Title: Polynomial approximation of one parameter families of (un)stable manifolds (II)

Abstract: I will talk about methods of parametrization of stable and unstable manifolds associated with a branch of hyperbolic fixed points or equilibria in a one parameter family of analytic dynamical systems. I will show how to approximate the invariant manifolds using polynomials and walk through the examples on Henon mapping and Lorenz system. An illustrative example will be shown. The talk will be based on the article "Polynomial approximation of one parameter families of (un)stable manifolds with rigorous computer assisted error bounds" by J.D. Mireles James, Florida Atlantic University, 2014.

Title: Metody parametryzacji torusów niezmienniczych.   

Abstract: Przedmiotem prezentacji będzie przedstawianie (bazując na rozdziale 3 książki „Jordi-Llu ́ıs Figueras, Alex Haro, The parameterization method for invariant manifolds”) metod parametryzacji torusów niezmienniczych. Główną część będzie stanowiło szczegółowe omówienie metody redukcji (Reducibility Method). Zostaną także zaprezentowany przykłady, w jaki sposób przy jej użyciu wyznaczyć całą rodziny takich torusów. 

Title: Tri-partitions and Bases of an Ordered Complex. (III)

Abstract: We will talk about polyhedral complexes. We will show that given polyhedral complex K and an ordering of p-cells, for every dimension p there exist a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. These partitions, called tri-partitions, are unique for a given ordering of the p-cells. We will present how this construction follows from a matrix reduction algorithm and show some examples of tri-partitions and canonical bases in Mathematica. 

This talk is based on the article Tri-Partitions and Bases of an Ordered Complex by H. Edelsbrunner, K. Ölsböck. arXiv:2103.10830 

Title: Polynomial approximation of one parameter families of (un)stable manifolds

Abstract: I will talk about methods of parametrization of stable and unstable manifolds associated with a branch of hyperbolic fixed points or equilibria in a one parameter family of analytic dynamical systems. I will show how to approximate the invariant manifolds using polynomials and walk through the examples on Henon mapping and Lorenz system. An illustrative example will be shown. The talk will be based on the article "Polynomial approximation of one parameter families of (un)stable manifolds with rigorous computer assisted error bounds" by J.D. Mireles James, Florida Atlantic University, 2014.

Title: Tri-partitions and Bases of an Ordered Complex. (II)

Abstract: We will talk about polyhedral complexes. We will show that given polyhedral complex K and an ordering of p-cells, for every dimension p there exist a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. These partitions, called tri-partitions, are unique for a given ordering of the p-cells. We will present how this construction follows from a matrix reduction algorithm and show some examples of tri-partitions and canonical bases in Mathematica. 

This talk is based on the article Tri-Partitions and Bases of an Ordered Complex by H. Edelsbrunner, K. Ölsböck. arXiv:2103.10830 

Title: Tri-partitions and Bases of an Ordered Complex. (I)

Abstract: We will talk about polyhedral complexes. We will show that given polyhedral complex K and an ordering of p-cells, for every dimension p there exist a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. These partitions, called tri-partitions, are unique for a given ordering of the p-cells. We will present how this construction follows from a matrix reduction algorithm and show some examples of tri-partitions and canonical bases in Mathematica. 

This talk is based on the article Tri-Partitions and Bases of an Ordered Complex by H. Edelsbrunner, K. Ölsböck. arXiv:2103.10830 

Title: Computation and analysis of Jupiter-Europa and Jupiter-Ganymede resonant orbits in the planar concentric circular restricted 4-body problem.

Abstract: Many unstable periodic orbits of the planar circular restricted 3-body problem (PCRTBP) persist as invariant tori when a periodic forcing is added to the equations of motion. In their study, Bhanu Kumar, Rodney L. Anderson, Rafael de la Llave and Brian Gunter, compute tori corresponding to exterior Jupiter-Europa and interior Jupiter-Ganymede PCRTBP resonant periodic orbits in a concentric circular restricted 4-body problem (CCR4BP). Motivated by the 2:1 Laplace resonance between Europa and Ganymede’s orbits, the authors then attempt the continuation of a Jupiter-Europa 3:4 resonant orbit from the CCR4BP into the Jupiter-Ganymede PCRTBP.

