organized by

Faculty of Mathematics and Computer Science

Jagiellonian University in Kraków

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**.

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

- Dynamics, Topology and Computations conferences in 2006, 2009, 2012, 2015
- Winter School on Computational Mathematics 2016
- Winter Workshop on Dynamics, Topology and Computations 2017

2018

Poland

Participants

Speakers

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 T_{0} 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 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.

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).

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

- Amadeu Delshams [slides]
- Mioara Joldes [slides 1] [slides 2] [slides 3] [code]

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

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).

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.

Mathematical Research and Conference Center

Ośrodek Konferencyjny IM PAN

60-060 Będlewo

ul. Parkowa 2

Phone: +48-61-813-5187

email

Prof. Marian Mrozek

e-mail: Marian.Mrozek@ii.uj.edu.pl

e-mail: Marian.Mrozek@ii.uj.edu.pl

Dr. Tomasz Kapela

e-mail: Tomasz.Kapela@uj.edu.pl

tel. (+48) 12 664 7540

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

Faculty of Mathematics and Computer Science

ul. Łojasiewicza 6, 30-348 Krakow, Poland