Title: Trapping regions and an ODE-type proof of the existence and uniqueness theorem for Navier-Stokes equations with periodic boundary conditions on the plane. 

Abstract: I will talk about the incompressible Navier-Stokes equation with periodic boundary conditions. I will present a proof for the existence and uniqueness of solutions in the plane using the so called “trapping regions” and discuss the limitations of the approach in the case of R^3. 

The talk will be based on “Trapping regions and an ODE-type proof of the existence and uniqueness theorem for Navier-Stokes equations with periodic boundary conditions on the plane” by Piotr Zgliczyński.

Title: 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 Mitchael T. Schaub, Yu Zhu, Jean-Baptiste Seby, T. Mitchell Roddenberry and Santiago Segarra.

Title: 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 algorithmfor 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.

Title: Bifurcation points induced by cyclic symmetries 

Abstract: Diblock copolymers are a class of materials formed by the reaction of two linear polymers. The different structures taken on by these polymers grant them special properties, which can prove useful in applications such as the development of new adhesives and asphalt additives. We consider a model for the formation of diblock copolymers first proposed by Ohta and Kawasaki, which is a Cahn-Hilliard-like equation together with a nonlocal term. Unlike the Cahn-Hilliard model, even on one-dimensional spatial domains the steady state bifurcation diagram of the Ohta-Kawasaki model is still not fully understood. We therefore present computer-assisted proof techniques which can be used to validate and continue its bifurcation points. This includes not only fold points, but also pitchfork bifurcations which are the result of a cyclic group action beyond forcing through $\Z_2$ symmetries.

Title: Conley Complexes for Parameterized ODE Systems 

Abstract: In this talk, we explore the challenges faced in analyzing time-varying systems with multi-scale dynamics. While traditional methods model these systems using ordinary differential equations (ODE), the direct analysis of such models is often difficult due to poorly measured parameters and numerous variables. To overcome these challenges, we propose a novel approach based on combinatorics and algebraic topology. We move away from classical ODE analysis and focus toward a more robust, scalable, and computable description of global dynamics in terms of annotated graphs (Morse graphs) and Conley complexes. 

Title: Variational integrators for stochastic dissipative Hamiltonian systems.

Abstract: Variational integrators are derived for structure-preserving simulation of stochastic forced Hamiltonian systems. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and considering a stochastic generalization of the deterministic Lagrange-d’Alembert principle. Our approach presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators satisfy a discrete version of the stochastic Lagrange-d’Alembert principle, and in the presence of symmetries, they also satisfy a discrete counterpart of Noether’s theorem. Furthermore, mean-square and weak Lagrange-d’Alembert Runge-Kutta methods are proposed and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to non-geometric methods. The Vlasov-Fokker-Planck equation is considered as one of the numerical test cases, and a new geometric approach to collisional kinetic plasmas is presented.

Title: Global dynamics for steep sigmoidal nonlinearities in two dimensions (II)

Abstract: I will present the article „Global dynamics for steep sigmoidal nonlinearities in two dimensions” written by T. Gedeon, S. Harker, H. Kokubu, K. Mischaikow and H. Oka. Its focus is the construction of mathematical, combinatorial model of dynamics given by biological structures known as regulatory networks. In the first seminar I will present motivation and goals. It will be followed by remarks on some basic concepts (lattice, Morse sets, multivalued maps) and lastly I will be focusing on building two concepts: switching system and associated continuous switching system. 

Title: Randomized algorithms approximating solutionsof ordinary differential equations.

joint work with Maciej Goćwin, Paweł M. Morkisz and Paweł Przybyłowicz (Faculty of Applied Mathematics, AGH University of Science and Technology, Kraków)

Abstract: We will discuss error bounds and optimality (in the Information-Based Complexity sense) forselected randomized schemes approximating solutions of ODEs.  In particular, we will investigate randomized  Euler  schemes  (explicit  and  implicit)  and  the  randomized  midpoint  scheme  under inexact information and mild assumptions about the right-hand side function (local Hölder and Lipschitz continuity in time and space variables, respectively), cf.  [3, 4, 5, 7].  Furthermore, we will discuss the concept of exceptional set for randomized schemes for ODEs [6] and we will establish an upper bound on the probability of exceptional set for a particular class of randomized methods, including Euler and midpoint schemes, under inexact information [2, 3].  We will report the results of numerical experiments which illustrate theoretical findings discussed in this talk.  Finally, we will discuss the probabilistic A-stability of randomized Taylor schemes [1, 3, 4].

