Stefan Banach International Mathematical Centre as a Centre of Excellence |

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

**Project Coordinator:**
Prof. Feliks Przytycki

- Public-Key Cryptography and Computational Number Theory
- Information Theory and its Applications to Physics, Finance and Biology
- Mathematical Modelling and Analysis of Cellular Populations
- Nonlinear Systems and Control
- Mathematics of Finance and Stochastic Control
- Approximation structures in Banach spaces
- Symplectic Singularities and Applications
- Visual Modelling

- cooperative research in above listed novel, and important topics, together with invited foreign scientists. Those projects will also exploit existing links with analogous institutes, including Max-Planck-Institut für Mathematik in Bonn, Institut des Hautes Etudes Scientifiques in Paris, Erwin Schrödinger International Institute for Mathematical Physics in Vienna, Mittag-Leffler Institute in Djursholm, Centre International de Rencontres Mathematiques in Marseille, and several first class universities. Such activities should attract bright young people and also help to establish collaboration with more applied research;
- organisation at the Banach Center of international conferences, workshops, semesters, research groups, schools for young researchers.

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

- H. Niederreiter (Institute of Discrete Mathematics, Vienna, Austria)
- G. Frey (Institut für Experimentelle Mathematik, Universität Essen, Germany)
- J. Brzezinski (Department of Mathematics, Chalmers Univ. of Technology, Göteborg, Sweden)

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

- to introduce young researchers to new developments in Information Theory with a special view to applications in Physics, Finance and Biology; this especially involves the Polish scientific community - which at present only has little activity and expertise in the area,
- research in an international setting with key specialists visiting, instructing at a workshop and lecturing at the conference which will mark the close of the activity.

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 https://www.math.ku.dk/IT-Banach2001

- Mark Broom (Sussex, BIOLOGY)
- Imre Csiszar (Budapest, INFORMATION THEORY)
- Denes Petz (Budapest, QUANTUM INFORMATION THEORY)
- Zhanna Reznikova (Novosibirsk, BIOLOGY)
- Boris Ryabko (Copenhagen, UNIVERSAL CODING, INFORMATION THEORY)
- Dominick Samperi (New Yourk, FINANCE THEORY)
- Lukasz Stettner (Warsaw, Director of the Banach Centre)
- Flemming Topsoe (Copenhagen, INFORMATION THEORY)
- Igor Vajda (Prague, STATISTICS)

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

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

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.

- Prof. Ovide Arino, Laboratoire de Mathématiques Appliquées, Université de Pau, Pau, France,
- Prof. Odo Diekmann, Utrecht University, Utrecht, The Netherlands,
- Prof. Eva Sánchez, Departamento de Matemática Aplicada, E.T.S. Ingenieros Industriales, Madrid, Spain,
- Laurent Pujo-Menjouet, Laboratoire de Mathématiques Appliquées, Université de Pau, Pau, France,
- Dr. N.E. Elhoussif, Utrecht University, Utrecht, The Netherlands.

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

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.

- Bernard Bonnard, Departement de Mathématiques, Université de Bourgogne, Dijon
- Witold Respondek, Institut National des Sciences Appliquees de Rouen, Rouen
- Michael Zeitz, University of Stuttgart, Institut für Systemdynamic und Regelungstechnik

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

- pricing of financial derivatives on incomplete market models,
- analysis of term structure models,
- construction of models with risk,
- application of stochastic control in mathematics of finance,
- asymptotic properties of stochastic control models.

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.

- from Italy: professors: W. Runggaldier, G.B. Di Masi, P. Dai Pra from Padova, M. Frittelli from Milano, G. Da Prato and Prattelli from Pisa,
- from France: professors: A. Sulem, Y. Kabanov (Besançon), N. Touzi, H. Pham, N. El Karoui from Paris VI, E. Jouini from ENSAE,
- from Germany: professors: K. Helmes and M. Schweizer from Berlin, M. Kohlmann from Konstanz, U. Rieder from Ulm, M. Schal from Bonn,
- from the Netherlands: prof. A. Bagchi from Twente.

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

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.

- prof. Stephane Jaffard (Paris)
- prof. Gilles Pisier (Paris VI)
- prof. S.J. Szarek (Paris and Cleveland)
- prof. Hermann König (Kiel)
- prof. Stephan Heinrich (Kaiserslautern)
- prof. Wolfgang Lusky (Paderborn)
- prof. Bernd Carl (Jena)
- prof. Wolfgang Sickel (Jena)
- prof. Hans Feichtinger (Wien)
- prof. Paul Müller (Linz)
- prof. O. Christiansen (Denmark)
- prof. J. Garcia-Querva (Universidad Autónoma de Madrid)
- prof. Gustav Gripenberg (Helsinki)

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

P. Giblin, J.W. Bruce, University of Liverpool;

J.C. Sikorav, Ecole Normale Superieure Lyon;

Dierk Siersma, Utrecht Univ.

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

- Organization of a workshop in visual modelling
- Assigning a staff member for consulting work
- Assigning necessary computational and network resources
- Organizing a conference summarizing developments of the project
- Inviting several specialists for lectures, workshops and cooperative work
- Creation of an integrated, distributed visual programming environment for rapid prototyping of algorithmic solutions

2. Institute of Applied Mathematics, University of Freiburg