse-Conley equation and Morse inequalities along persistence paths are derived. A scheme for computing path-wise barcodes is proposed. At the end, I will summarise main results of the joint work [3], which is a completion of the work done in [1], inspired by smooth analogies from [2].

[1] M. Allili, T. Kaczynski, C. Landi, and F. Masoni, Acyclic partial matchings for multidimensional persistence: algorithm and combinatorial interpretation, J Math Imaging Vis (61) (2019) 174-192, DOI 10.1007/s10851-018-0843-8.

[2] R. Budney and T. Kaczynski, Bi-filtrations and persistence paths for 2-Morse functions, arXiv:2110.08227 [math.AT] Oct 2021, to appear in Algebraic & Geometric Topology.

[3] M. Allili, G. Brouillette, and T. Kaczynski, Multidimensional discrete Morse theory, arXiv:2212.02424 [math.GT] Dec 2022. - 10/05/2023, 12:15

R.U. Gobithaasan: Exploring Malaysian Rainfall Time Series Patterns with Topological Data Analysis: A Case Study in Terengganu

**Abstract:**Due to its equatorial location, Malaysia has a tropical climate characterized by sunshine and rainstorms throughout the year. Identifying patterns in the Malaysian rainfall time series (RTS) can be a challenging task, especially with the impact of climate change. Recently, several reports have demonstrated the success of Topological Data Analysis (TDA) in extracting meaningful insights from incomplete, noisy, and high-dimensional datasets. In this study, we utilized two TDA methodologies, Persistent Homology (PH) and the Mapper algorithm, to analyze the RTS collected from 66 rainfall stations in Terengganu, Malaysia. Firstly, we employed Dynamic Time Warping (DTW) and PH for classification and clustering tasks and compared the results with satellite gridded data. Secondly, we used Mapper to create a spatiotemporal network that revealed the connection between RTS and interannual and annual events causing floods in Terengganu, Malaysia. This case study is a work-in-progress identifying the advantage of TDA methodologies for analyzing time series datasets and, hence providing a foundation for informed decision-making. - 26/04/2023, 12:15

David Hien (TU Munich): Topological Signatures for Analyzing Oscillations in Time Series Data

**Abstract:**Nonlinear dynamical systems often exhibit rich and complicated recurrent dynamics. Understanding these dynamics is challenging, especially in higher dimensions where visualization is limited. Additionally, in many applications, time series data is all that is available. This motivates our TDA-based approach to study such systems. More precisely, we introduce the cycling signature which is constructed by taking persistent homology of time series segments in a suitable ambient space. Oscillations in a time series can then be identified by analyzing the cycling signatures of its segments. We demonstrate this through several examples. In particular, we identify and analyze 6 oscillations in a 4d system of ordinary differential equations. - 19/04/2023, 12:15

Jacek Cyranka (UW): Contractibility of a persistence map preimage

**Abstract:**I will introduce our result on the contractibility of the persistence map preimage from our recent work with K. Mischaikow and Ch. Weibel https://arxiv.org/abs/1810.12447 . Such result is applied in answering the following question in dynamics: given a dynamical system that is observed via a time series of persistence diagrams that encode topological features of solutions snapshots, what conclusions can be drawn about solutions of the dynamical system? Also, I will present some recent generalizations of the result by other researchers and discuss future research. - 05/04/2023

Group meeting in Bedlewo - 12/04/2023, 12:15

Waclaw Marzantowicz: The covering type = a homotopy invariant useful also in the study of the problem of minimal triangulation

**Abstract:**In 2017 M. Karoubi and Ch. Weibel introduced a notion of “the covering type” ${\rm ct}(X)$ of a space $X$ as the minimal cardinality of a good cover of $X$ minimized over all $Y$ which are homotopy equivalent to $X$. They showed a few fundamental properties of this invariant. In particular it estimates from below the number of vertices of any triangulation $K$ of $X$. Later, other authors in a couple of papers presented next properties of this invariant, in particular giving an its estimate from below by the maximal weighted length of a multiply in the cohomology ring of $X$. This let to get estimates of the minimal number of vertices which are necessary to triangulate a space including such spaces for which this problem was not studied by direct combinatorial methods. However, there are also other attempts to study the problem of estimate the number of vertices. They are based on a computer assisted proofs which derive of some homotopy invariants of a complex provide the number of vertices is small. We also present briefly one of them. - 22/03/2023, 12:15

Mateusz Przybylski (Jagiellonian University): Topological methods in dynamics reconstructed from the data.