Title: Morse-based Fibering of the Persistence Rank Invariant   

Abstract:  Although there is no doubt that multi-parameter persistent homology is a useful tool to analyze multivariate data, efficient ways to compute these modules are still lacking in the available topological data analysis toolboxes. Other issues such as interpretation and visualization of the output remain difficult to solve. One of the simplest invariants for a multi-parameter persistence module is its rank invariant, defined as the function that counts the number of linearly independent homology classes that live in the filtration through a given pair of values of the multi-parameter. We show how discrete Morse theory may be used to compute the rank invariant, proving that it is completely determined by its values at points whose coordinates are critical with respect to a discrete Morse gradient vector field. These critical points partition the set of all lines of positive slope in the parameter space into equivalence classes, such that the rank invariant along lines in the same class are also equivalent. We show that we can deduce all persistence diagrams of the restrictions to the lines in a given class from the persistence diagram of the restriction to a representative in that class. An algorithm that chooses a representative line for each equivalence class in the case of 2-parameter persistence modules is presented.

This is joint work with  Asilata Bapat,  Robyn Brooks,  Celia Hacker, and Barbara I. Mahler.

Title:

Localization and the Szymczak category 

Abstract:

Localization of a category is a process of associating with the category and given collection of morphisms S a certain category and a functor F which makes each morphism in S an isomorphism. Moreover, the process involves the determination of a unique functor through which functor F factorizes. In this context, the Szymczak functor and category, which are tools used to construct the Conley index for discrete-time dynamical systems, can be seen as a localization of some category. In this talk, I will recall the notion of localization and compare it with Szymczak's construction in the setting of the category of finite sets and relations as morphisms.  

Title: Connection matrices in combinatorial dynamics (IV)

Abstract: Connection matrices constitute a generalization of Morse Theory to the setting of non-necessarily gradient dynamics. They may be viewed as a dynamically oriented homology theory. In the talk I will first recall the main ideas of the classical theory together with an example showing that the classical approach cannot be extended to multivalued dynamics. This is an obstacle in combinatorial dynamics which is inherently multivalued. I will continue by showing how this problem may be resolved. 

Title: Connection matrices in combinatorial dynamics (III)

Abstract: Connection matrices constitute a generalization of Morse Theory to the setting of non-necessarily gradient dynamics. They may be viewed as a dynamically oriented homology theory. In the talk I will first recall the main ideas of the classical theory together with an example showing that the classical approach cannot be extended to multivalued dynamics. This is an obstacle in combinatorial dynamics which is inherently multivalued. I will continue by showing how this problem may be resolved. 

Title: Connection matrices in combinatorial dynamics (II)

Abstract: Connection matrices constitute a generalization of Morse Theory to the setting of non-necessarily gradient dynamics. They may be viewed as a dynamically oriented homology theory. In the talk I will first recall the main ideas of the classical theory together with an example showing that the classical approach cannot be extended to multivalued dynamics. This is an obstacle in combinatorial dynamics which is inherently multivalued. I will continue by showing how this problem may be resolved. 

Title: Connection matrices in combinatorial dynamics

Abstract: Connection matrices constitute a generalization of Morse Theory to the setting of non-necessarily gradient dynamics. They may be viewed as a dynamically oriented homology theory. In the talk I will first recall the main ideas of the classical theory together with an example showing that the classical approach cannot be extended to multivalued dynamics. This is an obstacle in combinatorial dynamics which is inherently multivalued. I will continue by showing how this problem may be resolved. 

Title: Bi-Filtrations for 2-Morse Functions

Abstract: We study the homotopy type of bi-filtrations of compact manifolds $M$ induced as the pre-image of filtrations the plane for generic smooth functions $f : M \to \R^2$. Our primary goal is to provide a simple geometric description of the multi-graded persistent homology associated with such filtrations. The main result is a description of the evolution of the bi-filtration of $f$ 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. This is a joint work with Ryan Budney.

Title: 

Inner and outer enclosures for discrete and continuous dynamical systems via ellipsoid approach. 

