Goal

The aim of this winter workshop is to prepare PhD students and young scientists to undertake research in the areas of Computational Mathematics on the intersection of Dynamics, Topology and Computations.

The first goal is to provide series of mini-courses given by leading scientists working in the Computational Mathematics.

The second goal is to give opportunity to exchange ideas from various fields and to start new projects and collaborations.

Themes

  • Computational Dynamics
  • Combinatorial Topological Dynamics
  • Topological Data Analysis
  • Dynamics of Time Series

Previous events

Jan 28 - Feb 3
2018

Będlewo
Poland

40-60
Participants

4-6
Speakers

Speakers

Topologies on finite sets, homotopy and fixed points

Finite metric spaces are useful when studying infinite spaces, for instance in Data Analysis. However the topology of a finite metric space is not interesting: two such spaces with the same cardinality are necessarily homeomorphic (indistinguishable from a topological viewpoint). A topological space with finitely many points is a much richer object. Finite spaces can be used to model well-known Hausdorff spaces, such as manifolds or polyhedra. Given any triangulated space, there is a finite space with the same homology and homotopy groups. In contrast to simplicial complexes, there exists an algorithm due to Stong which decides whether two finite spaces can be deformed one into the other (homotopy equivalence). These ideas are used to prove that an action of a group G on a contractible finite T0 space always has a fixed point. This is not true for contractible compact polyhedra. The homotopy theory of finite spaces can be used to study a conjecture by Quillen about the poset of p-subgroups of a group G.
In this course we will see how finite spaces can be studied combinatorially using posets, and we will present the results of McCord which relate finite spaces and polyhedra. We will study homotopy types of finite spaces and establish connections between homotopy and fixed point properties.

Global instability and scattering maps

Global instability is a difficult phenomenon to be proven. It is typically associated to a sequence of homoclinic connections between landmarks (invariant objects) plus an adequate shadowing. A Normally Hyperbolic Invariant Manifold (NHIM) is a relevant typical example of a large landmark, and any map of connections to itself is called scattering or outer map. Since the NHIM has an inner motion, the combination of these two dynamics on the NHIM, inner and outer, provides several designs of instability paths.

We will review the computation and use of such scattering maps for Hamiltonian systems to prove global instability in several relevant problems, like periodic or quasi-periodic perturbations of geodesic flows, a priori unstable Hamiltonian systems, as well as to several applications to Restricted Three Body problems in Celestial Mechanics.

Some validated symbolic-numeric approximation algorithms

In computer-aided mathematical proofs, a basic, yet critical, building block is the problem of actually obtaining numerical values. In practice, one strives to achieve precise and/or guaranteed results without compromising efficiency. For this, we combine symbolic and numerical computation, which leads to the development of specific new arithmetic and approximation algorithms. Firstly, we focus on effectively computing polynomial approximations together with validated error bounds. We discuss Taylor series expansions as well as series expansions based on orthogonal polynomials and associated approximation algorithms (Remez' algorithm, Taylor and Chebyshev models). Secondly, we exploit approximation algorithms mainly related to D-finite functions i.e., solutions of linear differential equations with polynomial coefficients. This property allows for developing a uniform theoretic and algorithmic treatment of these functions, an idea that has led to many symbolic computation applications in recent years. Finally, all these techniques are illustrated with some applications related to the efficient finite precision evaluation of numerical functions (some of which appear in practical space mission analysis and design).

Rigorously verified numerics for infinite dimensional nonlinear problems: a functional analytic approach

Studying and proving existence of solutions of nonlinear dynamical systems using standard analytic techniques is a challenging problem. In particular, this problem is even more challenging for partial differential equations, variational problems or functional delay equations which are naturally defined on infinite dimensional function spaces. The goal of this course is to present rigorous numerical techniques relying on basic functional analytic tools to prove existence of steady states, time periodic solutions, traveling waves and connecting orbits for the above mentioned dynamical systems. We will present computer-assisted proofs of existence of halo periodic orbits in the circular restricted four body problem and in the Navier-Stokes equations.

Schedule

[ Schedule in PDF format ]

  • Afternoons will be left for private discussions, collaboration and small mini-workshops.
  • We do not plan any contibuted talks.

Slides

Registration & Pricing

The registration fee is 950 PLN. It covers accommodation and full board.
According to the current exchange rates (as of July 6, 2017) 950 PLN is equivalent to 230 EUR or 260 USD.
For the details on how to pay see the registration fee Event FAQs below.

We anticipate that we will be able to offer some number of grants to cover registration fee, mainly for young researchers in particular PhD students and post-docs. If you are interested in financial support please contact organizers by email providing your research interests and short recommendation letter from your scientific advisor.

If you would like to participate in the Winter Workshop on Dynamics, Topology and Computations please fill the following

REGISTRATION FORM

Event FAQs

From Poznań the best option is to take a taxi.
To avoid excessive taxi fares (even up to 400PLN) we recommend booking a taxi in advance through conference center in Będlewo. Please contact them by email (bedlewo@impan.pl) providing flight details and mobile number. The cost is 110 PLN and the taxi can be shared by up to 3 people. Taxi driver will wait at the airport (close to cash exchange office) or at the railway station (close to ticket office no. 1).

The payment of the registration fee can be made by transfer to the one of the following bank accounts (preferred) or on place in cash or using a credit card.

PLN currency account: PL 48 1130 1017 0020 1467 1520 0002
USD currency account: PL 37 1130 1017 0020 1467 1520 0006
EUR currency account: PL 80 1130 1017 0020 1467 1520 0008
Swift code: GOSKPLPW
Address of the bank: Bank Gospodarstwa Krajowego,
Aleje Jerozolimskie 7, 00-955 Warszawa, Poland
Owner of the account: Instytut Matematyczny PAN,
Sniadeckich 8, 00-956 Warszawa
Reason for transfer: WWoDTC18 + name of the participant.

Please do not forget to fill in the reason for the transfer.

DO NOT MAKE ANY PAYMENTS if you do not have the confirmation of participation. There will be no refunds.

Event Location

Mathematical Research and Conference Center

Ośrodek Konferencyjny IM PAN
60-060 Będlewo
ul. Parkowa 2
Phone: +48-61-813-5187
email

Contact Persons

Prof. Marian Mrozek
e-mail: Marian.Mrozek@ii.uj.edu.pl
Dr. Tomasz Kapela
e-mail: Tomasz.Kapela@uj.edu.pl
tel. (+48) 12 664 7540
Jagiellonian University
Faculty of Mathematics and Computer Science
ul. Łojasiewicza 6, 30-348 Krakow, Poland