Goal

Winter school is devoted to computational aspects of dynamics and topology.

The first aim is to provide series of lectures given by leading scientists working in the computational mathematics.

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

Themes

  • Computational dynamics
  • Computational topology
  • Topological Data Analysis

2016
Feb 7-13

Będlewo
Poland

Up to 30
Participants

4-6
Speakers

Speakers

Roberto Barrio

Roberto Barrio

Computational Dynamics group (CoDy)

University of Zaragoza, Spain

Experimental Mathematics in Dynamical Systems

Recent years have seen the flowering of “experimental” mathematics, namely the utilization of modern computer technology as an active tool in mathematical research. This development is not limited to a handful of researchers nor to a handful of universities, nor is it limited to one particular field of mathematics. Instead, it involves hundreds of individuals, at many different institutions, who have turned to the remarkable new computational tools now available to assist in their research, whether it be in number theory, algebra, analysis, geometry, or even topology. These tools are being used to work out specific examples, generate plots, perform various algebraic and calculus manipulations, test conjectures, and explore routes to formal proof. Using computer tools to test conjectures is by itself a major timesaver for mathematicians, as it permits them to quickly rule out false notions.

In these three talks we introduce the use of several computational techniques in the study of different problems in Dynamical Systems. We briefly detail how the mixed use of techniques like chaos detection techniques, continuation techniques, computer-assisted proofs, very high-precision numerics and specially designed methods permits to locate and study the origin of some interesting phenomena in Dynamics, as the global organization in neuron models, the global homoclinic structure of Lorenz-like systems, the coexistence of chaos-hyperchaos behavior, the existence of safe regions in open Hamiltonian systems, and so on.

We will cover these topics in three lectures.

  • Numerical methods for solving ODEs: high-precision and non-standard demands.
  • Playing with Hamiltonian Dynamics: chaos indicators, bifurcations, "safe-regions", ...
  • Playing with dissipative systems: chaos indicators, bifurcations, global bifurcations and global parametric pictures

Logics for discrete gene regulatory networks

The beginning of the course presents the basic modelling approach defined by René Thomas (Brussels) in the 70's. We firstly explain how the space of possible gene expression levels can be decomposed into several intervals in order to obtain a discrete qualitative description of gene networks. Then, we show how this discrete approach can be formalized and how the minimal set of Thomas' parameters is used to build an "asynchronous" automaton from the gene interaction graph. This automaton models the dynamic behaviour of the regulatory network (chronological evolution of the gene expression levels).

The main difficulty when modelling gene networks is the identification of the parameters that govern the dynamics. We show how to use temporal logic in order to extract unknown parameter values from the observed behaviours. We firstly give an overview on CTL (Computation Tree Logic) and its associated model checking methods. We show how it can be used in order to find the set of possible parameter values, i.e., parameters that are consistent with the known qualitative behaviours. Some complementary reasonings, taking into account the biological question under interest, can also help reducing the size of the gene network model.

We also present a new approach based on Hoare logic and an associated weakest precondition calculus that generates constraints on the parameter values. Once proper specifications are extracted from biological traces (e.g. based on transcriptomic data), they play a role similar to programs in the classical Hoare logic. The method is correct and complete w.r.t. the Thomas' semantics.

Lastly, we sketch how some logical considerations can be used in order to generate interesting "wet" biological experiments, starting from the formal descriptions of the interaction graph and the biological hypotheses under consideration.

A few biological examples of small size will illustrate the notions defined during the course.

Tamal Dey

Ohio State University, USA

Topics in Topological Data Analysis

The new emergent area of topological data analysis(TDA) investigates topological features hidden in data. This requires structural analysis of various complexes built on top of the data including their relationship to the ground truth and design of efficient algorithms to extract these structural information. Persistent homology and algorithms to compute them are central to this development. In this course, after introducing the general framework for persistence and TDA we focus mainly on two topics: (i) Sparsification (ii) Multiscale mapper. Specifically, we cover the following materials:

  • Persistent homology, stability of persistence diagrams, efficient algorithms for computing persistence under inclusions and simplicial maps.
  • Sparsification of data using Witness complex, Sparsified Rips complex, and Graph Induced complex.
  • Exposition of Multiscale Mapper (MM) as a tool for investigating multivariate maps, Stability of MM, Algorithms to compute persistence diagrams of MM.
We will cover these three topics in three lectures each devoted to one topic.

Jordi-Lluís Figueras

Uppsala University, Sweden

Computational Dynamics: invariant manifolds and beyond

This course aims at an exposition of the basic techniques for the computation of invariant manifolds in Dynamical Systems.

The contents would contain the following topics:

  • Numerical methods for the solution of zeros of nonlinear systems. The continuation method.
  • Fixed and periodic points and their stability.
  • Analytical and numerical approximation of attached invariant manifolds.
  • Computation quasi-periodic orbits.
  • Invariant objects in infinite dimensional systems (Delay equations, PDEs): Fixed points, travelling waves, periodic orbits, invariant tori.