References:

[1]  T. Bochacik. A  note  on  the  probabilistic  stability  of  randomized  Taylor  schemes. ElectronicTransactions on Numerical Analysis, 58, 101–114, 2023.

[2]  T. Bochacik. On the properties of the exceptional set for the randomized Euler and Runge-Kutta schemes. Advances in Computational Mathematics, 2023 (accepted).

[3]  T. Bochacik. Randomized algorithms approximating solutions of ordinary differential equations. PhD Thesis, 2023+ (in preparation).

[4]  T.  Bochacik,  M.  Goćwin,  P.  M.  Morkisz,  and  P.  Przybyłowicz. Randomized  Runge-Kutta method  –  Stability  and  convergence  under  inexact  information. Journal  of  Complexity,  65,101554, 2021.

[5]  T. Bochacik and P. Przybyłowicz. On the randomized Euler schemes for ODEs under inexact information. Numerical Algorithms, 91, 1205–1229, 2022.

[6]  S.  Heinrich  and  B.  Milla. The  randomized  complexity  of  initial  value  problems. Journal  of Complexity, 24, 77–88, 2008.

[7]  R.  Kruse  and  Y.  Wu. Error  analysis  of  randomized  Runge-Kutta  methods  for  differential equations  with  time-irregular  coefficients. Computational Methods in Applied Mathematics,17, 479–498, 2017.

Title: Global dynamics for steep sigmoidal nonlinearities in two dimensions (I)

Abstract: I will present the article „Global dynamics for steep sigmoidal nonlinearities in two dimensions” written by T. Gedeon, S. Harker, H. Kokubu, K. Mischaikow and H. Oka. Its focus is the construction of mathematical, combinatorial model of dynamics given by biological structures known as regulatory networks. In the first seminar I will present motivation and goals. It will be followed by remarks on some basic concepts (lattice, Morse sets, multivalued maps) and lastly I will be focusing on building two concepts: switching system and associated continuous switching system. 

Title: An Introduction to the concept of Vietoris-like maps and Vietoris-like multivalued maps. (III)

Abstract: Vietoris-like maps and Vietoris-like multivalued maps are an adaptation of Vietoris maps, to the setting of finite T0 topological spaces. They have the advantage that they induce morphism in homology and are a generalization of Strongly Upper SemiContinuous maps with acyclic value. During our talk, we will present some recent results about those special maps.

This talk is based on a preprint of Pedro J Chocano,  "Coincidence theorems for finite topological spaces".

Title: An Introduction to the concept of Vietoris-like maps and Vietoris-like multivalued maps. (II)

Abstract: Vietoris-like maps and Vietoris-like multivalued maps are an adaptation of Vietoris maps, to the setting of finite T0 topological spaces. They have the advantage that they induce morphism in homology and are a generalization of Strongly Upper SemiContinuous maps with acyclic value. During our talk, we will present some recent results about those special maps.

This talk is based on a preprint of Pedro J Chocano,  "Coincidence theorems for finite topological spaces".

Title: An Introduction to the concept of Vietoris-like maps and Vietoris-like multivalued maps. (I)

Abstract: Vietoris-like maps and Vietoris-like multivalued maps are an adaptation of Vietoris maps, to the setting of finite T0 topological spaces. They have the advantage that they induce morphism in homology and are a generalization of Strongly Upper SemiContinuous maps with acyclic value. During our talk, we will present some recent results about those special maps.

This talk is based on a preprint of Pedro J Chocano,  "Coincidence theorems for finite topological spaces".