**Abstract:**In this talk, I would like to present an approach to sampled dynamics through index theory. Dynamics reconstructed from the data can be represented by a multivalued map. There is well-developed Conley theory for such maps. This theory has slightly relaxed conditions imposed on the multivalued map than previous approaches. These weaker conditions allow us to apply index methods to broader class of multivalued maps (i.e. dynamics generators). Moreover, the theory allows to obtain richer dynamical phenomena (e.g. chaotic behavior) without increasing the resolution of the representation. The results will be presented on a well understood theoretical model and on data from a physical experiment. - 01/03/2023, 12:15

Lukasz Michalak (AMU): Reeb graph invariants of Morse functions, manifolds and groups

**Abstract:**The Reeb graph of a smooth function on a closed manifold is obtained by contracting each connected component of a level set of the function. There are two necessary and sufficient conditions for a finite graph to be realized as the Reeb graph of a smooth function with finitely many critical points: it needs to have the so-called good orientation and its first Betti number cannot exceed the corank of the fundamental group of a given closed manifold. Moreover, any free quotient of this group can be represented as the Reeb epimorphism of a Morse function which is induced on fundamental groups by the quotient map from the manifold to the Reeb graph. However, the realization of a graph as the Reeb graph of a Morse function is possible only up to a homeomorphism of graphs in general. The minimum number of degree 2 vertices in Reeb graphs of Morse functions is a strong invariant of the topology of manifold. It has three essentially different lower bounds in terms of the fundamental group, homology groups and Lusternik-Schnirelmann category. In the case of orientable 3-manifolds all of them can be improved by the inequality involving the Heegaard genus, and there is also another lower bound by a new invariant defined in terms of finite presentations of the fundamental group. We use Freiheitssatz, a fundamental fact from one-relator groups, to calculate it in some cases. The equalities in these bounds are closely related with the problem of finding a function such that the first Betti number of its Reeb graph is equal to corank. It is a one of potential geometric methods of calculating the corank, which is quite a complicated task in practise. - 25/01/2023, 12:15

Daniel Wojcik (Nencki Institute): Current sources of electrical activity in the brain: modeling and reconstruction

**Abstract:**Recordings of extracellular electric potential on the scalp or within the brain are commonly used in brain studies. Since the electric field is long range every recording reflects activity of multiple sources often quite remote from the recording site. In my talk I will present some of our efforts to better understand the activity behind the recordings using source reconstruction methods, combining them with independent component analysis and supporting by modeling. These results will serve me as a canvas to discuss our strategy of brain data analysis and model-based validation of such methods. I will also indicate some challenges which I think are worth addressing and describe some of our publicly available modeling data which may be used for validation of different methods of analysis of multimodal brain data. - 01/02/2023, 12:15

Julian Brüggemann (MPIM Bonn): Discrete Morse Functions and Merge Trees

**Abstract:**Discrete Morse Theory is a versatile tool to analyze regular CW complexes and, in particular, simplicial complexes in many contexts. We focus on the sublevel filtration induced by a discrete Morse function and on the persistent connectivity of said filtration. The combinatorial data of persistent connectivity can be tracked by the induced merge tree. In this talk, I will give a short introduction to merge trees and discuss the case of discrete Morse functions on multigraphs. For that case, I will discuss sublevel-symmetry equivalences and component-merge equivalences under which the induced merge tree turns out to be invariant. These notions of equivalence lead to a classification result for discrete Morse functions on multigraphs: Any discrete Morse function on a tree is up to matched cells, symmetry equivalences, and component-merge equivalences uniquely determined by the isomorphism class of its induced Morse-labeled merge tree. If time permits, we take a look at under which conditions this result can be restricted to the case of simple graphs and how the induced merge tree can be used to find cancellations of critical cells. - 08/02/2023, 16 (!)

Irina Gelbukh (Centro de Investigación en Computación of the Instituto Politécnico Nacional at Mexico City): Reeb graphs of arbitrary smooth functions (not necessarily Morse)

**Abstract:**Despite its name, the Reeb graph of a smooth function may not be a graph, it may even be a non-Hausdorff or non-one-dimensional topological space; I will give some examples. However, for a sufficiently wide class of functions, namely, smooth functions with a finite number of critical values defined on a closed manifold, the Reeb graph is a finite graph. In this case, one can study such functions using graph theory. We will consider graph-theoretic properties that characterize the Reeb graphs defined by functions of a given class (say, Morse or Morse-Bott), on a given manifold. In particular, can a given graph be the Reeb graph of a Morse function or a function of a given class, or even some smooth function? We will discuss these and some similar problems. That is, properties of the Reeb graph encode important information on the function and the manifold. - 11/01/2023, 12:15

Jose Carlos (CIMAT): Foams/2-stratifolds, 3-manifolds and TDA.