Eric Goubault

Ecole Polytechnique, France

Validation and control of hybrid systems

The aim of this course is to introduce the audience to the field of validation (or static analysis) of programs, and hybrid systems and its relationship to dynamical systems and control theories. Programs can be modeled as discrete dynamical systems, and hybrid systems are combinations of these discrete dynamical systems with continuous time dynamical systems (modeled for instance by ordinary differential equations, or differential inclusions).

In the lectures we would cover

  • The basics of static analysis of programs and hybrid systems : set-based dynamical systems, (polyhedric and semi-algebraic) abstractions, and calculations of fixed points of such set-based dynamical systems
  • Proving stability of such systems, in some region of the state space : classical Lyapunov approaches and elaborations of such, and a topological approach based on Wazewski’s property, and algorithms for determining isolating blocks
  • Viability and controllability of such systems, still using a topological approach on differential inclusions ; some algorithmics.

Thomas Wanner

George Mason University, USA

Rigorous Level Curve Enclosures - From Isolating Blocks to Bifurcation Diagrams

In this lecture series, we present two rigorous computational methods for enclosing level curves and level sets of smooth functions, as well as a number of associated applications in dynamical systems and partial differential equations. The first of these methods consists of an adaptive randomized subdivision algorithm which can be used to rigorously determine the topology of sub- and super-level sets of smooth functions, and it will be shown that it can be extended to find isolating blocks in dynamical systems. In this way, methods from Conley index can be used to derive global dynamical objects. The second method is centered around a numerical version of the implicit function theorem, which can be used to verify equilibrium bifurcation diagrams for a wide variety of evolution equations. This approach will be illustrated in the context of concrete examples from high-dimensional lattice dynamical systems, as well as for the infinite-dimensional diblock copolymer model in materials sciences.

Schedule

Sunday (7th February 2016) is an arrival day. Lectures will start on Monday morning. Winter school will end with a lunch around noon on Saturday (13th February, 2016).

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

Monday

9.00-10.00 Thomas Wanner [ SLIDES ]
10.00-11.00 Jordi-Lluis Figueras [ SLIDES ]
11.30-12.30 Tamal Dey [ SLIDES ]

Tuesday

9.00-10.00 Tamal Dey [ SLIDES ]
10.00-11.00 Gilles Bernot [ SLIDES ]
11.30-12.30 Roberto Barrio [ SLIDES ]
15.00Marcel Falkiewicz [ SLIDES ]
"Imaging of the working brain with Magnetic Resonance Imaging"

Wednesday

9.00-10.00 Jordi-Lluis Figueras [ SLIDES ]
10.00-11.00 Thomas Wanner [ SLIDES ]
11.30-12.30 Gilles Bernot [ SLIDES ]

Thursday

9.00-10.00 Tamal Dey [ SLIDES ]
10.00-11.00 Eric Goubault [ SLIDES ]
11.30-12.30 Roberto Barrio [ SLIDES ]
15:00-16:00 Thomas Wanner [ SLIDES ]

Friday

9.00-10.00 Gilles Bernot [ SLIDES ]
10.00-11.00 Eric Goubault [ SLIDES 1 2]
11.30-12.30 Roberto Barrio [ SLIDES ]

Saturday

8.00-9.00 Eric Goubault [ SLIDES ]
9.00-10.00 Jordi-Lluis Figueras [ SLIDES ]

Meals

Meals will be served at
8.00-9.00 Breakfast
11.00-11:30 Coffe break
13.00 Lunch
19.00 Dinner

Registration & Pricing

Participation in Winter School on Computational Mathematics is subject to personal invitation.

Young researchers interested in participation, in particular PhD students and post-docs, can contact organizers by email. Please provide your research interests and short recommendation letter from your scientific advisor.
We anticipate that we will be able to offer some number of grants to cover registration fee. If you are interested in financial support please indicate this when sending the above email.

The registration fee is 800 PLN. It covers accommodation and full board.
According to the current exchange rates (as of November 26, 2015) 800 PLN is equivalent to 190 EUR or 200 USD.
For the details on how to pay see the registration fee Event FAQs below.

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 90 1060 0076 0000 3210 0014 0962
USD currency account: PL 54 1060 0076 0000 3210 0014 1028
EUR currency account: PL 17 1060 0076 0000 3210 0014 1015
Swift code: BPHKPLPK
Address of the bank: Bank BPH SA, Oddz. w Warszawie,
Aleje Jerozolimskie 27, 00-508 Warszawa
Owner of the account: Instytut Matematyczny PAN,
Sniadeckich 8, 00-956 Warszawa
Reason for transfer: WSoCM16 + 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.

Please contact Tomasz Kapela by email (Tomasz.Kapela@uj.edu.pl)

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