Abstract: 

The talk is based on several articles and preprints by Andreas Rauh, Luc Jaulin, Auguste Bourgois. The authors propose a new algorithm for computation of inner and outer enclosures for solutions to initial value problems for ODEs. The method applies to discrete dynamical systems, as well. The test provided by the author shows superiority of the method over the available algorithms, including those implemented in the CAPD library. I'd like to revise these tests.

Title:

Morse pre-decomposition 

Abstract: 

In this talk I will introduce the concept of Morse pre-decomposition, a

generalization of Morse decomopsition, obtained by weakening the

requirement of an partial order of Morse sets to a preorder. The

construction allows for an additional description of an internal

structure of a recurrent Morse set. I will prove that a Morse

decomposition is indeed a special case of a Morese pre-decomposition.

Work in progress. 

Title:

Wasserstein Stability for Persistence Modules 

Abstract: 

In this talk, I will introduce a result on Wasserstein stability for persistence modules. Wasserstein or optimal transport distances have been used in applications of TDA for some amount of time but have been missing a stability proof.  I will present an elementary proof in the case of finite complexes, which also provides a simple proof of bottleneck stability, followed by a discussion of applications and generalizations. In particular, I will discuss the algebraic version of stability as well as discuss open questions. This is joint work with Katharine Turner. 

Title:

Languages of higher dimensional automata and Kleene theorem

(joint work with U. Fahrenberg, C. Johansen, G. Struth) 

Abstract: 

Higher dimensional automata (HDA) are a formalism for modeling

concurrent systems introduced by Pratt and van Glabbeek. I will present

several ways to define languages of HDA, which are collections of

interval pomsets closed under subsumption. Then I will define regular

interval pomset languages and formulate Kleene theorem for HDA. Finally,

I will sketch its proof, which employs some ideas derived from topology

and geometry. 

Title:

An Introduction to the concept of complete Lyapunov functions. 

Abstract: 

Ordinary differential equations arise in a variety of applications, and they can exhibit highly complicated dynamical behaviour. Complete Lyapunov functions capture this behaviour by dividing the phase space into two disjoint sets: the chain-recurrent part and the transient part. If a complete Lyapunov function is known for a dynamical system the qualitative behaviour of the system’s solutions is transparent to a large degree. The computation of a complete Lyapunov function for a given system is, however, a very hard task. In this talk we will try to introduce a method for computing complete Lyapunov functions. This talk is based on a paper on a paper of Carlos Argáez, Peter Giesl and Sigurdur Freyr Hafstein published in 2018 "Computational Approach for Complete Lyapunov Functions".

Title:

An Introduction to the concept of complete Lyapuniv functions. 

Abstract: 

Ordinary differential equations arise in a variety of applications, and they can exhibit highly complicated dynamical behaviour. Complete Lyapunov functions capture this behaviour by dividing the phase space into two disjoint sets: the chain-recurrent part and the transient part. If a complete Lyapunov function is known for a dynamical system the qualitative behaviour of the system’s solutions is transparent to a large degree. The computation of a complete Lyapunov function for a given system is, however, a very hard task. In this talk we will try to introduce a method for computing complete Lyapunov functions. This talk is based on a paper on a paper of Carlos Argáez, Peter Giesl and Sigurdur Freyr Hafstein published in 2018 "Computational Approach for Complete Lyapunov Functions".

Bogdan Batko (Jagiellonian University, Kraków, Poland)

Title:

Gaussian process-based surrogate models in dynamics. (III)

Abstract: 

We present a novel approach to the study of dynamical systems known only from data. The method allows for strict probabilistic conclusions about the global dynamics on the basis of finite samples.

Identification of the space from data, recently a subject of intensive study, is only part of the challenge of understanding dynamics; we also need to capture the behavior of the nonlinear map that generates the dynamics. The development of a theory for that step is in its early stages.  The common technique is a discretization of the space, which leads to a multivalued map. The advantage is that the homological computations are feasible given time and memory constraints. But there are also disadvantages. On one hand, if the grid is too grainy then we wastefully use the data and the multivalued representation gives a very coarse description of dynamics. On the other hand, if the grid is too fine then the domain of the multivalued model becomes a collection of isolated cubes given data amount constrains, and it cannot properly approximate the phase space which is a continuum.

As a remedy, in our approach we adapt Gaussian process theory to the needs of sampled dynamics. Our heuristic is that the accuracy of well fitted Gaussian process surrogate model can meet the robustness of topological tools. The significance of our approach is that we do not need to directly identify the underlying map. Moreover, we can quantify confidence of our results.

Talk based on joint work with M. Gameiro, Y. Hung, W. Kalies, K. Mischaikow and E. Vieira 

Bogdan Batko (Jagiellonian University, Kraków, Poland)

Title:

Gaussian process-based surrogate models in dynamics. (II)

Abstract: 

We present a novel approach to the study of dynamical systems known only from data. The method allows for strict probabilistic conclusions about the global dynamics on the basis of finite samples.

Identification of the space from data, recently a subject of intensive study, is only part of the challenge of understanding dynamics; we also need to capture the behavior of the nonlinear map that generates the dynamics. The development of a theory for that step is in its early stages.  The common technique is a discretization of the space, which leads to a multivalued map. The advantage is that the homological computations are feasible given time and memory constraints. But there are also disadvantages. On one hand, if the grid is too grainy then we wastefully use the data and the multivalued representation gives a very coarse description of dynamics. On the other hand, if the grid is too fine then the domain of the multivalued model becomes a collection of isolated cubes given data amount constrains, and it cannot properly approximate the phase space which is a continuum.

As a remedy, in our approach we adapt Gaussian process theory to the needs of sampled dynamics. Our heuristic is that the accuracy of well fitted Gaussian process surrogate model can meet the robustness of topological tools. The significance of our approach is that we do not need to directly identify the underlying map. Moreover, we can quantify confidence of our results.

Talk based on joint work with M. Gameiro, Y. Hung, W. Kalies, K. Mischaikow and E. Vieira 

Bogdan Batko (Jagiellonian University, Kraków, Poland)

Title:

Gaussian process-based surrogate models in dynamics. 

Abstract: 

We present a novel approach to the study of dynamical systems known only from data. The method allows for strict probabilistic conclusions about the global dynamics on the basis of finite samples.

Identification of the space from data, recently a subject of intensive study, is only part of the challenge of understanding dynamics; we also need to capture the behavior of the nonlinear map that generates the dynamics. The development of a theory for that step is in its early stages.  The common technique is a discretization of the space, which leads to a multivalued map. The advantage is that the homological computations are feasible given time and memory constraints. But there are also disadvantages. On one hand, if the grid is too grainy then we wastefully use the data and the multivalued representation gives a very coarse description of dynamics. On the other hand, if the grid is too fine then the domain of the multivalued model becomes a collection of isolated cubes given data amount constrains, and it cannot properly approximate the phase space which is a continuum.

As a remedy, in our approach we adapt Gaussian process theory to the needs of sampled dynamics. Our heuristic is that the accuracy of well fitted Gaussian process surrogate model can meet the robustness of topological tools. The significance of our approach is that we do not need to directly identify the underlying map. Moreover, we can quantify confidence of our results.

Talk based on joint work with M. Gameiro, Y. Hung, W. Kalies, K. Mischaikow and E. Vieira 

Daniel Wilczak (Jagiellonian University, Kraków, Poland)

Title:

TBA

Abstract: 

TBA

Title: The level set method applied to structural optimization. (II)

Abstract: We are going to show how to optimize a functional, dependent on the solution of the boundary value problem, by modifying the domain on which it is defined. We will represent a domain implicitly, using a level-set function. Then we embrace to derive a transport equation, which describes the domain deformation by a vector field that is the descent direction for a given functional. Next, we discuss two methods of solving the transport equation numerically.

[1] Gregoire Allaire, Francois Jouve and Anca-Maria Toader (2004), Structural optimization using sensitivity analysis and a level-set method, Journal of Computational Physics, 194: 363-393 https://doi.org/10.1016/j.jcp.2003.09.032

[2] Wang, S. and Wang, M.Y. (2006), Radial basis functions and level set method for structural topology optimization. Int. J. Numer. Meth. Engng., 65: 2060-2090. https://doi.org/10.1002/nme.1536.