**Abstract:**I will define what we mean by Foams and 2-stratifolds. These spaces appear in several applications. Then, I will relate these spaces with the study of 3-manifolds. Finally, I will indicate how TDA could be used to study 3-manifolds. All this is joint work with F. Gonzalez-Acuña (UNAM), W. Heil (FSU) and J. Frias (CIMAT). - 13/12/2022, 14:15(!)

Leonid Fedorov (MPI Tuebingen): Structure in free viewing of natural scenes

**Abstract:**When one looks at a picture, for e.g. a natural scene, eye-movement trajectories appear random, even upon repeated viewing, and vary from subject to subject. Previous studies of the oculomotor system have shown that much of these eye movements are ballistic jumps between fixation points. Around every fixation point, part of a viewed scene is a signal transmitted through the fovea, which is then transformed and processed further along the visual system. Specific transformations of this raw signal are often modeled in various ways depending on part of the visual system. Further, a full set of these signals is often analyzed using some metric. The models, the metric, the eye movement trajectories, and a set of image patches have been subjects of prior research. Here we show that given a set of such signals from any viewing, and assuming that some metric (any metric) would be used in analysis, there always exists a partition of this set. Assuming a point-wise transformation on the set of signals, every subset forms an order complex, which is invariant under this transformation. Therefore, eye movements in free viewing can be thought of as a collection of order complexes, with vertices being the signals, irrespective of how diverse the eye movement trajectories appears in data. We then provide a deterministic algorithm to calculate such a partition and a data structure that shows how distance data is organized, when partitioned. We discuss that either a finite number of partitions can be calculated based on number of fixations, or further modeling assumptions need to be made for the partition to also be unique. The implementation of the algorithm and an example on data are in progress. - 23/11/2022

Rafał Topolnicki (Dioscuri TDA/ Wrocław): Topology-Driven Goodness-of-Fit Tests in Arbitrary Dimensions

**Abstract:**In the talk a novel application of topology to the statistical hypotesis testing is presented. The method we developed adopts the Euler characteristic curve (ECC) to one- and two-sample goodness of fit tests. The presented tests, which we call TopoTest, work for samples in arbitrary dimension, having often higher power than the state of the art statistical tests. It is demonstrated that the type I error of TopoTests can be controlled and their type II error vanishes exponentially with increasing sample size. Results of extensive numerical simulations of TopoTest are shown to demonstrate its power. Joint work with Paweł Dłotko, Niklas Hellmer, Łukasz Stettner. - 09/11/2022

Anna-Laura Sattelberger (MPI MIS/ KTH Stockholm): An Algebraic Invariant of Multiparameter Persistence Modules

**Abstract:**In the one-parameter case, persistence modules naturally are graded modules over the univariate polynomial ring and hence perfectly understood from an algebraic point of view. By a classical structure theorem, one associates the so-called “barcode” to the module, from which one reads topological features of the data. Generalizing persistent homology to a multi-filtration and its homology allows for the extraction of finer information from data, but its algebraic properties are more subtle. In this talk, I discuss the shift-dimension, a stable invariant of multiparameter persistence modules which is obtained as the hierarchical stabilization of a zeroth total multigraded Betti number of the module. This talk is based on work with Wojciech Chachólski and René Corbet (arXiv preprint) - 26/10/2022

Benjamin Brück (ETHZ): tba

**Abstract:**tba - 19/10/2022 3:15 pm

Radmila Sazdanovic

**Abstract:**tba - 12/10/2022

Aldi Hoxha: Simulation and measurements of a rotary inductive position sensor

**Abstract:**PCB-based inductive position sensors (IPS) are a valid alternative in the measurement of the position in all those applications where the sensor has to work in harsh environmental conditions. The main components of the sensor are the stator, which contains two receivers and a transmitter, and the conductive target, which moves at a predefined air gap from the stator. Having established within the region where the moving conductor relies a known induced magnetic field, eddy currents are generated inside it, as the Faraday-Newmann law predicts. The receivers are able to encode this motion in electrical signals, thus providing the position of the target. The aim of the presentation is to show the working principle of the sensor, and how to solve numerically the eddy currents problem on the target in order predict the signals on the receivers. - 05/10/2022 3:15 pm

Alexander D Smith: Data analysis via Topology and Geometry with Applications in Chemical Engineering

**Abstract:**This talk is focused on the development and application of topological and geometrical methods for the analysis of chemical engineering data. Topology and geometry allow us to view data as a shape and provide us tools to quantify this shape. Abstracting data as a shape captures intrinsic characteristics of the data that are independent of the environment and methods used to obtain the data. These representations (e.g., graphs, manifolds, and point clouds) also provide means for integrating domain knowledge into data analysis that can strengthen connections between theory and experiment. We present a thorough review of applied topology and geometry methods in chemical engineering through applications on real datasets such as: molecular dynamics simulations, dynamical systems, process systems, and soft matter systems. - 21/09/2022

