Referaty:

When studying algebraic invariants of topological spaces, such as homology groups or the fundamental group, it is common practice to replace spaces with related combinatorial structures, such as simplicial complexes or simplicial sets. These structures are both easier to analyze using combinatorial methods and possess better categorical properties (such as cartesian closedness and local presentability). Moreover, these counterparts of standard topological spaces can be endowed with forms of homotopy theories (usually formalized via the notion of model category structures) that encode different types of equivalences than usual homotopy equivalences, resembling some properties of equivalences of categories. The most notable example of such a structure is the Joyal model structure for quasicategories on simplicial sets, also known as a model structure for $\infty$-categories.

The category $pOpe_\iota$ of positive opetopes with $\iota$-contractions was introduced by Zawadowski in \cite{Z} as a tool to capture some higher categorical phenomena, particularly the homotopical behavior of CW-complexes and $(\infty, 1)$-categories and their connections to rewriting theory. It consists of certain combinatorial shapes together with rules for embedding and collapsing them. In my talk, I will define the category $\widehat{pOpe_\iota}_{EZ}$ of Eilenberg-Zilber opetopic sets, which consists of those presheaves on $pOpe_\iota$ that can be thought of as analogues of simplicial sets. I will describe a modification of Cisinski's theory, which can be applied to $\widehat{pOpe_\iota}_{EZ}$, yielding the existence of a model structure and sketch the proof that this model structure is Quillen equivalent to the Kan-Quillen model structure. If time permits, I will discuss another model structure and the possibility of comparing it with the Joyal model structure on simplicial sets.

References:

M. Zawadowski, Positive Opetopes with Contractions form a Test Category, ArXiv:1712.06033 [math.CT], (2017).

Zigzag persistence, an advanced extension of persistent homology, has emerged as a versatile tool for data analysis across various domains. This talk will explore the theoretical foundations of zigzag persistence and demonstrate its practical applications. We will begin by discussing the fundamental concepts, including its ability to provide a more flexible analysis of topological features over time.

The application of zigzag persistence in detecting Hopf bifurcations in dynamical systems will be highlighted, showcasing its capability to identify critical transitions in system behavior. Additionally, we will explore its use in temporal network analysis, where it effectively captures and analyzes evolving structures and temporal patterns. The potential of zigzag persistence in cybersecurity, particularly in detecting malicious cyber activity, will also be examined, illustrating its utility in identifying and characterizing complex patterns within cyber data.

This presentation aims to provide a comprehensive overview of zigzag persistence, demonstrating its theoretical and practical contributions to various fields and emphasizing its potential for interdisciplinary research.

References:

1. Carlsson, G., de Silva, V., Zigzag Persistence, *Foundations of Computational Mathematics*, 2010.

2. Tymochko, S., Munch, E., Khasawneh, F.A., Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems, *Algorithms*, 2020.

3. Myers, A., Muñoz, D., Khasawneh, F.A., et al., Temporal Network Analysis Using Zigzag Persistence, *EPJ Data Sci.*, 2023.

4. Myers, A., Bittner, A., Aksoy, S., et al., Malicious Cyber Activity Detection Using Zigzag Persistence., 2023 IEEE Conference on Dependable and Secure Computing (DSC), Tampa, FL, USA, 2023

Porous metals are increasingly important in technology. Due to their tunable mechanical properties, they are promising candidates in various emerging applications such as metallic scaffolds for load-bearing bones and lightweight structures for transport technologies. The aim of this study is to create topological descriptors of porous materials allowing fast prediction of their mechanical properties. At present, the main focus is on Young’s modulus. The topological properties of an object do not change when the object is rotated, while Young’s module may depend on direction. To construct direction-aware descriptors we encoded direction-dependent information in filtration values. We combine topological data analysis with theoretical models based on material's porosity for better results. In this talk, we will present new topological descriptors of porous materials and discuss the effectiveness of regression models based on them.

In this talk, we present a novel approach to understanding phase transitions through the exploration of topological properties, moving beyond the traditional Hamiltonian-based framework. Classical models such as the Ising model, percolation theory, and Erdős–Rényi random graphs serve as foundational examples for investigating how topological invariants, like Betti numbers, change during phase transitions. Our preliminary results suggest that these topological signatures not only align with the known critical points of classical systems but also offer a new framework for detecting phase transitions in systems where a Hamiltonian is either unknown or cannot be defined. We propose the concept of "topological Hamiltonians" as universal descriptors that capture the essence of phase transitions across various domains, including physics, biology, and social sciences. This approach opens up the possibility of extending phase transition theory beyond traditional one-dimensional parameter spaces to more complex, multidimensional systems. Initial calculations using percolation theory will be presented as proof of concept, with further applications to follow.

In topology, we often encounter problems of given types: we have some space, and we want to know if we could enrich our space with some additional structure, for example, the structure of a topological monoid or even a loop space of another space. It turns out that there are tools, named operads, that enable us to answer, or at least translate, topological problems into the language of operads and action of operads. Turning one problem into a new, even more abstract one doesn't look promising. Still, it turns out that we have functor from the category of operads to the category of monads, and then the action of operad on a given space X is equivalent to X having an algebra structure over the associated monad. In my talk, I will briefly describe the way, from operads to monads. Also, I will show the really important family of monads, namely the little cubes operads, which detect the structure of iterated loop spaces.

In this poster, I will present fundaments of topological deep learning, a field that tries to use topological data analysis in deep learning techniques. It is an approach that delves into the shape of data. This can be used, for example, in the design of neural networks (NNs). We look at as a combinatorial complex, a far generalization of NNs. I will also present the use of topological deep learning in, among other things, regularization, NNs pruning, or predicting model performance in classification problems.