IMPAN-BC
Institute of Mathematics -
Stefan Banach International Mathematical Centre
as a Centre of Excellence

This programme finished its activities in September 2004

Main pages of Institute of Mathematics, Stefan Banach International Mathematical Centre, Mathematical Conference Centre in Będlewo.


Project Coordinator: Prof. Feliks Przytycki

Objectives:

The Centre of Excellence will stimulate and coordinate research and activities at the border of pure mathematics and applications, as well as to stimulate interdisciplinary research. What is lacking in Poland (and also in other preaccession phase countries), it is a link between pure research and applications. It is a general trend in world (especially in highly developed countries), to eliminate middle stages between pure sciences and applications. More and more often the so-called "pure scientists" could communicate with people from the very application process. Unfortunately, such approach falls very short in Poland. It is mostly due to inheritance of communism, which destroyed enterprise and competitiveness culture at the borderline between sciences and industry and it is our aim to re-establish with EU help such relations in the field of mathematical sciences, and to employ mathematics to needs of the society.

Description of work

Activities will be divided into eight packages:
  1. Public-Key Cryptography and Computational Number Theory
  2. Information Theory and its Applications to Physics, Finance and Biology
  3. Mathematical Modelling and Analysis of Cellular Populations
  4. Nonlinear Systems and Control
  5. Mathematics of Finance and Stochastic Control
  6. Approximation structures in Banach spaces
  7. Symplectic Singularities and Applications
  8. Visual Modelling
Each topic will last between twelve and twenty four months. Forms of activities will include

We expect that our approach, following established tradition of the Banach Center, will bring new qualitative and quantitative results, both in pure and applied mathematics. Activities at the Center, including such important topics like elliptic curves cryptosystems, new algorithms for wavelet compression techniques, stochastic analysis of financial derivatives, risk theory, multi-scale biological visual modelling should have long lasting impact, contributing to our knowledge, and increased intellectual potential. Should also bring new impulses for modern technology based on applied mathematics.

We expect several mathematicians to take part in realization of the program of the Centre, including distinguished specialists as well as post-doctorate researchers.


Core for European Excellence Networks (agreement between the Institutes of Mathematics of the Academies of Bulgaria, Hungary, Poland and Romania


Work packages progress report: Dec. 2000 - May 2003, as Word document

Package 1.
Public-Key Cryptography and Computational Number Theory

Start: December 2000
Timetable:
1) Invitation of specialists in the field during December 2000 - November 2001
2) Organization of a workshop - 11 - 15 June 2001
3) Organization of a conference - November 2001
Coordinator: Jerzy Urbanowicz

Objectives:

Development of faster encryption algorithms with shorter keys and lower memory and processing power requirements for smart cards applications. Analysis of provable security protocols for new cryptosystems based on elliptic curves and group algebra and better understanding of underlying mathematics to assure a high cost of cipher-breaking.

Description of the contents, the workplan, the steps, the approach or the methodology:

Cryptographic technologies are at the heart of information society. The rapid growth of Internet and electronic commerce in Europe is directly related to widespread use of cryptography as the preferred means to ensure authenticity and confidentiality in electronic communications. However, despite more than 50 years of mathematical cryptography, there is no publicly known complete mathematical proof of strength for any practical cipher. Also, despite rapid development of cryptography for practical applications there are only very few mathematicians involved in cryptography studies in Central Europe. We propose to enhance both the research in the field and multilateral interaction in the Central Europe region using the Centre of Excellence as a platform initiating, stimulating and proliferating cryptography research. We plan to organise a mini-school of cryptography for young researchers from Central Europe and to give them an opportunity to work together with, and to listen lectures by prominent European experts in the field and representatives of the industry using cryptography for electronic trade and banking. The Centre will promote new ideas and will help young scientists to transfer their results into commercial applications. The Centre will also, by public lectures and open panel discussions, attempt to increase a general cryptography literacy level of various professional groups (medical doctors, lawyers, banking and insurance industry professionals, etc.) and high school students. The conferences, devoted to "The mathematical aspects of public key cryptography" is planned.

Partners involved:


Package 2.
Information Theory and Applications to Physics, Finance and Biology

Type of activity: workshop, conference, research visits
Start: December 2000
Timetable:
Activity for this package occupies months December 2000 - November 2002
Information Theory Days, 23 -29 April 2001
Conference Applications of Information Theory in Biology, Finance and Physics - 21 - 26 May 2001
Workshop Information Theory, Algorithms and Applications to Probability and Statistics - 15 - 20 October 2001
Workshop Holomorphic Iteration, Non-uniform Hyperbolicity, 22 - 25 May, Warszawa
Coordinators: Flemming Topsøe, Feliks Przytycki

Objectives:

Description of the contents, the workplan, the steps, the approach or the methodology:

The purpose of the activity is to combine two approaches to the information theory, the one based on universal coding and prediction developed by the Russian, Israeli, American and other schools with another game theoretical approach. It is expected to obtain applications to cryptoanalysis, parts of quantum physics, finance theory (e.g. portfolio management), statistics, and biology.

An international conference will be held in May. A further major meeting is expected in October. Extended studies at the Banach Center will also take place.

For further details, we refer to http://www.math.ku.dk/IT-Banach2001

Partners involved:


Package 3.
Mathematical Modelling and Analysis of Cellular Populations

Type of activity: research visits, conference, workshop
Start: June 2002
Timetable: 14 months,
School on Population Dynamics, Będlewo, Poland, June 17 - 21, 2002
Conference on Mathematical Modelling of Population Dynamics, Będlewo, Poland, June 24 - 28, 2002
Coordinator: Ryszard Rudnicki

Objectives:

  1. Mathematical description of theoretical functions of population dynamics.
  2. Qualitative properties of cellular populations: stability, periodicity, chaos.
  3. Application to periodic hematological diseases and tumor growth control.

Description of the contents, the workplan, the steps, the approach or the methodology:

The aim of our project is to build mathematical models describing theoretical foundations of population dynamics and study them to obtain qualitative and quantitative properties of populations. The main object of our study are bacteria populations, blood cells and tumour cells. The crucial role in the description of these populations is played by the cell cycle. A single cell is characterized by its maturity (or biological age) which governs the life history of any cell. Physiologists associate the maturation with the size of the cell or with the concentration of a special substance. Models of this type have been yet studied but it has been assumed that reproduction occurs by fission into two equal parts.

This assumption is only a simplification of the real process of replication because it does not take into consideration the stochastic character of the distribution of maturation into daughter cells. We want to study models based on the hypothesis that the maturity of the daughter cells is described by means of the probabilistic measure which depends on the maturity of the mother cell.

The biological assumptions on rates of division, mortality and maturity growth lead to complicated partial differential equations with integral perturbation. Our team has developed some methods of study of equations of this type. In particular methods based on the spectral analysis and the theory of Markov semigroups lead to interesting mathematical and biological results as asymptotic stability or chaotic behaviour of the population. Our research is inspired by new trends and results of theoretical biology. We want to apply our results to study periodic haematological diseases and to tumor growth control.

Partners involved:

  1. Prof. Ovide Arino, Laboratoire de Mathématiques Appliquées, Université de Pau, Pau, France,
  2. Prof. Odo Diekmann, Utrecht University, Utrecht, The Netherlands,
  3. Prof. Eva Sánchez, Departamento de Matemática Aplicada, E.T.S. Ingenieros Industriales, Madrid, Spain,
  4. Laurent Pujo-Menjouet, Laboratoire de Mathématiques Appliquées, Université de Pau, Pau, France,
  5. Dr. N.E. Elhoussif, Utrecht University, Utrecht, The Netherlands.
Also Prof. M.C. Mackey, McGill University, Montreal, Canada will participate in the project.


Package 4.
Banach Center Semester on Nonlinear Systems and Control

Type of activity: workshop, Ph.D. courses, conference, research visits
Start: December 2002
Timetable:
1. Invitation of specialists in the field during December 2002 - November 2003
2. Preparatory courses for Ph.D. students or young researchers, introducing main tools - December 2002 - February 2003
3. A workshop, survey and expository lectures presented by best specialists in the field
4. Research seminars and research on new results/techniques presented during the workshop
5. A specialized conference focused on the most active areas of the research
Coordinator: Bronisław Jakubczyk

Objectives:

Introducing the techniques offered by recent advances in nonlinear system and control theory to candidates for researchers and to young researchers. Emphasis on applications. Improving techniques of stabilization (static and dynamic), observability and observers, optimal control and feedback modification. Applications in robotics, batch reactors, destilation columns, biological systems.

Description of the contents, the workplan, the steps, the approach or the methodology:

In the first part we intend to organize a summer school consisting of 3-4 parallel advanced mini-courses which will give an introduction to the subject and present basic techniques and results which were obtained during last 10-20 years. Topics covered in the courses should include: controllability, observability, realization theory, stabilization, adaptive control, filtration, feedback equivalence and linearization, decoupling and disturbance decoupling, optimal control and examples of applications, nonholonomic systems, systems appearing in robotics, aerospace engineering, chemistry and biology.

The second part of the semester is intended to gather specialists working in the field of nonlinear system theory and control theory, and in areas where applications of this field appear. In the last month a conference is planned.

Partners involved:


Package 5.
Finance Mathematics and Stochastic Control

Type of activity: workshop, summer school, research visits
Start: June 2001
Timetable: 12 months
1. Invitation of specialists in the field during 12 months period,
2. International Workshop on Mathematics of Finance01School.htm - June 4-10, 2001,
3. Summer School on Mathematics of Finance - July 2-10, 2001,
4. International Workshop on Stochastic Control and its Applications - June 3-8, 2002.
Coordinator: Łukasz Stettner

Objectives:

The purpose of the project is to stimulate research on the following topics: A special attention will be devoted to educate young mathematicians in direction of mathematics of finance on various levels: Ph.D.'s in mathematics of finance, lecturers ready to teach mathematics of finance.