Rob Scharein: Introducing KnotPlot (IMPAN R. 106)

**Abstract:**The speaker will introduce his software KnotPlot, a tool for mathematicians, biologists, physicists and others to explore computational knot theory. A brief review of knot theory will be made and example applications will be shown. Also knots as artistic and cultural entities will be discussed. - 01/06/2022

Matt Burfitt (Aberdeen): Topological approaches to the analysis of Fast Field-Cycling MRI images

**Abstract:**Fast Field-Cycling MRI (FFC MRI) has the potential to recover new biomarkers form a range of diseases by scanning a number of low magnetic field strengths simultaneously. The images produced by an FFC scanner can be interpreted in the form of a sequence times series of 2-dimensional grey scale images with each times series corresponding to each of the magnetic field strengths. I will investigate the applications of topology and machine learning to brain stroke images obtained by the FFC MRI scanner. A main obstacle to achieving good results lies in multiplicative brightness errors occurring in the data. A simple solution might be to consider pixelwise image feature vectors initially only up to multiplication by a constant. This can be thought of as splitting the data point cloud within a product by first embedding into standard n-simplices. We observe that this point cloud embedding can provide good information on tissue types which when modelled against the other component of the data point cloud can be used to highlight stroke damaged tissue. A drawback of the first method is that it discarded the pixel spatial locations information of the image. However, this can be captured by persistent homology in a parameter choice free process. A direct comparison between pixel intensity histograms and the Betti curves reveals that homology captures and emphasises tissue signals. Moreover, addition geometric and topological information about the images is captured with persistent homology, in particular by considering changes between images at differing fields and times. From here the ultimately aim is to extract persistent homology features useful for machine leaning and develop new visual diagnostics for the FFC data. - 06/06/2022, 10-12 (!), tbc

Yonatan Gutman: Optimal Embedding of Topological Dynamical Systems

**Abstract:**Given a metrizable space $X$ of dimension $d$ it is a classical fact that the minimal $n$ which guarantees that $X$ may be embedded topologically in $[0,1]^n$, is $n=2d+1$.An analogous problem in the category of dynamical systems, is under what conditions one can guarantee that a topological dynamical system $(X,T)$ embeds in $([0,1]^n)^\mathbb{Z}$ under the shift action. This problem witnessed a rapid development in the last 20 years starting with Gromov's introduction of the invariant of mean dimension (1999) and the breakthrough results of Lindenstrauss and Weiss (2000). We will present our joint result with Tusukamoto concerning optimal embedding of minimal systems (2020) whose proof offers a surprising connection to signal analysis. - 25/05/2022

Bernd Sturmfels (MPI MiS, Leipzig): Geometry of Dependency Equilibria

Abstract: An n-person game is specified by n tensors of the same format. Its equilibria are points in that tensor space. Dependency equilibria satisfy linear constraints on conditional probabilities. These cut out the Spohn variety, named after the philosopher who introduced the concept. Nash equilibria are tensors of rank one. We discuss the real algebraic geometry of the Spohn variety and its payoff map, with emphasis on connections to oriented matroids and algebraic statistics. - 11/05/2022

Justyna Signerska-Rynkowska: Embedding theorems and various notions of entropy in time-series analysis

**Abstract:**I will start with presenting Takens' embedding theorem and its various generalizations. Next, I will discuss different notions of entropy commonly used in time-series analysis such as permutation entropy, sample entropy, approximation entropy, and directional entropy. We will see how these ideas can be applied for determinism testing and reconstruction of the dynamics of the underlying dynamical system. - 04/05/2022, 4:15 pm (!)

Michał Lipiński: Morse pre-decomposition of an invariant set

**Abstract:**An invariant set is a central object in the study of dynamical systems. We can reveal the internal structure of the set by decomposing it into gradient and recurrent parts. In particular, the recurrent parts (stationary points, periodic orbits, etc.) are sealed together with gradient-like trajectories. As a result, we get a Morse decomposition, a collection of isolated invariant sets (Morse sets) partially ordered by connecting trajectories. In this talk, I will present some results concerning the extension of the idea of Morse decomposition. The refined construction will allow studying the internal structure of recurrent parts of the decomposition. It is achieved by replacing the partial order with a preorder. - 27/04/2022

Federica Buccino: Micro-cracks and lacunae in human bones: a debated mechanobiological link