Title: Applied and computational topology in DioscuriTDA Centre - a brief overview 

Abstract: In this talk I will present a selection of topics we are working on in the Dioscuri Centre in TDA at IMPAN. Among others, I will mention the new goodness of fit tests, referred to as TopoTests, discuss their properties and show how they advance the state of the art hypothesis testing in statistics. I will mention new, knot theory inspired, developments in mapper type algorithms, showing how they apply to a wide range of topics. Last but not least I will discuss distributed algorithms to compute Euler characteristic curves and profiles; show their stability and applicability in a number of practical problems.

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.

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

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.

Title: Chain Representations for Attractor Networks (III)

Abstract: Following a paper by Joel W. Robbin and Dietmar A. Salamon, we will construct a connection matrix for an attractor network of a dynamical system both in continuous and discrete time. We will define it using a chain representation constructed through the direct limit functor, especially in the discrete time case. We will also examine, whether it is possible to replace the direct limit functor with a different normal functor. 

Title: Chain Representations for Attractor Networks (II)

Abstract: Following a paper by Joel W. Robbin and Dietmar A. Salamon, we will construct a connection matrix for an attractor network of a dynamical system both in continuous and discrete time. We will define it using a chain representation constructed through the direct limit functor, especially in the discrete time case. We will also examine, whether it is possible to replace the direct limit functor with a different normal functor. 

Title: Chain Representations for Attractor Networks (I)

Abstract: Following a paper by Joel W. Robbin and Dietmar A. Salamon, we will construct a connection matrix for an attractor network of a dynamical system both in continuous and discrete time. We will define it using a chain representation constructed through the direct limit functor, especially in the discrete time case. We will also examine, whether it is possible to replace the direct limit functor with a different normal functor. 

Title: Efficient and Reliable Algorithms for the Computation of Non-Twist Invariant Circles. (II)

Abstract: The presentation is about methodology of non-twist invariant circles and their bifurcations for area preserving maps, which give sufficient conditions guaranteeing the existence of a true non-twist invariant circle, provided an approximate invariant circle is known. Hence, one can compute confidently even very close to breakdown.Non-twist invariant circles are characterized not only by being invariant, but also by having some specified normal behavior. The methodology leads to efficient algorithms to compute and continue, with respect to parameters, non-twist invariant circles.Seminar focus on present defintions of invariant rational circles with non-twist characterization and algorithm to compute non-twist invariant circles.

Title: Efficient and Reliable Algorithms for the Computation of Non-Twist Invariant Circles. (I)

Abstract: The presentation is about methodology of non-twist invariant circles and their bifurcations for area preserving maps, which give sufficient conditions guaranteeing the existence of a true non-twist invariant circle, provided an approximate invariant circle is known. Hence, one can compute confidently even very close to breakdown.Non-twist invariant circles are characterized not only by being invariant, but also by having some specified normal behavior. The methodology leads to efficient algorithms to compute and continue, with respect to parameters, non-twist invariant circles.Seminar focus on present defintions of invariant rational circles with non-twist characterization and algorithm to compute non-twist invariant circles. 

Title: Conservation laws respecting the discreteness of nature. (III)

Abstract: The approximation of materials as continuous solids leads to the classical formulations of conservation of scalar (mass, energy, charge) and vector (linear and angular momentum) quantities in terms of their densities. Such formulations do not allow for describing localisations of properties and processes by underlying material structures. One approach to address the deficiency of the classical formulations is to consider explicitly the internal structures of engineering materials as assemblies of discrete, finite cells. Calculus on such cell complexes was born at the same time as our current students. It offers great opportunities for revisiting and reformulating the descriptions and analyses of real materials. This series of talks introduces gradually the building blocks required to constructs mathematical formulations of the physical conservation laws, where the discrete, finite nature of materials’ structures is strictly respected.