Description of the contents, the workplan, the steps, the approach or the methodology:

We plan to organize first two international workshops. The first one on recent developments in mathematics of finance, in particular: on martingale and stochastic control methods in pricing of financial derivatices, term structure models and stochastic partial differential equations corresponding to these models. The second workshop will be devoted to stochastic control methods and their applications with an emphasis on mathematics of finance (in particular the study of various risk cost functionals).

The purpose of a summer school on mathematics of finance is to give a number of lectures how to teach mathematics of finance. It is expected to invite 5-6 lecturers who are experienced in teaching mathematical finance, and will be able to deliver series of selfcontained lectures as well as to share their teaching experiences. The following series of lectures are considered: static portfolio analysis (Markowitz theory), dynamic portfolio selection models - Bellman equations, introduction to pricing of financial derivatives in discrete time, stochastic calculus and Blacke-Scholes model, term structure models, financial engineering.

Partners involved:


Package 6.
Approximation Structures in Banach Spaces

Type of activity: workshop, conference, research
Start: December 2001
Timetable: 24 months
1. Invitation of specialists in the field
2. Workshop for graduate students
3. International Conference
Coordinator: Przemysław Wojtaszczyk

Objectives:

Many topics in functional analysis like bases in Banach spaces, local theory or theory of s-numbers provide a conceptual basis for many applied or computational methods. As examples let us mention theory of computational complexity, wavelet methods for signal compression and analysis or Gabor analysis. The objective of the program is to provide a framework and opportunity for interaction of researchers in those fields of functional analysis with specialists in numerical or computational fields.

Description of the contents, the workplan, the steps, the approach or the methodology:

The aim of the program is to show to the pure functional analyst that his field of research has close ties with many applied theories and those theories may serve as great source of important questions.

Quantitative estimates of geometry of convex bodies which are essential to local theory of Banach spaces (like various s-numbers, projection constants etc.) have also direct bearing on estimates of computational complexity or efficiency of numerical algorithms, especially non-linear. Various bases in Banach spaces like wavelet bases, spline bases, polynomial bases, Walsh and trigonometric systems form a foundation of various numerical methods. Especially important are trash-holding methods for data analysis and compression based on wavelet bases. The fact that those bases are unconditional bases in many function spaces ensures the great efficiency of those algorithms. This connects numerical methods, functional analysis and function theory. On the other hand ridge approximation which is an active area of research in function theory and is directly motivated by computer tomography and neural networks should lead to important questions in functional analysis.

Partners involved:


Package 7.
Symplectic Singularities and Applications

Type of activity: workshop, conference, research visits
Start: December 2001
Timetable: 24 months
1. Invitation of specialists in the field
2. Workshop
3. Conference
Coordinator: Stanisław Janeczko

Objectives:

Formulation of a new approach to the collective phenomena, braking of symmetry. Classification of symplectic spaces and structures.

Description of the contents, the workplan, the steps, the approach or the methodology:

  • Classification of caustic invariant with respect to the compact symmetry group action;
  • Finding of symplectic invariants of systems of surfaces and composed symplectic relations;
  • Recognition of integrability of the implicit differential systems - generalized Hamiltonian dynamics;
  • Diffraction patterns and extension of A, D, E classification of simple singularities;
  • Application of modern symplectic topology results and techniques - Gromov-Witten invariants. Extension of pseudoholomorphic curves methods to the singular symplectic structures.
  • Partners involved:

    H. Farkas, Technical University of Budapest;
    P. Giblin, J.W. Bruce, University of Liverpool;
    J.C. Sikorav, Ecole Normale Superieure Lyon;
    Dierk Siersma, Utrecht Univ.


    Package 8.
    Visual Modelling

    Type of activity: workshop, conference, research visits
    Start: December 2000
    Timetable:
    1) Invitation of specialists in the field - December 2000 - May 2002
    2) Organization of a workshop - May 2001
    3) Organization of a conference - April 2002
    Coordinator: Marek Niezgódka

    Objectives:

    Facilitating the researchers participating in the project to verify their solutions in a distributed visual modelling environment with the use of advanced computing infrastructure.

    Description of the contents, the workplan, the steps, the approach or the methodology:

    Computational resources and expertise of ICM will be used for development of visual modelling environments for various field of application oriented research, including biomolecular, atmospheric and medical science. The developed solutions form an extensive set of numerical, programming and visualization tools that can be used for conducting, verifying and analyzing of mathematical and numerical models of complex, multi-scale physical, biological and economical systems. ICM plans to allow the participants of the project the access to these resources by:
    1. Organization of a workshop in visual modelling
    2. Assigning a staff member for consulting work
    3. Assigning necessary computational and network resources
    4. Organizing a conference summarizing developments of the project
    5. Inviting several specialists for lectures, workshops and cooperative work
    6. Creation of an integrated, distributed visual programming environment for rapid prototyping of algorithmic solutions

    Partners involved:

    1. IWR (Interdisciplinary Scientific Computing Centre), University of Heidelberg
    2. Institute of Applied Mathematics, University of Freiburg