**Abstract:**The increase in fragility fractures and the related dramatic impact on the healthcare and financial system, raise a big red flag, especially in the approach to the clinical treatment. With the rise in life expectancy, the prevalence of chronic conditions, such as osteoporosis, is also set to escalate. All these contributing factors highlight the need and urgency to address the fragility fracture crisis with a critical eye on the intricate multi-scale arrangement of bone structure. Indeed, despite the evident health- and economic-related interests about bone fracture, its comprehension is still limited to macro-(50 cm-10 mm) and meso-scale (10 mm-500 µm) level, where its occurrence is however catastrophic. It is still unclear what befalls at the micro-scale (500 µm -1 µm) from a mechanobiological point of view and what is the role of microscopic bone porosities, named lacunae, during the damage processes. This is mainly due to the complexity of the parameters to be considered when dealing with bone fractures, such as bone's structural and morphological features and their cross-relationships with damage initiation and propagation. More in depth, there exists a dualism behind the role of lacunae: they seem to play an antithetical mechanical contribution, having an effect on both strength and toughness. For these reasons, recent studies address the issue from a multi-scale perspective, elevating the micro-scale phenomena as the key for detecting early damage occurrence. However, several limitations arise specifically for defining a quantitative framework to assess the contribution of lacunar micro-pores to fracture initiation and propagation. Moreover, the need for high resolution imaging imposes time-demanding post-processing phases. To overcome these issues, we exploit synchrotron scans in combination with micro-mechanical tests, to offer a fracture mechanics-based approach for quantifying the critical stress intensification in healthy and osteoporotic trabecular human bones. This is paired with a morphological and densitometric framework for capturing lacunar network differences in presence of pathological alterations. To address the current time-consuming and computationally expensive manual/semi-automatic segmenting steps, we implement convolutional neural network to detect the initiation and propagation of micro-scale damages. The results highlight the intimate cross talks between toughening and weakening phenomena at micro-scale as a fundamental aspect for fracture prevention. - 20/04/2022

Bartosz Naskręcki: Crystallographic tesselations and growth polynomials

**Abstract:**We consider the Euclidean space R^N with a CW-tesselation which is transformed regularly by a crystallographic group. Such a group is naturally associated with a flat orbifold, obtained from the N-dimensional torus under the action of a certain finite group. I will explain the relation of this orbifold with the polynomials that count the number of cells in our tesselation treated as an expanding CW-complex. The natural choice of the vertex of the tesselation leads to growth functions which are polynomials of degree N for every fixed cell dimension. The leading terms of these growth polynomials form a vector which up to constant characterizes the flat tesselation orbifold. I will also explain the role of the lower order terms in these polynomials based on some calculations with fundamental domains of the crystallographic group. In practice, this leads to a simple concrete algorithm which allows one to recognize the crystallographic group from a partial data about tesselation. - 06/04/2022

Daniel Wilczak: The method of covering relations in (in)finite dimension.

**Abstract:**The method of covering relations [1,2] proved to be a very efficient topological tool in qualitative analysis of finite dimensional dynamical systems. It has been used in validation of the existence of various types of dynamics, including chaotic dynamics, connecting orbits, even some specific global bifurcations. In this talk I will show that the method can be adopted to a class of infinite dimensional problems. The main motivation for this research was the rigorous numerical analysis of dissipative PDEs. We have shown the existence of chaos [3] and infinite number of connecting orbits [4] between periodic solutions in one-dimensional Kuramoto-Shivashinsky PDE. Presented results are based on two articles [3,4] written with Piotr Zgliczyński.

[1] P. Zgliczyński, Computer assisted proof of chaos in the Rossler equations and the Henon map, Nonlinearity {\bf 10} (1997), 243-252.

[2] P. Zgliczyński, M. Gidea, Covering relations for multidimensional dynamical systems, J. Diff. Eq., 202/1(2004), 33--58.

[3] D. Wilczak, P. Zgliczyński, A geometric method for infinite-dimensional chaos: symbolic dynamics for the Kuramoto-Sivashinsky PDE on the line, J. Diff. Eq., Vol. 269 No. 10 (2020), 8509-8548.

[4] D. Wilczak, P. Zgliczyński, A rigorous C1-algorithm for integration of dissipative PDEs based on automatic differentiation and the Taylor method, in preparation. - 28/03/2022, 9 am Purdue = 3 pm Warsaw

Tamal Dey (Purdue): Computing Generalized Rank Invariant for 2-parameter Persistence Modules via Zigzag Persistence

**Abstract:**The notion of generalized rank invariant in the context of multiparameter persistence has become an important ingredient for defining interesting homological structures such as generalized persistence diagrams. Naturally, computing these rank invariants efficiently is a prelude to computing any of these derived structures efficiently. We show that the generalized rank over a finite interval I of a Z^2-indexed persistence module M is equal to the generalized rank of the zigzag module that is induced on a certain path in I tracing mostly its boundary. Hence, we can compute the generalized rank over I by computing the barcode of the zigzag module obtained by restricting the bifiltration inducing M to that path. If I has t points, this computation takes O(t^\omega) time where \omega\in[2,2.373) is the exponent of matrix multiplication. Among others, we apply this result to obtain an improved algorithm for the following problem. Given a bifiltration inducing a module M, determine whether M is interval decomposable and, if so, compute all intervals supporting its summands.

Joint work with: Woojin Kim and Facundo Memoli - 18/03/2022, 10 am

Witold Rudnicki: All-relevant feature selection in Machine Learning: methods and applications.

**Abstract:**In the talk I will present short overview of general approaches to feature selection and explain the need for the all-relevant feature selection. I will give short motivation for the need of the feature selection methods that take into account synergistic interactions. Then I will describe two algorithms that I have proposed, Boruta and MDFS. Finally I will show some examples of real life applications and discuss caveats and limitations of these methods. - 14/03/2022, 8:30-10:30

Joint workshop with Lodz University - 02/03/2022

Davide Gurnari: ToMATo Clustering

**Abstract:**Topological Mode Analysis Tool is a clustering algorithm that exploits topological persistence to guide the merging of candidate clusters. After a brief overview of the most famous clustering schemes, I will present the original paper by Chazal et al. and provide additional examples using current off-the-shelf implementations. - 23/02/2022

Anastasios Stefanou (Uni Bremen): Multiparameter persistence meets Gröbner bases.

**Abstract:**Ordinary persistent homology extends from one parameter to multiparameter filtrations. The resulting algebraic methodology is called multiparameter persistent homology. Although multiparameter persistence modules of finite type admit a unique decomposition into indecomposables, the indecomposable modules do not always have the structure of a generalized interval module. Thus, to study these multigraded module structures one has to use alternative algebraic tools to present them. One such tool is called minimal free resolutions. A known result in homological algebra is that any $N^n$-graded module admits a unique minimal free resolution (up to a non-canonical notion of equivalence). Although these resolutions are not unique for the module in the standard sense, there are ways on which one can utilize the theory of Gröbner bases to explicitly construct a minimal free resolution for the persistent module of a given multifiltration. The output minimal free resolution depends on (i) choosing a monomial order in the polynomial ring R:=K[x1,\ldots,x_n] (e.g. the lexicographic order) and (ii) on other things, e.g. choosing an ordering on the simplices of the input multi-filtered complex. In this talk I will give an introduction on Gröbner bases over R^l, and discuss how Gröbner bases can be useful in multiparameter persistence. - 16/02/2022

Kunal Dutta (University of Warsaw): On Dimensionality Reduction for Persistent Homology

**Abstract:**Given a point cloud in a high-dimensional space, we are interested in computing the persistent homology of the Cech filtration while avoiding the curse of dimensionality, and investigate the effectiveness of dimensionality reduction for this problem. This line of work was initiated by Sheehy [SoCG 2014] who showed that any map preserving pairwise distances between points up to a $(1\pm \varepsilon)$ factor, also preserves the Cech filtration up to a $(1 \pm 4\varepsilon)$ factor. We'll follow the work of Sheehy [2014] and Lotz [Proc. Roy. Soc., 2019], before arriving at the result of [Arya, Boissonnat, D., Lotz, J. App. Comp. Topol., 2021], which answers an open question of Sheehy regarding dimensionality reduction for persistent homology computed using the \emph{$k$-distance}, or \emph{distance to a measure}. We show that any linear transformation that preserves pairwise distances up to a $(1 \pm \varepsilon)$ multiplicative factor, must preserve the persistent homology of the Čech filtration, computed using the $k$-distance, up to a factor of $(1 - \varepsilon)^{-1}$. Our results also hold for the Vietoris-Rips and Delaunay filtrations for the $k$-distance, as well as the Čech filtration for the approximate $k$-distance of Buchet et al. [Comput. Geom., 2016]. We'll also look at several extensions of this result, for point sets (i) lying in a region of bounded Gaussian width or (ii) on a low-dimensional manifold, obtaining the target dimension bounds of Lotz [Proc. Roy. Soc., 2019] and Clarkson [Proc. SoCG, 2008 ] respectively. Joint work with S. Arya, J.-D. Boissonnat, and M. Lotz. - 19/01/2022

Wojciech Chachólski (KTH Stockholm): Why should one care about metrics on (multi) persistent modules?

**Abstract:**What do we use metrics on persistent modules for? Is it only to asure stability of some constructions? In my talk I will describe why I care about such metrics, show how to construct a rich space of them and illustrate how to use them for analysis. - 05/01/2022 and 14(!)/01/2022

Justyna Signerska-Rynkowska: Entropy and entropy-like quantities in time series analysis

**Abstract:**In the first part of the talk I will present the concepts of topological and metric entropy in dynamical systems theory and relations between them. Having introduced the mathematical background, we will explore these notions and their various counterparts applied to time series analysis, namely approximate entropy, sample entropy, permutation entropy and transfer entropy used, among others, in heart-rate variability analysis, EEG research, cluster analysis or testing determinism. - 08/12/2021 and 15/12/2022

Michał Lipiński: An introduction to the selected topics on analysis of a time series.

**Abstract:**During this working seminar I will present some aspects, classic ideas and approaches in analysis of a time series. The talk is based on Nonlinear Time Series Analysis handbook by H.Kantz and T.Schreiber (2004). - 01/12/2021

Workshop on TDA and Medical Data in Gdansk, no seminar

- 24/11/2021

Niklas Hellmer: Bottleneck Profiles and Prokhorov Metrics for Persistence Modules

**Abstract:**In TDA, Bottleneck and interleaving distances and their variants are common tools to compare persistence diagrams or modules. However, they only capture the single, most extreme difference. We introduce bottleneck profiles, a way to study all the differences at varying scales. Algebraically speaking, this construction can be thought of measuring how far two persistence modules are from being interleaved. In addition, one easily obtains a rich family of (extended pseudo)metrics for persistence modules, mimicking the classical Prokhorov metric from probability theory. As an application, we describe proxies for the Wasserstein distance, which allow for fast computations. - 18/11/2021, 4 pm (note the unusual time and date)

Simon Rudkin (Swansea): Hidden in Plain Sight: The Real Messages of Economic Data

**Abstract:**Recognising the powerful messages that may be drawn from the shape of data, we ask how inference on well-studied datasets in economics and finance may be enriched through topological data analysis. Examples considered include the application of ball mapper algorithms in credit scoring, regional growth trajectories, analysis of stock returns and to the political economy of Brexit. In each case we demonstrate how anomaly phenomenon are better understood when viewed through the multidimensional lens; policy and practice then being more effectively directed as a consequence. Discussing the ways in which developments of theory may support research agendas across economics and finance, we signpost where the tools of TDA themselves can be developed to embolden the synergies between disciplines. Consequentially, we signpost an exciting research agenda that further reveals the information that an inability to meaningfully capture the multidimensional nature of data had veiled for so long. - 03/11/2021, 2:30 pm

Wolfram Bosbach: TDA in medical engineering

**Abstract:**Medical research can greatly benefit from advanced mathematical methods. There are patterns of higher order in human anatomy which are not intuitively recognisable by the human eye. Trabecular bone, neuronal networks in the brain, or lung bronchi exhibit interconnectivities which allow analyses about their physiological or pathological state. Numerical finite elements can extend pattern analyses and give insights into mechanical tissue behaviour. Radiology is the medical field which can provide 3-dimensional imaging of anatomical body parts. Funded by the DFG, we are now working towards applying topological data analysis (TDA) to existing and new classification problems.

Funding: Deutsche Forschungsgemeinschaft (DFG) grant no-BO-4961/11 - 03/11/2021

Davide Gurnari: Distributed algorithms for Euler Characteristic Curves

**Abstract:**The Euler characteristic of a simplicial complex is the alternate sum of its Betti number, or equivalently the alternating sum of the number of simplices of following dimensions. For a filtered complex the Euler Characteristic Curve is a function that assigns an Euler number to each filtration level. ECCs are closely related to persistence diagrams via the fundamental theorem of persistent homology and stability with respect to the 1-Wasserstein distance can be proven under certain conditions. The first part of this talk we will introduce the concept of ECC and prove its stability with respect to the 1-Wasserstein distance. In the second part we will present new techniques to compute the ECC of filtered Vietoris-Rips complexes. By following a distributed approach, the contributions to the ECC can be computed locally without having to explicitly build up the whole complex. This allows us to significatively reduce both time and memory requirements, giving us the opportunity to tackle much larger datasets compared to, for instance, persistent homology. An implementation of such algorithms is available as a scikit-learn-compatible Python package. Finally, we will discuss how such ideas and algorithms can be adapted to work in the multiparameter persistence setting. - 27/10/2021

Michał Lipiński: Morse-Conley-Forman theory for generalized combinatorial multivector fields on finite topological spaces

**Abstract:**In the talk, I will present a generalization of the theory of multivector fields which can be seen as a combinatorial counterpart of vector fields induced by continuous dynamical systems. The theory has its roots in the theory of combinatorial vector fields by Robin Forman and was later extended by Marian Mrozek to multivector fields for Lefschetz complexes. Our generalization involves three fundamental changes, which will be explained during the talk. Consequently, the theory becomes more flexible and handy in applications. Moreover, we can introduce a new interpretation of a multivector as a dynamical "black box." With the new settings, we define combinatorial counterparts of multiple objects from the classical theory of dynamical systems, among others: isolated invariant set, index pair, Conley index, limit set, attractor, or Morse decomposition. We also show that the desirable properties as additivity of a Conley index and Morse inequalities hold. Furthermore, we use persistent homology to study the robustness of the structure of Morse sets. In particular, we construct a zigzag persistence module for Morse decomposition for multivector fields. Finally, we will present some numerical experiments based on the presented theory. - 20/10/2021

Jan F. Senge: Topological descriptors for surface roughness of shot peened surfaces

**Abstract:**Surface topography and roughness evaluation plays an important role in many problems such as friction, contact deformation or coat adhesion. Impact surface treatments can be used to improve the properties of the treated surface in the cases mentioned. Different roughness parameters like the arithmetic average height deviation, root mean square height or skewness are used to describe the topography of a surface and constitute the most used features to describe changes occurred during impact treatment. Depending on the surface treatment process considered, standard roughness parameters may struggle to exhibit differences between surface samples of the same material at different stages of the process. As an extension the work done to improve existing methods and find new descriptors, we consider methods from computational topology, namely persistent homology to capture meaningful geometric features of the surface and use that to construct new surface roughness parameters better able to distinguish surface samples. Furthermore, invariants such as the persistence diagram have been studied extensively in the last years and provide a rich structure which can be exploited using a plethora of different machine learning techniques. In the talk, I give an introduction into existing standards for surface roughness measurement and compare how well persistence-based parameters perform for classification and clustering tasks in the case of the shot peening surface treatment process opposed to conventional roughness parameters mentioned in these standards. - 13/10/2021, 5 pm

Anastasios Stefanou: Persistent Homology of Phylogenetic Networks

**Abstract:**We devise a hierarchical representation of phylogenetic networks over a set of taxa X, called the cliquegram, which generalizes the dendrogram representation of phylogenetic tree ultrametrics. We show that cliquegrams form a lattice. Furthermore, we show that the cliquegram lattice model is a faithful model for solving the Phylogenetic Network Reconstruction Problem (PNRP): Given a set of trees there is a unique minimal network that contains them. Moreover, we develop a zigzag persistent homology theory for cliquegrams relative to a choice of a taxon x in X, called taxa-decorated persistent homology. Generalizations of these ideas (i.e. the cluttergram) from the setting of networks to the setting of filtrations are also developed. The algorithms implementing those constructions are discussed and their implementations are provided in Python. - 02/12/2020

Artem Dudko: On computability and computational complexity of Julia sets.

**Abstract:**In this talk I will give an introduction to computability and computational complexity of sets with the focus on Julia sets of polynomials. For a polynomial p(z) its Julia set is the set of points on a complex plane near which the iterations of p(z) behave chaotically. Roughly speaking, a set is called computable if there is an algorithm which can produce an approximation of this set (by a finite set of pixels) with any given precision. Computational complexity measures how hard it is to produce such approximations. In the talk I will show that already in the class of Julia sets of quadratic polynomials there are various interesting computational phenomena, including:

- large classes of functions having Julia sets of low (polynomial) computational complexity;

- functions with non-computable Julia sets;

- large classes of functions having Julia sets of arbitrarily high computational complexity. - 25/11/2020

John Harvey (Swansea University): Estimation of dimension

**Abstract:**I will survey various established and new techniques to estimate the dimension of a submanifold of Euclidean space from a sampled point cloud, how we might investigate underlying assumptions of the estimators, and how they relate to testing for manifoldness - 18/11/2020

Paweł Dłotko (Dioscuri TDA, Warsaw): Mild introduction to TDA

**Abstract:**In this talk I will provide a general introduction to methods of TDA - concentrated mostly on persistent homology and mapper algorithm. I will present the basic theory, highlight the main application. But, I will mostly encourage questions and discussion. - 11/11/2020

**No seminar due to public holiday!** - 04/11/2020

Niklas Hellmer (Dioscuri TDA, Warsaw): Nonembeddability of Persistence Diagrams with p>2 Wasserstein Metric.

**Abstract:**To apply kernel methods on persistence diagrams, they are mapped into a Hilbert space. The question is how much information we lose by doing so. I present recent work by Alexander Wagner that shows that persistence diagrams with p-Wasserstein metric do not admit a coarse embedding into a Hilbert space for p>2.

### Talks of center members in other seminars

- 26/02/2021

Niklas Hellmer: "Discrete Prokhorov Metric for Persistence Diagrams" at APATG.