The first talk will focus on the motivations for developing discrete formulations: practical and fundamental. It will introduce examples of materials’ structures at different length scales and discuss how their components might control different physical and mechanical macroscopic (engineering scale) properties. It will then show the existing methods for description and analysis of physical and mechanical processes in solids to clarify the points where they become incapable of, or inappropriate for, faithful analysis of real materials. The fundamental motivation comes from two directions: the basic requirement for a rational physical theory, background-independence, leads logically to discreteness of nature; and the continuum assumption leads to a paradoxical conclusion of infinite entropy at any continuum point.

The second talk will be dedicated to the mathematical apparatus for describing materials as geometric realisations of discrete topological spaces, or cell complexes, and for analysis of processes in such spaces. First, it will introduce potentially familiar notions, such as chains and cochains, and boundary and coboundary operators on the spaces of chains and cochains, respectively. Second, it will introduce the notions of combinatorial vector fields and differential forms and their algebraic representations. Third, it will describe the construction of (topologically) unique extended complex, where the spaces of 1-chains and p-cochains are isomorphic to the spaces of combinatorial vector fields and p-forms in the original (physical) complex, respectively, and the action of the coboundary operator on p-cochains is identical to the exterior derivatives of p-forms. This will be used to form all topological operations with forms and vector fields in the original complex as operations with cochains and 1-chains in the extended complex – exterior and interior product, Dirac and Lie derivatives. Finally, metric operators in the spaces of cochains will be introduced and used to derive canonically metric-dependent operators including inner products, (geometric) Hodge-stars, and Laplacians.

The third talk will discuss the formulation of conservation of scalar quantities and balance of momenta in discrete spaces. First, it will introduce physics-modified Hodge-stars that include material properties as relations between cells of different dimensions. Second, it will show the general conservation of a scalar quantity and make it specific to several cases – diffusion of substance or charge, heat conduction, and transport through porous medium. Third, it will show the balance of momenta and the complete formulation of elasticity including kinematics and constitutive relations. Finally, the talk will discuss the application of boundary conditions.

Title: Conservation laws respecting the discreteness of nature. (II)

Abstract: The approximation of materials as continuous solids leads to the classical formulations of conservation of scalar (mass, energy, charge) and vector (linear and angular momentum) quantities in terms of their densities. Such formulations do not allow for describing localisations of properties and processes by underlying material structures. One approach to address the deficiency of the classical formulations is to consider explicitly the internal structures of engineering materials as assemblies of discrete, finite cells. Calculus on such cell complexes was born at the same time as our current students. It offers great opportunities for revisiting and reformulating the descriptions and analyses of real materials. This series of talks introduces gradually the building blocks required to constructs mathematical formulations of the physical conservation laws, where the discrete, finite nature of materials’ structures is strictly respected.

The first talk will focus on the motivations for developing discrete formulations: practical and fundamental. It will introduce examples of materials’ structures at different length scales and discuss how their components might control different physical and mechanical macroscopic (engineering scale) properties. It will then show the existing methods for description and analysis of physical and mechanical processes in solids to clarify the points where they become incapable of, or inappropriate for, faithful analysis of real materials. The fundamental motivation comes from two directions: the basic requirement for a rational physical theory, background-independence, leads logically to discreteness of nature; and the continuum assumption leads to a paradoxical conclusion of infinite entropy at any continuum point.

The second talk will be dedicated to the mathematical apparatus for describing materials as geometric realisations of discrete topological spaces, or cell complexes, and for analysis of processes in such spaces. First, it will introduce potentially familiar notions, such as chains and cochains, and boundary and coboundary operators on the spaces of chains and cochains, respectively. Second, it will introduce the notions of combinatorial vector fields and differential forms and their algebraic representations. Third, it will describe the construction of (topologically) unique extended complex, where the spaces of 1-chains and p-cochains are isomorphic to the spaces of combinatorial vector fields and p-forms in the original (physical) complex, respectively, and the action of the coboundary operator on p-cochains is identical to the exterior derivatives of p-forms. This will be used to form all topological operations with forms and vector fields in the original complex as operations with cochains and 1-chains in the extended complex – exterior and interior product, Dirac and Lie derivatives. Finally, metric operators in the spaces of cochains will be introduced and used to derive canonically metric-dependent operators including inner products, (geometric) Hodge-stars, and Laplacians.

The third talk will discuss the formulation of conservation of scalar quantities and balance of momenta in discrete spaces. First, it will introduce physics-modified Hodge-stars that include material properties as relations between cells of different dimensions. Second, it will show the general conservation of a scalar quantity and make it specific to several cases – diffusion of substance or charge, heat conduction, and transport through porous medium. Third, it will show the balance of momenta and the complete formulation of elasticity including kinematics and constitutive relations. Finally, the talk will discuss the application of boundary conditions.

Title: Conservation laws respecting the discreteness of nature. (I)

Abstract: The approximation of materials as continuous solids leads to the classical formulations of conservation of scalar (mass, energy, charge) and vector (linear and angular momentum) quantities in terms of their densities. Such formulations do not allow for describing localisations of properties and processes by underlying material structures. One approach to address the deficiency of the classical formulations is to consider explicitly the internal structures of engineering materials as assemblies of discrete, finite cells. Calculus on such cell complexes was born at the same time as our current students. It offers great opportunities for revisiting and reformulating the descriptions and analyses of real materials. This series of talks introduces gradually the building blocks required to constructs mathematical formulations of the physical conservation laws, where the discrete, finite nature of materials’ structures is strictly respected.

The first talk will focus on the motivations for developing discrete formulations: practical and fundamental. It will introduce examples of materials’ structures at different length scales and discuss how their components might control different physical and mechanical macroscopic (engineering scale) properties. It will then show the existing methods for description and analysis of physical and mechanical processes in solids to clarify the points where they become incapable of, or inappropriate for, faithful analysis of real materials. The fundamental motivation comes from two directions: the basic requirement for a rational physical theory, background-independence, leads logically to discreteness of nature; and the continuum assumption leads to a paradoxical conclusion of infinite entropy at any continuum point.

The second talk will be dedicated to the mathematical apparatus for describing materials as geometric realisations of discrete topological spaces, or cell complexes, and for analysis of processes in such spaces. First, it will introduce potentially familiar notions, such as chains and cochains, and boundary and coboundary operators on the spaces of chains and cochains, respectively. Second, it will introduce the notions of combinatorial vector fields and differential forms and their algebraic representations. Third, it will describe the construction of (topologically) unique extended complex, where the spaces of 1-chains and p-cochains are isomorphic to the spaces of combinatorial vector fields and p-forms in the original (physical) complex, respectively, and the action of the coboundary operator on p-cochains is identical to the exterior derivatives of p-forms. This will be used to form all topological operations with forms and vector fields in the original complex as operations with cochains and 1-chains in the extended complex – exterior and interior product, Dirac and Lie derivatives. Finally, metric operators in the spaces of cochains will be introduced and used to derive canonically metric-dependent operators including inner products, (geometric) Hodge-stars, and Laplacians.

The third talk will discuss the formulation of conservation of scalar quantities and balance of momenta in discrete spaces. First, it will introduce physics-modified Hodge-stars that include material properties as relations between cells of different dimensions. Second, it will show the general conservation of a scalar quantity and make it specific to several cases – diffusion of substance or charge, heat conduction, and transport through porous medium. Third, it will show the balance of momenta and the complete formulation of elasticity including kinematics and constitutive relations. Finally, the talk will discuss the application of boundary conditions.

Title: Algorytm całkowania równań wariacyjnych dla pewnej klasy problemów. 

Abstract: Opowiem o zastosowaniu algorytmu obliczania oszacowywań na rozwiązania inkluzji różniczkowych (Kapela, Zgliczyński 2009) do całkowania równań wariacyjnych dla pewnej klasy problemów, w szczególności dysypatywnych równań cząstkowych. 

Title: Connection matrix algorithm. (III)

Abstract:

Title: Connection matrix algorithm. (II)

Abstract: 

Title: Connection matrix algorithm. (I)

Abstract: