IMPAN-BC

Work packages progress report 1-5 (cumulative)

Package No. 1. “Public Key Cryptography and Computational Number Theory”

(coordinators J. Urbanowicz A. Wittlin).

The following activities took place:

Personnel:

First half - year period

- Professor Jacques Patarin, University of Versailles, December 4-14, 2000. He gave a sequence of lectures for specialists concerning his recent results, including: “Efficient Algorithms for Solving Over-defined Systems of Multivariate Polynomial Equations”, “Unbalanced Oil and Vinegar Signature Schemes”, “Flash, a fast Multivariate Signature Algorithm”.
- Prof. Peter Hegarty
*, Uni. Goeteborg, Sweden*stayed at IMPAN-BC for 2 months, in April-June, 2000. He worked with J. Urbanowicz, R. Nessel, S. Spież and K. Alster. He gave a course on elliptic curves cryptography, see details in WPPR. - Prof. Jacques Patarin, France, 7-19 January, 2002.

Prof. Peter Hegarty, April – June 2001. Prof. Hegarty gave a series of lectures on Elliptic Curve Cryptography.

Third half - year period

Professor Patarin gave two talks on:

- “Generic Attacks on Feistel Schemes”, on January 11,
- “Proofs of Security on Feistel Schemes (i.e. on encryption schemes in cryptography)”, on January 16.

During his stay at IMPAN-BC professor Patarin worked on algebraic aspects of cryptography.

Fourth half - year period

- Prof. Gheorghe Paun, from Romanian Academy of Science, Romania, visited

IMPAN-BC, Warsaw in 15 September – 15 October, 2002.

During the visit he wrote the following two joint papers with dr. A. Obtułowicz:

- “(In Search of) Probabilistic P systems”, to appear in BioSystems,
- lecture ‘’On Three Classes of Automata-Lik P systems”, presented on International conference MolCoNet, Budapest, 26-29 November, 2002.

He gave a lecture entitled “Membrane Computing. An Introduction” at the Institute of Mathematics of Warsaw University. Prof. G. Paun discussed the problem of molecular computing and cryptography during seminar meetings and informal meetings of the research group of molecular computing led by dr. A. Obtulowicz. Prof. G. Paun presented this research group during the meeting of the directors of *European Molecular Computing Consortium* in November 2002 in Budapest to introduce the group to this consortium, and the group has been accepted to be a member.

Fifth half –year period

- Prof. Jacques Patarin, Bull, France, January 2003 (two weeks).

Presented talks:

- "About the XL algorithm over GF(2)" (January 10),
- "Algebraic attacks on symmetric ciphers" (January 17)

Paper completed (acknowledge to the Center):

J. Patarin "Luby-Rackoff: 7 Rounds are Enough for 2n(1-?) Security" accepted for Crypto 2003.

f) Prof. H. C. Williams, Uni. Calgary, Canada, May, 2-17, 2003.

He collaborated with Professors A. Schinzel and J. Urbanowicz discussing

some computational aspects of Number Theory and Cryptography and gave two lectures :

- "Some Results Concerning Periodic Continued Fractions".
- "Some Contributions of Cryptography to Number Theory"

Abstract:

Since the mid-seventies many advances have been made in cryptography through the utilization of results from number theory. In this talk I will describe several techniques, which owe their origin to the application of number theory to cryptography, that have been successfully applied to classical problems arising in computational number theory. In particular, I will discuss the integer factoring problem, the discreet logarithm problem, and the problem of solving the Pell equation.

Professors S. Spiez, J. Urbanowicz and H.C. Williams discussed some unconventional methods in Cryptography such that higher quadratic cryptography (HQC) based on K-groups of rings of the integers as well as cryptosystems related to non-commutative algebraic structures.

g) Dr Nicolas Courtois (Schlumberger Smart Cards Paris) , Louveciennes, France, May 2003, 1 month.

He presented several talks (in Polish) covering algebraic attacks on symmetric

cyphers, structure and security of AES cypher, and a general talk " Modern

Cryptology" presented at the Institute of Physics of P.A.S.

h) Dr. Christine Swart, Mathematics Department, Royal Holloway, Egham, Surrey,

UK, May, 2003.

She gave two lectures:

- "Elliptic Divisibility Sequences and Samos Sequences
*"* - ”Elliptic Divisibility Sequences and Cryptography”

An elliptic divisibility sequence is a sequence *h _{n}* of integers satisfying

*h _{m+n} h_{m-n} = h_{m+1} h_{m-1} h_{ n²} - h_{n+1} h_{n-1} h_{ m²}* for all

such that *h _{n}* divides

She gave some symmetry properties of elliptic divisibility sequences, and explained how these have been used by Shipsey and Stephens to derive simple and elegant attacks on two weak cases of the elliptic curve discrete log problem. In particular, if # *E(F _{p}*) divides

Short visits:

First half - year period

- Mr. Rafał Nessel, and dr. Aleksander Wittlin attended IACR Eurocrypt 2001
- Dr Aleksander Wittlin visited in December 2001 Cryptology Center at the
- Dr Aleksander Wittlin attended conference "Eurocrypt 2002", in Amsterdam, April 27
^{th}– May 2^{nd,}2002. - Dr Aleksander Wittlin attended “STORK Meeting”, Brugge, Belgium, 29
- Dr Petra Wolhmacher, from PKI (Public Key Infrastructure) Reg TP, Mainz, Germany, visited IMPAN-BC, Warsaw in 21 –24 September, 2002.
- Prof. Maurice Mignotte, Pasteur Uni., Strasbourg, France, May 4-14, 2003
- Dr. Adam Obtulowicz, IMPAN-BC, participated in the conference Brainstorming Week on Membrane Computing, Universitat Rovira i Virgili, Tarragona (Spain), 05.02 - 11.02.2003. He delivered the lecture "Mathematical models of uncertainty with a regard to membrane systems" (60').
- Adam Obtulowicz, IMPAN-BC, participated in the European Molecular Computing Consortium (EmCC) Spring meeting 2003, Abo Akademi and Turku Center for Computer Science, Turku (Finland), 15.05 - 17.05.2003. Lecture "Algebraic P systems" (30') was given by A. Obtulowicz and he presented at the General Assembly of EMCC the research group 'Mathematical aspects and applications of membrane computing' at IMPAN, associated to EMCC.

Conference in Innsbruck, Austria, May 6-10 2001. __http://ec2001.ocg.at/__ (next Eurocrypt Conference will take place in Banach Center in 2003).

Third half - year period

University of Bochum – Germany (host prof. dr Hans Dobbertin). The visit included seminar, discussions related to public key cryptography, and organization of future Europe cryptology research network within the VI Framework Programme of the European Community.

The next Eurocrypt conference, in May 2003, will be organized in Warsaw. Together with prof. Jerzy Gawinecki (Military Technical Academy) dr A. Wittlin also participated on behalf of IMPAN-BC in organization meetings with IACR officials and with organizers of Eurocrypt 2002, concerning the Warsaw Eurocrypt conference.

Fourth half - year period

November, 2002. (STORK is a cryptology consortium).

During “STORK Meeting” the Institute of Mathematics formulated its views on priorities in the cryptology research for the forthcoming 4 years. In particular, it includes: fundamental research in foundations of cryptology, including various aspects of "difficult problems" underlying basic algorithms; studies of new tools to design and analyse stream ciphers; development of new, non-obtrusive reliable personal identification techniques; efforts for general improvement of crypto literacy of citizens, in particular among legislators and lawyers, proposal of European university CV in cryptology, and better understanding of challenges and difficulties concerning the introduction and dissemination of electronic signature , in particular in Central European countries.

The visit of dr. P. Wohlmacher was related to recent introduction of electronic

signature law in Poland. Presented by dr Wohlmacher German perspective, based on six years experience in that field allowed us to better understand issues related to design, establishment, and operation of PKI, in particular problems of system security, and demands related to cryptographical aspects of electronic signatures.

Fifth half –year period

)

He collaborated with Professors A. Schinzel and J. Urbanowicz discussing

some computational aspects of Number Theory and gave a lecture:

“On the Diophantine equation

Abstract:

We first give a short survey of the results known on the "Nagell- Ljunggren" equation . Then we present a proof of a result of Bugeaud-Mignotte-Shorey-Roy, namely there is no non-trivial solution for which x is a square. This result was obtained independently by M. Bennett, with another proof, but we insist on the computational aspects of our proof.

Conferences and Workshops:

Second half – year period

a)** The first German-Polish Workshop in Cryptology** was held in The Mathematical Research and Conference Centre, Bedlewo, Poland, June 11-15, 2001. The meeting was very successful and gave the first impulse for the future cooperation. It is planned to organize the next workshop in July this year.

Topics:

- symmetric cryptosystems
- public key cryptography
- history of cryptography
- hash-functions and data integrity
- identification and entity authentication
- digital signatures
- key establishment protocols
- key management techniques
- power attacks.

About 50 participant all together.

Fourth half - year period

b) “II German-Polish Workshop on Cryptology”, Bedlewo, 29 July – 2 August, 2002

c.a. 20 participants, (Speakers: Z. Kotulski, K. Kulesza, R. Dwilewicz, S. Mueller, P. Felke, M. Daum).

Organized together with group of Professor Dobbertin in Bochum (Germany) and with Dr. Siguna Mueller in Klagenfurt (Austria).

Fifth half –year period

- Workshop “Computational Number Theory and Cryptography in the Honour of the 60
^{th}Birthday of Professor Hugh C. Williams”, held in Warsaw, May 12 – 14, 2003. - Workshop "Cryptology - fundamentals and frontiers", Bedlewo, April 28 - May 3, 2003.

The workshop gathered 31 scientists: 15 from Poland and 16 from abroad (from 11 countries). The workshop was devoted to computational number theory and its applications to cryptography. 14 lectures were presented. The following problems were considered:

class fields; computations and constructions

computational aspects of diophantine equations

factoring polynomials and finite polynomial orbits

semigroup technique in number theory

exponential sums and estimates of pseudopowers

elliptic divisibility sequences

CM curves in cryptography

application of norm forms in cryptography

decisional number theoretical assumptions in cryptography

contributions of cryptography to number theory

Number of participants - 55, including approx. 40 students and PhD students from 15 Universities in Poland and abroad; eight speakers from Europe and Australia.

Number of lectures 21, - total number of lecture hours 44.

Speakers:

dr Nicolas T. Courtois (Schlumberger Smart Cards Paris)

prof. dr Hans Dobbertin (U. Bochum)

prof. dr Andreas Enge (Ecole Polytechnique Paris)

dr Florian Hess (U. Bristol)

dr Phong Nguyen (ENS Paris)

prof. dr Jozef Pieprzyk (Macquire University, Sydney)

prof. dr Bart Preneel (K.U. Leuven)

dr Christine Swart (Royal Holloway London)

Each speaker gave 4 - 8 hours of tutorial lectures covering a broad range of topics, from fundamentals of number theory and cryptology to "Collisions detection", "Attacking RSA", "Autentication Codes", "Algebraic Attacks on Block Ciphers", "Mobile Network Security", "Elliptic Curve Cryptography", and "Lattices in Cryptoanalysis".

Other activities:

First half - year period

- Preparation of Proceedings of Public Key Cryptography and Computational Number Theory Conference which was held in Warsaw in September 2000.
__Editors K. Alster, J. Urbanowicz, H. Williams, Walter de Gruyter, Berlin, New Jork, 2001.__ - Public lectures and open panel discussions to increase literacy level of various

professional groups (banking, insurance, lawyers, scientists, telecommunication industry professionals, etc.):

- “Public Key Infrastructures”, December 7-8, 2000; the main lecturer professor Jacques Patarin.
- “Enigma and Science”, March 7, 2001; the lecturer dr Arkadiusz Orłowski.
- “Around the Electronic Signature Act”, March 22-23, 2001; the main lecturer professor Mirosław Kutyłowski.
- “Security Policy for Companies”, March 28-29, 2001; the main lecturer dr. Janusz Cendrowski.

These lectures gathered all together approx. 400 participants.

- Intensive course on “Elliptic Curves Cryptography”, May 21-23, 2001; speakers list includes professor Peter Hegarty, dr. Jerzy Konarski, and dr. Jan Spalinski. The course was followed by 20 participants.
- A biweekly Cryptography seminar has started in November. Talks in November included:

- Paszkiewicz - Numerical studies on smallest prime roots modulo prime.

M. Kuś - Quantum cryptography.

Second half - year period

- Prof. dr Jerzy Browkin (Warsaw University) and dr Aleksander Wittlin run at
- Friday seminars, (J. Browkin, A. Wittlin) 15-40 participants.
- Eurocrypt 2003. The Institute co-organized the Eurocrypt 2003 conference in Warsaw (May 5-9) 2003. The conference was attended by approx. 400 participants from all over the world.
- ECRYPT - The Institute's Cryptology Group was accepted as a member of ECRYPT consortium, an Europe-wide research in cryptology network project coordinated by K.U. Leuven (prof. B. Preneel), submitted to the Information Society Technologies Priority of the EU VI Framework Programme, Call FP6-2002-IST-1. The project hearing was held on June 11, 2003, evaluation result (June 24, 2003): 4.9 (24.5/25).

IMPAN-BC the weekly Warsaw Cryptology Seminar. Between November 2001 and May 2002 seventeen talks have been presented by speakers from Polish and foreign research institutions. The talks covered broad range of topics including general cryptography, number theory, combinatorics and electronic commerce. (The whole list with abstracts is published here.

Fourth half - year period

One day tutorial course "Electronic economy, possibilities and challenges", September 23, 2002, Warsaw (speakers J. Sojka (Poznan), W. Chocianowicz (Szczecin), J. Przetocki (Lublin), P. Wohlmacher (Mainz)).

Fifth half –year period

__ Summary__. This package has almost finished its activitis, very sucessfully, in particular joning the ECRYPT consortium, being clasified on the first place in the adequate FP6/IST competition. Still a small school/workshop and some YR visits remain to be organized.

Package No. 2 “Information Theory and its Applications to Biology, Finance, and Physics”

(coordinators F. Topsoe and F. Przytycki).

The following activities took place:

Personnel:

Second half – year period

a) Prof. Genadi Levin, Hebrew Uni., Jerusalem, Israel, August – Oct, 2001.

He continued his study of dynamical systems with chaotic behaviour, and the dependence of the dynamics on parameters. A simplest, still very non-trivial, example is the following family of dynamical systems, either of the real line or the complex plane:

- z → z ² + c

- This family is known in biology as logistic model of the population growth, and is studied since 1980's.
- The paper "Making explicit connections between dynamical and parameter spaces" in cooperation with Feliks Przytycki has been written.
- G. Levin delivered talks on seminars at IMPAN and Warsaw University.

b) Dr. Marian Grendar, Slovak Academy of Sciences, Slovakia, October, 2001. He participated in the workshop "Information Theory, Algorithms and Applications" where he presented a work 'Why Maximum Entropy? A Non-axiomatic Approach' - He benefited from discussions with Flemming Topsoe, Jonathan D. H. Smith, Imre Csiszar, Peter Harremoes, Andras Gyorgy, Kazimierz Sobczyk. During his stay he attended seminars on Probability and Statistics. He is preparing a paper on Chernoff's bonds.

Third half – year period

- Marian Grendar, Slovak Academy of Sciences, Slovakia, April, 2002. This was an extension of his visit in Fall 2001. It will be continued in July/August 2002.

During his two weeks stay he presented a seminar "Randomness as an Equilibrium" at the meeting of the Probability and Statistics group of IMPAN. Also he continued to work on Chernoff's

bound (edited as a preprint of IMPAN).

d) Boris Ryabko, University of Novosibirsk, Russia, University of Copenhagen,

Denmark, January, 2002.

Topics of lectures and research during the stay:

- The nonprobabilistic approach to learning the best prediction (lecture at dynamical systems seminar)

- Fast enumeration for combinatorial problems (lecture at IMPAN colloquium)

- The imaginary sliding window: Randomness vs memory space

- The simple ideal cipher system

- Fast coding of low-entropy sources.

e) Jon Aaronson, University of Tel-Aviv, Israel, February 17^{th} -March 2^{nd}, 2002.

During this visit, he gave four talks:

- Ergodic and Exact Cocycles for Probability Preserving Transformations

(IMPAN colloquium 2^{nd} Feb.)

- Conformal measures, de Finetti's theorem, and invariant measures for

skew products and horocycle flows on "abelian covers". (dynamics seminar, 1st March)

- Infinite ergodic theory – a minicourse(complex dynamics seminar, 20th and 27th Feb.).

- There were also promising mathematical discussions with the Warsaw dynamics

people (especially Kotus, Przytycki and Rams) and with Thouvenot (also

visiting).

Prof. Jon Aaronson, continued his visit visiting mainly IMPAN Torun branch and UMK.

In collaboration with M. Lemańczyk, a class of skew product extensions, so called Rokhlin cocycle extensions, was studied.

The basic problem of lifting ergodic properties to extensions was solved in case of exact endomorphisms (giving rise to the answer to some old question by

I. Meijlison) as well as in case of weak mixing property of the base.

f) Jean-Paul Thouvenot, University of Paris 6, France. His stay at the Center (22

January-22 February) was divided in two periods of two weeks. In the first two weeks, he has been visiting the Mathematics Department of the University of Torun, UMK, cooperating with IMPAN -BC. He gave there a lecture on positive entropy transformations. He worked with Professor M. Lemanczyk on the following question: Is it true that a compact extension of a transformation which is mixing relative to a factor will in fact, if it is relatively weakly mixing with respect to this* *same factor, stay relatively mixing? They also made corrections on their joint paper "relative discrete spectrum and joinings". He then spent two weeks at the Institute of Mathematics in Warsaw (IMPAN). He gave two lectures, one on a new proof using joinings of the Furstenberg-Roth theorem, the other on "Information theory and joinings". He is presently writing a paper on the topics discussed in this lecture which will also include some work which he achieved during his stay at IMPAN.

Fourth half – year period

- Dr Marian Grendar, from Slovak Academy of Sciences, Slovakia, visited
- Prof. Joerg Schmeling, Uni. Lund, Sweden, April 11- May 11, 2003.
- M. Sc. Steffi Burmeister, Uni. Goettingen, Germany.

MPAN- BC in 10 July – 9 August, 2002.

This was a continuation of his previous visits, to work with the Probability and Statistics people.

Fifth half –year period

He worked with mathematicians from the dynamical systems groups at IMPAN, Warsaw University and Warsaw University of Technology on geometric measure

theory in relation with solenoidal fractals.

He gave a 4-hour mini-course

"Dimension Theory in Dynamical Systems".

Abstract:

The dimension of invariant sets or invariant measures plays an important role in dynamcal system theory. It is an essential invariant that encodes geometric and

complexity information. It is closely related to other characteristics like Lyapunov exponents or entropy. In this series of lectures we will give an introduction

to the dimension theory of dynamical systems. First we will explore the theory of hyperbolic systems. We will start with the theory of low-dimensional systems and

then develop the general theory in higher dimensions. In particular we will discuss the existence of pointwise dimensions for hyperbolic measures. While the dimension theory is better understood for hyperbolic systems this theory is almost undeveloped

for systems of low complexity. We will emphazise on the different theory for zero entropy systems and show the connection to number theory on some examples.

*Finally we will introduce to the theory of multifractal formalisms. This theory gives a deep insight to fine aspects of dimension theory in low dimensions. In particular it* *helps to understand some classical problems from number theory. If there is some time left we will also discuss some non-standard multifractal scalings which ave implications to function theory.*

She stayed at IMPAN-BC for 2 months (1 Dec. 2002-31 Jan 2003, and the following 4 months as the Marie Curie training Site BANACH fellow.

During the IMPAN-BC months she has been attending 2 Dynamical Systems seminars at Warsaw University and at IMPAN, and the Stochastic Processes seminar led by Prof. Zabczyk at IMPAN. She gave a talk: "Almost sure local Limit Theorems". She discussed and worked in this topic in relation with Dynamical Systems with Prof. J. Zabczyk and Prof. M. Denker (who spent Spring semester at IMPAN, within another programme). She listened to M.Denker course on Martin Boundary for Sierpinski gasket.

Short visits:

First half – year period

- Prof. Flemming Topsoe, University of Copenhagen, Denmark, visited IMPAN-BC,

Warsaw in Jan 25 –28, 2001. F. Topsoe and P Harremoes presented two 2 hours talks on 27 Jan, 2001, Information Theory Friday.

- F. Topsoe, “Maximum Entropy versus Minimum Risk and Applications to Geometric, Binomial and Poisson Distributions”;
- Dr. Peter Harremoës “Kuhn-Tucker condition in information theory”.

- Dr. Peter Harremoës, University of Copenhagen, Denmark, visited IMPAN-BC,
- Karoly Simon, from Techn. University, Budapest, Hungary, visited IMPAN-BC, Warsaw in 8-15 August, 2002.
- Prof. Michel Zinsmeister, Uni. Orleans, France, visited IMPAN-BC for 3 days 7-9 March, 2003. He took part in preparation of the Conformal Structures and Dynamics (CODY) RTN FP6 proposal, had fruitful scientific discussions with dynamical systems people in Warsaw and gave a lecture:
- Prof. Feliks Przytycki, IMPAN-BC, (coordinator of the Centre) and K. Nowinski (in charge of the Package 8) participated in the IST, DG Information Society, Unit F2,
- Prof. Genadi Levin, Hebrew Uni, Jerusalem, Israel, visited IMPAN-BC on 13-19 April, 2003.

Warsaw in Jan 25 –28, 2001. See Prof. Flemming Topsoe.

Fourth half – year period

He worked on solenoids together with dr Michal Rams, on the possible applications and generalisations of their joint paper on this subject, concerning fractal dimensions.

Fifth half –year period

"Stochastic and deterministic growth models".

He presented a survey on the growth models introduced by the physicists: stochastic ones: Eden model, DLA (diffusion-limited aggregation), DBM (dielectric breakdown model) and the deterministic ones like Hele-Shaw flows. He also showed how Loewner differential equation allowed to unify them.

workshop in Brussels, 29-30 January 2003, "Grids for complex problem solving".

F. Przytycki at one of the thematic sessions presented the IMPAN-BC Centre of Excellence. Several informal discussions related to the possibilities to participate in the FP6 took place, in particular with a representative of the Heidelberg University Centre.

For more details on the meeting see www.cordis.lu/ist/events/29-01-2003.htm

He continued working with people in dynamical systems group, in particular with Feliks Przytycki, discussing conformal measures and fractal dimensions of some invariant sets for infinitely renormalizable maps of interval.

He gave a talk:

"Universality and dynamics of unimodal maps with infinite criticality" (Joint work with Greg Swiatek).

Abstract.

The universality in one-dimensional dynamics is described by fixed points of renormalization operators. We study the limiting behavior of these fixed-point maps

as the order of the critical point increases to infinity. It is shown that a limiting dynamics exists, with a critical point that is flat, but still having a well-behaved analytic continuation to a neighborhood of the real interval pinched at the critical point. We study the dynamics of limiting maps and prove their rigidity. In particular, the sequence of fixed points of renormalization converges, uniformly on the real domain, to a mapping of the limiting type, as the criticality tends to infinity along the reals.

Conferences and workshops:

First half – year period

- The international conference “Information Theory and its Applications to Biology,

Finance and Physics” was organized in Warsaw from May 21 to May 26, 2001. The conference was by nature inter-disciplinary. There was an emphasis on instructional lectures and survey lectures. More specialist lectures were presented with due regard to the professional, but mixed, audience being addressed. The Conference gathered approx. 35 participants.

Conference program:

__http://www.math.ku.dk/IT-Banach2001/may.html__

Conference Report:

__http://www.mdpi.org/entropy/htm/e3020064.htm__

b) A week long, April 23-30, 2001 event was organized together with the Institute of

Physics E.U. Center of Excellence “CELDIS”- __http://info.ifpan.edu.pl/celdis/__

__The first part of the workshop__ “The Information Theory Days” - was devoted to fundamental aspects of the Information Theory and it included seventeen presentations. The purpose of these presentations was to give an overview, and practical applications of classical information theory, computational biology, and molecular modeling of complex bio-molecular systems and processes. Then, fundamental of quantum information theory with applications to quantum computing, quantum cryptography and quantum teleportation have been subsequently discussed. Lectures have been given by leading specialists in molecular biology, physics, mathematics and bio-informatics.

In particular the following topics have been presented:

- Relations between DNA and m-RNA sequences with coded protein structures.
- Similarity and homology analysis of protein sequences.
- Practical algorithms in comparing sequences.
- Selected topics in structure of genomes.
- Genome rearrangements and evolution trees.
- Predicting 3D protein structures from sequences and international competitions in this area.
- From finite and discrete structure of the genetic information carriers to complexity of brain structure.

Howard Barnum and Martin Plenio provided tutorial lectures for non-specialists, presenting introductions to quantum information theory from mathematical and physical points of view, respectively. Howard Barnum gave a lecture on using non-local quantum correlation to the problem of data authentication. Nicolas Cerf started by recalling basic principles of quantum non-ideal cloning for the discrete case and reported on recent progress in cloning of continuous variables. Pawel Horodecki presented methods and results regarding nontrivial problem of entanglement of mixed states. Martin Plenio investigated asymptotic limits of multiparticle entanglement. Arkadiusz Orlowski gave an introduction to quantum teleportation and quantum cryptography. Primers were followed by more advanced presentations which allowed non-specialists to better understand the problems and to participate in discussions. The Workshop gathered approx. 40 participants.

The program : __http://www.icm.edu.pl/infodays.html__

__The second part of the__ “Workshop on Solid State Quantum Computing” was organized by CELDIS and was devoted to basic concepts and more specialized applications of quantum computing.

The program: __http://www.ifpan.edu.pl/~orlow/celdis/Workshop.html__

Second half - year period

A small workshop "Information Theory, Algorithms and Applications to Probability and Statistics", **package 2,** took place in October in Warsaw, in relation to a similar meeting in Centre of Excellence MATHIAS in Budapest. There were 10 participants including: F. Topsoe, P. Harremoes from Copenhagen, I. Csiszar, A. Gyorgy from Budapest and M. Grendar from Bratislava.

Third half – year period

- "Holomorphic Iteration, Non-uniform Hyperbolicity", 22-25 May, 2002 (in cooperation with the European Science Fundation). 64 mathematicians took part, including 45 from outside of Poland. Several PhD students participated. Talks were provided by outstanding specialist, including three lectures course "Non-uniformly hyperbolic horseshoes" by Fields medallist J. -C.Yoccoz.
- The meeting of the research group on "Characterizations and Bounds for Ordered Statistical Data" held in Banach Center in Warsaw on May 12-17, 2003. This concerns also package 5.

Fifth half –year period

The purpose of the working group was to analyze the state of art in order statistics, in particular its characterizations and bounds. The bounds appear to be useful in various applications, in particular in insurance and biology. A special panel discussion was devoted to specify the areas of possible further studies and extensions.

Organizers: T. Rychlik (IMPAN), J. Wesolowski (Warsaw University of Technology).

__ Summary.__ This package has had a rich history. Its activities started with several visits of F. Topsoe, who organized successful courses and 2 workshops and 1 month visit of B. Ryabko. However there were troubles with the Polish audience; few Polish YR were interested, as the topic had little traditions in Warsaw. Later the scientific area has been extended to Ergodic Theory and Dynamical Systems and in cooperation with the PRODYN program of the European Science Foundation and Polish national programmes has greatly influenced progresses of Warsaw and Katowice group in Dynamical Systems and applications in biology and statistics. One joint workshop on a high level with physicists (CELDIS Centre of Excellence) including quantum information and cryptography was organized. Recently (June/July 2003) a course by Tyll Krueger was a highlight. Still some visits by YR remain to be organized, together with a workshop bridging us to FP6 programmes in Marie Curie actions in which we have applied, RTN CODY and ToK BANACH SPADE, if successful.

Package No. 3 "Mathematical Modelling and Analysis of Cellular Populations"

(coordinator R. Rudnicki).

Personnel:

Fourth half - year period

- Prof. Ovide Arino, from University de Pau, France, visited IMPAN-BC, Silesian branch, in 20 November – 20 December, 2002.
- Dr M. Molina Becerra, from University Sevilla, Spain, visited IMPAB-BC in 20 November– 19 December, 2002.
- Dr. Mostafa Bachar, Uni. Graz, Austria, April 24 – May 24, 2003.

He worked together with the team of prof. R. Rudnicki. A preliminary version of the paper O. Arino and R. Rudnicki "Stability of phytoplankton dynamics" has been prepared.

O. Arino gave a lecture at the meeting of the Institute of Mathematics at the

Silesian University: "A mathematical derivation of size spectra in fish populations".

Stay in Silesian branch of IM PAN and in Silesian University in Katowice.

Scientific cooperation with the team of prof. R. Rudnicki and with prof. O. Arino.

Under the direction of O. Arino and R. Rudnicki, she studied structured population models. She worked on generalization of M.C. Mackey and R. Rudnicki theorem on stability of partial differential equations with retarded space argument and

delay time argument.

Fifth half –year period

Stay in Katowice Branch of IM PAS and in Silesian University in Katowice.

Scientific cooperation with the team of prof. R. Rudnicki.

He studied a model of neutrophil dynamics. Such a model is described by a differential equation with delay which is distributed with some density. It is observed that some hematological diseases display a distinct of periodic nature. Bachar tried to confirm this fact mathematically and proved the existence of periodic solutions of this equation.

During his visit he had an opportunity to discuss with M.C. Mackey and scientists from the Silesian University medical aspects of hematological diseases, the way of treatment etc.

Short visits:

Third half – year period

- Prof. Ryszard Rudnicki, Torino, Italy, 25
^{th}February -3^{rd}March, 2002. - Prof. Michael Mackey, Mc Gill Uni., Montreal, Canada, May 17- 30, 2003. Stay in Katowice Branch of IM PAS and in Silesian University in Katowice.
- Prof. Ryszard Rudnicki, IMPAN-BC, visited France, UR GEODES, Institut de Recherche pour le Dèveloppement (IRD), Bondy, February 2 – 23, 2003.

Scientific collaboration with Prof. Hisao Fujita Yashima.

During the visit he has prepared a preliminary version of the paper "Long-time behaviour of a prey-predator model". He gave a lecture on the "Seminari di analisi matematics" on the subject "Markov semigroups and their applications", 26^{th} February, Department of Math. Univ.Torino.

He also gave a series of "working lectures” connected with the above publication.

Another aim of the visit has been to prepare a conference at IMPAN-BC in June.

Fifth half –year period

Scientific cooperation with the team of prof. R. Rudnicki and with prof. A. Lasota.

During his visit M.C. Mackey discussed the questions connected with dynamic behaviour in mathematical models of the tryptophon and lactose operons. The general theory of operon postulates that all genes are controlled by means of operons subject to a single regulatory mechanism known as repression. There are several models of operons: starting from the system of three ordinary differential equations to the system of five differential equations with delay. The most important question of global stability for this model were discussed.

Lecture on the meeting of the Institute of Mathematics Silesian University:

"Modelling molecular central systems: two examples and open problems".

He cooperated with Prof. Ovide Arino. He gave two lectures.

- Lecture on the meeting of the Institut de Recherche pour le Dèveloppement:

Chaos for PDE.

- Lecture on the meeting at Universite Pierre et Marie Curie (Paris 6), Faculte de la Medecine: Markov semigroups and their applications.

The work on phytoplankton dynamics and on Individual Base Models (IBM) has been continued. This kind of models describes many interesting phynomena: from aggregates of mammalian cells to self-organisation of herds of wildbeast and fish schools. The problem of the passage from IBM to Eulerian models which describes the evolution of densities has been studied.

Conferences and Workshops:

Fourth half – year period

a) “School in Population Dynamics”, 17 –21 June, 2002, in IMPAN-BC, Będlewo.

There were primary courses 4 hours each by:

- Ovide Arino
*,*“Delays Induced in Population Dynamics”, - Odo Diekmann, “Adaptive Dynamics”,
- Mimmo Iannelli, “Modelling Epidemics”,
- Michael C. Mackey, “Mathematics, Dynamics and Medicine: The role of mathematics and physics in biological and clinical research for the new millennium”,
- Glenn F. Webb, “Structured Population Dynamics”,

suplementary courses:

- Marek Kimmel, “Why Mathematics Is Needed to Understand Complex Genetic Diseases”,
- Andrzej Lasota, “Fractals, Semifractals and Population Dynamics”,
- Philip K. Maini, “Using Mathematics to Help Understand Pattern Formation”,
- Ziad Taib, “On the Galton-Watson Process”,
- Branka Cajavec, “Getting Started with Molecular Biology”,

and several lectures and short communications. 87 mathematicians (most young researchers) from 13 states participated.

b) Conference on “Mathematical Modelling of Population Dynamics”, Będlewo, 24 –

28 June, 2002.

The organizers are preparing Proceedings of the Conference and School to be published in "Banach Center Publication" and "Comptes Rendus de L'Academie des Sciences".

There were the following plenary lectures:

- Pierre Auger, “Aggregation methods in population dynamic:

from individuals to populations and communities”,

- Karl P. Hadeler, “Is Hutchinson's equation a population model, Rigorous derivation of population models in the form of standard or even neutral delay equations from structured population models and renewal equations’’,
- Uwe an der Heiden, “New scenarios of nonlinear growth models with delays”,
- Horst R. Thieme, “An endemic model with variable re-infection rate: an
- example for some problems and their solution arising in general structured population theory”.

Moreover 60 invited talks took place and several short communications were delivered. 90 mathematicians from 20 states participated.

A special session celebrating the 70 th birthday of Prof. A. Lasota was organized with 1 hour talks by: J.A. Yorke, M.C. Mackey, J. Myjak, R. Rudnicki.

__ Summary__. This package, which started only 1 year ago, is in a half way. It organized very impresive Schoola dn Conference on population dynamics, with a participation with biologists. We do hope it will bridge us to FP6. One of outcomes will be a creation of a bioinformatics seminar at IMPAN next year.

Package No. 4. ,,Nonlinear Systems and Control’’

(coordinator B. Jakubczyk).

The following activities took place:

Personnel:

First half – year period

a) Prof. Witold Respondek, V1, Institute National des Sciences Appliquees de

Rouen, France, 2 weeks in April-May 2001.

b) Dr Issa Tall, Institute National des Sciences Appliquees de

Rouen, France, 3 months in April-July 2001.

W. Respondek and Issa Tall together with Prof. B. Jakubczyk from IMPAN worked on canonical forms for single-input control systems under the feedback group action. They atacked also the fundamental problem of convergence of the normalizing procedure. An IMPAN preprint is in preparation. They participated in a workshop: Control and Stabilization of Dynamical Systems, 17 April - 2 May, organized at Banach Center by B. Jakubczyk (IMPAN) and prof. A. M. Kovalev (Donetsk, Ukraine). W. Respondek gave at the workshop the talk "Geometry of Control Systems Equivalent to Canonical Contact Systems", I. Tall gave the talk "Symmetries of single-input control systems". I. Tall took part also in the 33th Symposium on Mathematical Physics in Torun, Poland, where he gave the talk "Symmetries of systems with corank 1 affine nonholonomic constrains".

Fourth half – year period

c) Professor Witold Respondek from INSA de Rouen, France, visited IMPAN-BC in

August and September 2002 (2 months). In this period he wrote lecture notes for his course "Geometry of static and dynamic feedback" at the Summer School on Mathematical Control Theory (they were printed and distributed to participants). During the School (September 2-20) he gave a course of 9 lectures. In addition to this, he collaborated with Bronislaw Jakubczyk on bifurcation on control systems in the plane. A classification of generic 1-parameter local bifurcations was obtained. A paper entitled "Bifurcations of control-affine systems in the plane" is in preparation and should be finished at the beginning of 2003.

Fifth half –year period

- Prof. Michail Zhitomirski, Uni. Technion, Haifa, Israel, January 14- February 3,

2003 and February 12-21, 2003.

He gave a lecture on the Singularity Seminar "Symmetric centers".

He cooperated with W. Domitrz, S. Janeczko, B. Jakubczyk, P. Mormul (collaboration with package 7).

Two papers are in preparation:

- The relative Darboux theorem for singular varietes in a symplectic manifold and its applications.
- The relative Poincaré lemma for singular varietes and guasi-homogenity.

- Prof. W. Respondek, INSA-Rouen, France, visited the Banach Center in April 2003 (1 month). He collaborated with Prof. B. Jakubczyk on bifurcations of control systems. The result of this collaboration is the paper: "Bifurcations and phase portraits of control-affine systems in the plane" (41 pages) which was submitted for publication.

During his visit the final preparations were done for the CE Workshop "Geometry in Nonlinear Control" in Bedlewo, June 2003, organized by J.P. Gauthier (Dijon), B. Jakubczyk (Warsaw), W. Respondek (Rouen), H.J. Sussmann (New Brunswick) and M. Zeitz (Stuttgart) (CE, the packadge "Nonlinear Control").

Short visits:

Fourth half - year period

- Professor Fernand Pelletier, from Universite de Savoie, visited IMPAN-BC in 1-15

June 2002. He gave a lecture at the Banach Center Workshop "Caustics II". He also gave informal seminars and discussed recent research problems with Marek Grochowski, Bronislaw Jakubczyk, Stanislaw Janeczko, Tadeusz Mostowski, Piotr Mormul and other mathematicians.

Conferences and Workshops:

Fourth half - year period

- School on “Mathematical Control Theory”, Bedlewo, 2-13 September, 2002.

The following courses took place:

- V. Jurdjevic, “Variational problems and optimal control”, ( 7 lectures),
- W. Respondek, “Geometry of static and dynamic feedback”, ( 7 lectures),
- Jakubczyk, “Introduction to Hamiltonian formalism”, (4 lectures),
- J. Zabczyk, “Control and stabilization of linear systems with distributed parameters”, (5 lectures).

** Summary.** This package, which started only 1 year ago, is in a half way. It works in a cooperation with FP5 Marie Curie Control Theory Network, and developed an intensive cooperation with Trieste SISSA.

Package No. 5. ,,Finance Mathematics and Stochastic Control''

(coordinator Łukasz Stettner).

The following activities took place:

Personnel:

Second half – year period

- Nguen Pham,
*Uni. Paris 7, France*, on his 2 two-weeks visits in June and October 2001 delivered a series of lectures (10 hours) on pricing and construction of portfolio with constraints. - Prof. Z. Brzezniak, University of Hull, U. K., two 2 weeks visits in March and in May 2002. Prof. Brzezniak collaborated with prof. J. Zabczyk and Sz. Peszat on various aspects of stochastic analysis and stochastic partial differential equations with possible applications in finance;
- Prof. M. Jeanblanc, University d'Evry, France, one 2 weeks visit in May 2002 (the second 2 weeks visit is planned in the Fall 2002). Prof. M. Jeanblanc collaborated with prof. M. Rutkowski and L. Stettner on modelling of defaultable risk, and portfolios with random termination;

Third half – year period

- Both visitors had lectures on the seminars organized at IMPAN. Prof. M.
- Jeanblanc additionally gave lectures in Wroclaw and at the Warsaw Technical
- University.

Fourth half – year period

- Prof. Roger Tribe, from University of Warwick, England, visited IMPAN-BC in 19 August – 2 September, 2002, (shall continue in 2003), he worked with prof. J. Zabczyk on infinite particle systems.
- Prof. Remi Leandre, from Nancy University 1, France, visited IMPAN-BC in 28 May – 27 June, 2002, worked with prof. J.Zabczyk on stochastic systems on manifolds with applications to physics, prepared during his stay at IMPAN preprint “Hypoelliptic diffusions and cyclic cohomology”.
- Dr Miklos Rasonyi, form CAI, Hungarian Academy of Sciences, Hungary, in 10 October – 9 December, 2002, visited prof. L. Stettner, working in mathematics of finance. He presented a series of lectures on arbitrage of large financial matters. He worked with L. Stettner on utility maximization under general conditions.
- Prof. Giovanni di Masi, Uni. Padova, Italy, 1. April 2003 (one month), prof. G. Di
- Dr. Miklos Rasonyi, Computer and Automation Institute, Hungarian Academy of Sciences, Hungary, 2. April – May 2003 (two months) (earlier also November - December 2002), delivered a series of lectures on large financial markets and collaborated with prof. L. Stettner on maximization of a general utility function. The following paper has been written: M. Rasonyi, L. Stettner, On utility maximization in discrete - time market models, preprint.

He presented a talk on cohomology and hypoelliptic diffusions, presented a lecture on asymptotic behaviour of infinite particle systems on the Seminar at IMPAN.

Each visitor had a lecture or a series of lectures on the seminars held at IMPAN. Some results of the cooperation between M. Rasonyi and L. Stettner perhaps will be prepared in the form of a publication.

Fifth half –year period

Masi from Dipartimento di Matematica Pura ed Applicata Universita Degli Studi di Padova - visit in the category of senior professors; Prof. Di Masi collaborated with prof. L. Stettner on the problem of invariant measures for filtering processes. As a result of the cooperation the following paper was written:

G. Di Masi, L. Stettner, Ergodicity of Hidden Markov Models, preprint.

The paper contains necessary and sufficient conditions for the existence of invariant measures of filtering processes and attempts to fill (partially) the gap in the famous Kunita's papers from 1971.

The paper provides a conditions for the existence of an optimal strategy maximizing the expected terminal utility. Using optimal strategies the martingale measures are then constructed;

Short visit:

Fifth half – year period

- Dr. Enrico Priola, Uni. Torino, Italy, Oct. 28 – Dec. 7, 2002.
- Prof. Lukasz Stettner, IMPAN-BC.

He visited shortly Warsaw to cooperate with prof. J. Zabczyk and Sz. Peszat on various aspects of infinite dimensional stochastic analysis. He actively attended the seminar on stochastic processes organized regularly at IMPAN.

Prof. L. Stettner and Prof. St. Janeczko attended the meeting of ERCOM in Vienna on March 15 and 16th, 2003. The purpose of the meeting was to stimulate and extend cooperation between mathematical centers in Europe. During the meeting prof. L. Stettner discussed with the other directors on various aspects of mathematics of finance and stochastical analysis, in particular in the context of research and education in these fields. Prof. St. Janeczko talked to prof. P. Michor on possible applications of differential and partial differential equations, in particular theory of singularities in mathematical physics. He attended the "Discussion of research abilities in pure and applied mathematics" and "Projects for PhD training and exchange".

Conferences and Workshops:

Second half – year period

**"The Workshop on Mathematics of Finance**" was organized in Będlewo on

June 4-9, 2001.

The following problems were considered at the workshop:

- pricing of financial derivatives - 5 lectures,
- term structure models - 7 lectures,
- portfolio optimalization - 4 lectures,
- numerical methods - 4 lectures,
- arbitrage theory - 3 lectures,
- general theory of markets - 3.

**"Summer School on Mathematics of Finance**" in Będlewo on July 2-10 th, 2001.

The school gathered approximately 80 participants (including 7 lecturers). The following lecture were presented:

K. Krzyżewski (UW) - ,,Portfolio analysis" - 12 hours,

J. Jakubowski (UW) - ,,Introduction to stochastic analysis'' - 10 hours,

Ł. Stettner (IMPAN) - ,,Introduction to pricing of financial derivatives (in discrete time)" -10 hours,

- Palczewski (UW) - ,, Pricing of financial derivatives (continuous time) -

10 hours,

A. and R. Weron (PWr) - ,,Financial engineering" - 8 hours,

M. Rutkowski (PW) - ,,Interest rate models" -10 hours,

Other activity:

Seminar: (Ł. Stettner) Weekly working seminar devoted to mathematical methods of mathematics of finance took place. It has gathered all specialist in Poland as guest at CE IMPAN-BC including PhD and Masters thesis students.

Fourth half – year period

c) “Stochastic Control and Its Applications”, held in Bedlewo 2-8 June,

2002. Organizers: Lukasz Stettner and Jerzy Zabczyk.

The conference gathered 35 scientists: 10 from Poland, 25 from abroad. The conference was devoted mainly to stochastic control, stochastic analysis and mathematics of finance. 28 talks were presented. The following problems were considered:

filtering theory and control with partial observation - 4 talks,

fractional Brownian motion - 3 talks,

stochastic approximation - 2 talks,

hedging and pricing of financial derivatives - 5 talks,

interest rate models - 3 talks,

optimal investment portfolios - 4 talks.

d) Workshop on “Optimal Stopping and Stochastic Games”, Banach Centre,

coorganized by Wroclaw University of Technology, held in Bedlewo, 1 –7 July, 2002.

Organizers: Zdzislaw Porosinski, Krzysztof Szajowski (Wroclaw Univ. of

Tech.), Lukasz Stettner (IMPAN).

The workshop gathered 45 scientists: 23 from Poland, 22 from abroad. The workshop was devoted to optimal stopping problems and games with optimal stopping. 39 talks were presented. The following problems were considered:

optimal selection problems - 5 talks,

zero sum stopping games - 5 talks,

multiperson optimal stopping games - 6 talks,

prophet inequalities - 4 talks,

secretary problem - 6 talks.

**Fifth half –year period**

Participation of three scientists (prof. O.Boxma from Eindhoven Univ. of Technology, H. Daduna from Univ. of Hamburg and A. Muller from Univ. of Karlsruhe) The conference besides of queuing and communication theory was devoted also to the various aspects of risk arising in mathematics of finance and insurance theory and therefore closely related to the main topic of the package.

For more information see www.math.uni.wroc.pl./workshop/workshop.html

** Summary.** This package accomplished already most of its plans. It has been working in cooperation with a large Polish project ,,Risk Management in Complex Random Systems’’. On its basis the workshop ,,Analysis of Random Markets: Products and Prices” to bridge the package to this FP6 project will be organized in autumn 2003.

Package No. 6. ,,Approximation Structures in Banach Spaces''

(coordinator: Przemysław Wojtaszczyk).

The full activities have started only in January 2002.

Personnel:

First half – year period

- Dr. Ignacio VILLANUEVA (Univ. Complutense, Madrid) stayed at IMPAN-BC in 1 May -30 June 2001. He worked (under guidance of Prof. Pelczynski) on multilinear operations in Banach spaces. Jointly with prof. Pelczynski he obtained results on complements of rich, in the sense of Bourgain, translation invariant Banach spaces. He gave a 2 hours talk at the Functional Analysis Seminar, entitled: “Banach spac properties of the projective tensor products of Banach spaces”.

Fifth half –year period

- Prof. Mostafa Mbekhta, Uni. Lille 1, France, May 6 -18, 2003.
- Prof. Boryslav Boyanov, Uni. Sofia, Bulgaria, February 2 – April 2, 2003. He cooperated with Prof. P. Wojtaszczyk and prof. H. Wozniakowski group at Warsaw University, where he delivered a beautiful lecture: "Multidimensional quadrature formulas" and Functional Analysis seminar.
- Prof. Victor Kolyada, Uni. Autonoma, Madrid, Spain, March 21- April, 2003.

He gave a lecture on "Distance formulas for several spectra" at the Operator Theory Seminar at the IMPAN-BC and discussed with some participants, mainly PhD students from various cities.

Together with Prof. Jaroslav Zemanek he continued joint work on geometric characterizations of spectral quantities of linear operators (this seems to be nearly prepared for publication), and discussed other problems concerning operator ergodic theory that emerge from earlier papers.

About the same time the book "Operator Theory and Banach Algebras"

(Conference Proceedings, Rabat (Morocco), April 1999) has appeared in Theta, Bucharest 2003, distributed by the AMS. They were two of the editors of that book, and so they had an occasion to discuss also the problems of wider context arising from that book.

He visited Cracow branch of IMPAN and Prof. Plesniak group at Jagiellonian University. At IMPAN he gave a talk: "Landau-Kolmogorov inequality and the problem of Erdös".

He collaborated with Prof. P. Wojtaszczyk, Prof. A. Pelczynski and prof. Wozniakowski. He gave a lecture "Transplantation theorems for ultraspherical polynomials in ReH^1 and BMO".

Short visits:

Second half – year period

- Professor Gilles Pisier -correspondent member of the French Academy visited Poland from June 16 to June 20, 2001. On June 18 he received the Banach medal. His invited speech (one hour) concerned problems, some relations between Operator Theory and group Representation. He spoke also at the IMPAN Functional Analysis Seminar in Warsaw (2hours) on June 19. Title of his talk: "Multipliers of the Hardy space $H^1$ and power bounded operators"

b) Professor Szarek, Uni. Paris 6, visited Warsaw from Nov 7 to Nov 13 , 2001.

He spoke at the IMPAN Functional Analysis Seminar in Warszawa (2 hours)

on Nov. 13. Title of his talk:

"The hyperbolic plane, expander graphs and approximation of operators."

c) Prof. Aleksander Pelczynski, from IMPAN-BC, visited in Nov. 24- 30, 2001,

University of Kiel, Germany, for** **the invitation of Prof. Hermann Koenig. He had a lecture on spaces of functions with bounded variation, based on the results of the joint paper with Dr. hab. Michal Wojciechowski, accepted recently by J. Angew. Math. (Crelle). He had scientific discussions with: prof. prof. E. Koenig, D. Mueller, N. J. Nielsen on Sobolev spaces, local theory of Banach spaces and some conjectures on generalized Chinchin inequality.

Third half – year period

d) Marianna Csornyei, University College, London.

Talk "How to make Davies' theorem visible?", 16^{th} January, 2002.

- Pavel Shvartzman, Technion. Hajfa, Israel, talk "The Whitney extension problem
- Tomas Schlumprecht, University of Texas A&M, talk "On trees and Branches in
- Paul Mueller, University of Linz, research joint with A. Pelczynski, participation in
- Prof. Aleksander Pelczynski, from IMPAN-BC, visited Germany in 17–27 November, 2002). He participated in an international meeting "8-th Wangerooge-Meeting on Functional Analysis" (Nov. 17 - Nov. 21) organized by Mathematical departament of the Oldenburg University with about 35 participants from Belgium, Germany, Poland, Spain and Switzerland. He delivered an one hour talk "On strict Dvoretzky Theorem". From Nov. 21 to Nov. 27 he was in Kiel where he was an invited speaker on a weakly colloquium of the Department of Mathematics of the Christian Albrecht University in Kiel. The topic of the colloquium talk: "On elliptic sections of convex bodies". Toogether with Professor Hermann Kőnig he worked on the project concerning complex zonotops and zonoids.
- Prof. Evgenij Bereznoj, from University Jaroslaw, Russia, visited IMPAN-BC in 10-17 October, 2002.
- Prof. A. Montes Rodriguez, from University of Seville, Spain, visited IMPAN-BC in 1 –15 July, 2002.
- Prof. Zbigniew Ciesielski from IMPAN-BC, took part in “International Conference on Constructive Function Theory”, Varna, Bulgaria, in 19-23 June, 2002
**.** - Anna Kamont, from IMPAN BC, participated in the “Fifth International Conference on Curves and Surfaces”, Staint-Malo, France, June 27 - July 3, 2002.
- Prof. Sergei Konyagin, Moscow State Uni., Russia. He attended the Workshop “Computational Number Theory and Cryptography...” he gave also a lecture at the Functional Analysis Seminar "About exponential sums on groups"
- Prof. Dirk Werner, Freie Universitaet Berlin, visited IMPAN-BC in March 3-7, 2003.

and Helly's Theorem".

Banach spaces".

the functional analysis seminars.

Fourth half – year period

Most time he worked with prof. Mastylo in the IMPAN-BC branch in Poznan. The topic was extrapolation of positive operators between function Banach spaces. The starting point was Schur test. Applied to integral operators in Lp weighted spaces, it led to an extrapolation theorem. Schur test was studied also for Hardy operators in Orlicz spaces. A joint paper is in preparation.

E. Bereznoj gave a lecture at Math Dept of Adam Mickiewicz University, on 5th October, entitled: "Subspaces of C(0,1) and Lip(0,1) consisting of non-smooth functions".

He worked together with prof. J. Zemanek on various questions of operator theory, in particular, on hypercyclic and supercyclic operators. Some of the problems, concerning the Volterra operator, have been subsequently solved by prof. Alfonso Montes-Rodriguez and his former student dr. Eva Gallardo, and this work is accepted for publication in Integral Equations and Operator Theory, and was also presented at a conference in Luminy.

As a member of the Program Committee he took part in preparing the scientific part of the conference. This conference was the consecutive in the series of the well established of international conferences on Constructive Function Theory held in Bulgaria since 1970. It was dedicated to Blagovest Sendov on his seventies birth anniversary. On the occasion a volume Approximation Theory edited by B.D. Bojanov, DARBA, Sofia 2002 was published in this volume A. Kamont and Z. Ciesielski published “Survey on the orthogonal Franklin system”*.*

It was fifth in a series of conferences on Curves and Surfaces organized by Association Française d'Approximation since 1990. The topic of the conference was approximation theory and its applications. The conference included a series of mini-symposia. At the conference, she presented a talk "Greedy approximation and multivariate Haar system". The talk was a part of the mini-symposium on Nonlinear Approximation, organized by prof. V.N. Temlyakov from University of South Carolina.

Fifth half –year period

He worked together with A. Pelczynski, P. Mankiewicz and P. Wojtaszczyk.

He delivered a talk on the Functional Analysis Seminar:"On narrow operators on Banach spaces".

Conferences and Workshops:

Third half – year period

- "Operator Theory and its Applications in Mathematical Physics", 11-18 May, 2002, Bedlewo. 50 participants, in particular from: Austria, Belgium, Czech Republic, Finland, Greece, Spain, Romania, Sweden, Ukraine, Ireland, Germany, New Zealand, Mexico, Russia, USA.
- Workshop "Approximation Theory", Warsaw, 25-28 January, 2002. 22 participants.
- Conference on Functional Analysis, Bedlewo, 22-29 September, 2002.
- Conference on Topological Algebras, their Applications, and Related Topics. Bedlewo, 11--17 May, 2003.
- “
__Spaces of Analytic and Smooth Functions__”, 6 - 12 April, 2003,

The conference topics: singular perturbations, random Schrödinger operators, spectral theory of Jacobi operators.

Fourth half – year period

83 mathematicians participated.

Fifth half –year period

Organized jointly by IMPAN-BC and the Faculty of Mathematics

and Computer Science, Adam Mickiewicz University, Poznan.

Organizing Committee:

H. Garth Dales (University of Leeds, England) - the head of the Scientific

Committee,

Krzysztof Jarosz (Southern Illinois University at Edwardsville, USA), Andrzej Soltysiak (Adam Mickiewicz University, Poznan, Poland) -the local organizing committee.

61 scholars from 21 countries participated.

The number of lectures was 57; 14 of them were 40 minutes long, 30 - 30 minutes long, and 13 - 20 minutes long.

The talks were concerned with new results in general theory of Banach and topological algebras, spectral theory, functional calculi in Banach algebras,

operator theory on Hilbert and Banach spaces, C*-algebras and their generalizations, homologies of Banach algebras, Banach-Lie algebras and Lie algebras of operators, Choquet theory of integration, and topological dynamics. 70th birthday anniversary of one of the outstanding Polish mathematicians Professor Wieslaw Zelazko, working in the area of Banach and topological algebras was celebrated.

(Abstracts are available at IMPAN-BC).

The conference brought together researchers working in different branches of Functional Analysis but interested in or using spaces of analytic or smooth functions. Topics discussed included operator theory on Hilbert spaces (Zhu, Mityagin, Jasiczak), nuclear spaces (Zaharyuta), Sobolev spaces (Pelczynski), interpolation theory (Mastylo, Kislyakov) etc.

__ Summary__. This package is in its biggest momentum now. Several visits of personnel are planned in autumn 2003. The area of the package has been extended from Approximation Theory to various related fields in Functional Analysis.

Package No. 7. ,,Symplectic Singularities and Applications''

__(__*coordinator: Stanisław Janeczko).*

Personnel :

Fourth half – year period

- Prof. Bronislaw Wajnryb, from Institute of Techn., Hajfa Israel, stay at IMPAN-BC
- Prof. Joachim Rieger, Uni. Halle, Germany, April 23 – May 10, 2003.

15/09/2002 - 27/02/2003,(15 September – 31October, 2002 – he was paid by EC). He was teaching a course on Braid groups, 11 lectures. Several faculty members and over 10 students participated in the course and some of them will take exam. He gave several invited lectures:

“Artin groups and geometric monodromy”, Colloquium talk for graduate students at IMPAN, Warsaw.

“Artin groups and geometric monodromy”, at the seminar of Professor Janeczko at Warsaw Polytechnic Institute.

“Approximation with multivariate polynomials”, Colloquium talk at Department of Mathematics of Gdansk University.

“Approximation with multivariate polynomials”, at the meeting of Polish Mathematical Society at IMPAN, Warsaw.

“Braid groups and its applications”, at the meeting of Scientific Society of Lodz.

He was also invited to give a Colloquium talk at Bayreuth University, Germany, on January 23, 2003.

He is also working on a research paper “Configuration space of generic sets of points in C²”.

Fifth half –year period

Two lectures on the Singularity Theory Seminar:

- "Real deformation of A-simle rank p-1 germs From R
^{n}to R^{p}for n>p-1"

- "Criteria for A-finites of complex-analytic corank 1 germs from C
^{n}to

C^{p} for n<p+1"

Colaboration with W. Domitrz and S. Janeczko.

Classification of germs of mappings from *R ^{n}* to

Short visit:

Second half – year period

- Dr. Alain Joets gave a talk at the Seminar on Singularities Theory.
- Prof. Grigori Karlovich Giorgadze, from Georg. Academy of Scienc., Tbilisi, Georgia, visited IMPAN-BC, Warsaw in 1-12 June, 2002.
- Prof. Stanislaw Janeczko, from IMPAN-BC, visited Max Planck Institute, Bonn,
- Prof. Takuo Fukuda, Department of Mathematics, College of Humanities and
- Prof. Stanislaw Janeczko, IMPAN-BC, attended the meeting of ERCOM in
- Prof. Stanislaw Janeczko, IMPAN-BC, participated in the Conference on occasion of 50 Anniversary of Czech Academy of Sciences in Prague, Czech, December 11- 13, 2003.

He collaborated with Polish mathematicians in:

- Bifurcations of symmetrical caustics - the creation - annihilation of two symmetric butterflies.

- Classification of Lagrangian singularities invariant through the mirror symmetry.

- On some generalization of Gauss-Bonnet formula for singular discriminant surfaces.

Fourth half – year period

During his stay Prof. G. Giorgadze was working on the monodromy problems for Fuchsian differential equations. He gave several lectures on monodromy and Riemann-Hilbert boundary value problem and the relations of these topics with some analytical and topological aspects of the theory elliptic Carleman-Bers-Vekua systems discussed in some papers of Prof. B. Bojarski. As a result of this cooperation G. Giorgadze prepared the paper: “Lp – connections on holomorphic bundles over Riemann Surfaces”.

Germany in 4 –29 August, 2002.

During his stay at MPI prof. Janeczko worked on symplectic geometry together

with M. Marcolli, S.Stolz, and others. He gave a talk at Hirzebruch-Manin-Zagier Oberseminar entitled "Symplectic invariants of curves and bifurcations". He prepared Preprint 2002/132 (jointly with Z. Jelonek) "Linear automorphisms to become symplectomorphism" and Preprint 2002/62 (jointly with Fukuda) "Smooth integrability of differential systems" about generic (in the sense of Tougeron) properties of implicit systems to be smoothly integrable. This was applied to Hamiltonian systems of Dirac type. The visit was an element of our cooperation with MPI at Bonn continued later in November within the "Manifolds in Mathematics" symposium at IMPAN-BC in Warsaw.

Fifth half –year period

Sciences, Tokyo, Japan, visited IMPAN-BC in Dec 28, 2002 – Jan 7, 2003.

He worked, jointly with Prof. S. Janeczko on the paper "Integrability criteria for implicit differential systems" (submitted to Banach Center Publications) and "Classification of Dirac singularities for generalized Hamiltonian systems" (in preparation).

Vienna on March 15 and 16th, 2003. See In package 5 Prof. Lukasz Stettner.

He attended the Panel discussions on differential systems and applications to thermodynamics of continuous media.

Moreover he collaborated with Mathematical Applied Groups in Mathematical Research Institutes of Academies.

Conferences and Workshops:

Fourth half – year period

- “Geometry and Topology of Caustics”, Warsaw, 17-29 June, 2002.
- “Manifolds in Mathematics”, Warsaw, 6-9 November, 2002

- Lagrangian and legendrian singularities

- Asymptotic behaviour around caustics and wavefronts

- Singular symplectic, contact and Poisson spaces

- Systems of rays, optical caustics

- Bifurcations of caustics and wavefronts

- Singularities of J-holomorphic curves

II. Symplectic invariants

- Vassiliev invariants, Gromov-Witten invariants

- Lagrangian cobordism invariants

- Symplectic capacities

- Singular reduction, Hamiltonian systems

- Applications to physical systems and control theory

This has been a short symposium accompanying the programme “Science and Art in Europe: Max Planck Society and Scientific Institutions of Poland”, cosponsored by Centre of Excellence IMPAN-BC. Outstanding mathematicians from the Max-Planck-Institute in Bonn and from IMPAN delivered 6 lectures. About 100 mathematicians participated.

c) Polish-Japanese Singularity Theory Working Days, Bedlewo, 3-10 September, 2002.

Topics of the conference:

- Differential geometry and singularities
- Hamiltonian systems and generalizations
- Singularities of Lagrangian and Legendrian varieties
- Subanalytic and semialgebraic sets - Lipschitz stratification
- Blow analytic equivalence
- Singularities of smooth maps and differential forms

There were 29 lectures, 35 participants (17 from Poland, 15 from Japan, 2 from France, 1 from Germany). The Banach Center Volume Proceedings of this Conference “ Geometric Singularity Theory ” will be published. Most was financed from other source. IMPAN-BC financed Slodowy (Germany) and partially Fukuda (Japan).

**Summary.** This package, which started only 1 year ago, is in a half way. It develops very well and will (hopefully) be continued in FP6 ToK: BANACH SPADE, in relation to PDE's and Control Theory. A cooperation with Japanese, German and French groups has been developed.

Package No. 8. “Visual Modelling”

*(coordinated by ICM *http://www.icm.edu.pl)

Personnel:

First half – year period

- Prof. Gerhard Dziuk, Institute of Applied Mathematics, University of Freiburg, Germany, was a guest of the IMPAN-BC Centre of Excellence from 19 February 2001 to 2 March 2001.
- Prof. Alain Bossavit, Electricité de France, R&D Division, Clamart, France was a guest of the IMPAN-BC Centre of Excellence from: 17 April 2001 to 27 April 2001. He presented a series of lectures entitled "From cell complexes to 'mixed' and mixed-hybrid' numerical schemes"..
- Prof. Alain Bossavit, Nov 25, 2001 – Dec 10, 2001. The aim of the visit was to create a written version of the lectures entitled "From cell complexes to 'mixed' and mixed-hybrid' numerical schemes" and to make them available to a wider audience via the internet.

Prof. Dziuk presented a series of lectures devoted to “Computational Methods for Geometric Differential Equations”

The audience included University staff, research institutes staff, Ph.D. students and graduate students of various affiliations.

It is envisaged that Prof. Dziuk’s visit will result in further co-operation related to the software for scientific computations and applications Albert. Albert (Adaptive multiLevel finite element toolbox using Bisectioning refinement and Error control by Residual Techniques) was developed in the Institute of Applied Mathematics, University of Freiburg.

Three classes of model problems were used in order to provide illustrations (1) Electrodynamics, (2) Underground water dynamics, (3) Molecular dynamics. The course was addressed not only to mathematicians but code-developers and users of numerical methods of 'finite element' and/or 'finite volume' flavour as well.

A very important aspect of Prof. Bossavit visit was related to the co-operation between ICM and Netherlands Institute of Applied Geoscience TNO. From the mathematical point of view, the current objective of this co-operation is to implement edge-based elements, to compare with the algebraically equivalent face-based elements, and to compare with the node-based elements. As Prof. Bossavit is an expert in this field, his stay in Warsaw was an excellent opportunity for many inspiring discussions. First results of these discussions have already been summarised and became a section of a recently submitted paper. A support of the Banach Centre of Excellence was gratefully acknowledged.

(W.Zijl, A.Trykozko, Numerical Homogenization Based on Complementary Finite Elements Applied to Nonlinear Flow in Porous Media and Viscous Layers. Submitted to *Computational Geosciences* – a journal published by Baltzer Science Publishers)

It should be emphasised that this way a transfer of methods developed in the frames of computational electromagnetism to the field of flow-related processes occurred.

Second half –year period

- As a result of Alain Bossavit's visit the following web pages have been created:
- - Applied differential geometry - A compendium.
- - Discretization of electromagnetic problems.
- - Axial vectors.
- - Bilateral estimates on "homogenized" permeabilities.

Address**: **** http://www.icm.edu.pl/new_icm/edukacja/mat.html**.

Third half – year period

d) Dr. Robert Cunderlik, Slovak University of Technology, Bratislava, Slovakia, PhD

student visit, 2^{nd} January - 31^{st} March, 2002.

He has been preparing the PhD Thesis: "Boundary Element Method applied to

Geodetic Boundary Value Problem".

He has been trained in the mathematical background of Boundary element method and numerical analysis (consulting with dr. Anna Trykozko, dr. Kerstin Kantiem and dr. Krzysztof Nowinski).

e) Prof. Karol Mikula, Slovak University of Technology, Bratislava, Slovakia, April

2002. Seminar "Variational numerical methods in image processing and analysis".

Fourth half – year period

f) Prof. A. Bossavit, from EDF, France, visited IMPAN-BC in 1-30 November, 2002.

The main aims of his stay were as follows:

- To improve the electronic courses on
*Applied differential geometry*and*Discretization of electromagnetic problems*.

See http://www.icm.edu.pl/edukacja/mat.

- To work out the frames of a new lecture with a preliminary title “Convex analysis with applications to some physical problems (nonlinear magnetostatics, porous media, superconductivity)”

He had discussions with the staff members and PhD

students at ICM.

- Prof. Daniel Sevcovic, from University Comenius, Bratislava, Slovakia, visited
- Dr. Angela Handlovicova, from University of Technology, Bratislava, Slovakia, visited IMPAN-BC in 17-30 June and 7 – 19 October, 2002.
- Zuzana Kriva, from University of Technology, Bratislava, Slovakia, visited IMPAN-BC in 3 June – 2 July, 2002.

IMPAN-BC in 7 –19 October, 2002.

The research has been focused on the topic: “Mean curvature flows of curves and surfaces”. This included a theoretical study of the analytical and qualitative aspects of the flow and a design of an effective numerical scheme for computing mean curvature flows involving a nontrivial tangential redistribution.

Angela Handlovicova worked together with Z. Kriva on the article and computing results printed in proceeding ALGORITMY 2002. She had a lecture at ICM in October and worked on the new article “Error estimates for finite volume scheme for Perona-Malik Equation” (jointly with Z. Kriva).

Zuzana Kriva was completing her PhD thesis.

She gave the talk on seminar 21st June: “Finite Volume method and it coarsening strategy for Perona Malik equation in image processing”.

Short visits:

Second half – year period

- Dr. Anna Trykozko,
*ICM, Warsaw University*, visited TNO Institute of Applied - Dr. Krzysztof S. Nowiński, Birmingham AVS Users Group Conference and a
- Prof. Marek Niezgodka visited University of Heidelberg, 2-5 May, 2002.
- Dr Anna Trykozko, ICM, Warsaw University, visited Electricite de France (EdF), R&D Division, Clamart, France, 10 -17 Dec, 2001.
- Sönke Senff, Freie Universität Berlin, Germany, 16 January, 2002.
- Dr Marcus Elstner, University of Paderborn, Germany. Seminar "New Developments for Quantum Mechanical Simulations of Structure and Dynamics of Molecular and Biomolecular Systems".
- Prof. Frederrique Simondon, Uniwersited de Franche-Comt, Besamson, France, 19
^{th}December, 2001. - Prof. Wojciech Zajaczkowski, from IMPAN-BC, participated “French–Czech Conference in Fluid Mechanics”, Prague, in 6-11 July, 2002.
- Prof. Wojciech Zajaczkowski, IMPAN-BC, participated in the international conference “Navier-Stokes Equations” in St. Petersburg, 11-18 September, 2002. He delivered there a talk “On stability of axially symmetric solutions for the Navier –Stokes equations.
- Dr Franziska Matthaeus, from IMPAN-BC, participated in the Conference

Geosciences- National Geological Survey, Utrecht, The Netherlands, from 18 November 2001 to 25 November 2001.

Inviting person: Dr. Wouter Zijl.

visit to the Manchester Computing Centre/ Manchester Visualisation Centre (December 5-10, 2001).

**Third half – year period**

The main aim of his trip was to attend the meeting "Mathematical Methods for Modelling in Biology". Prof. Willi Jager (IWR), Marcus Kirkilionis (IWR and Warwick), Vincenze Capasso (MIRIAM Milan) and others participated in the meeting.

Inviting person: Prof. Alain Bossavit.

The stay in EdF resulted in several ideas of exploiting further analogies between the problems in computational electromagnetism to the field of the groundwater flow arising in these two fields of applications.

He is a graduate student of Prof. R.Kornhuber. He is cooperating with Maria Gokieli from ICM, A. Wittlin from IMPAN-BC and L. Marcinkowski from Warsaw Univ. on the project consisting in numerical implementation of the scheme presented in Discrete approximation of the Cahn-Hilliard/ Allen-Cahn model with logarithmical entropy (submitted to Japan Journal of Applied and Industrial Mathematics).

Lecture "Convergence or divergence in time for solutions of parabolic equations".

Fourth half – year period

He delivered a talk “Global special regular solutions to Navier – Stokes equations in axially symmetric domains”.

He worked together with:

Prof. G. Seregin, St. Petersburg,

Prof. D. Krőner, University of Freiburg,

Prof. P. Penel, Toulon University.

“Mathematical Modelling & Computing in Biology and Medicine ESMTB”, Milano, Italy, 2-6 July, 2002.

This was a large conference with over 400 participants and up to six parallel sessions.

k) Dr Franziska Matthaeus, participated in the Conference “Dynamics Days Europe”,

Heidelberg, Germany, 15-19 July 2002.

She used also the opportunity to meet students of Heidelberg University the Graduate College on "Complex and Optimization".

Fifth half –year period

l) Krzysztof Nowinski, participated in the IST, DG Information Society, Unit F2,

workshop in Brussels, 29-30 January 2003, "Grids for complex problem solving".

Several informal discussions related to the possibilities to participate in the FP6 took place, in particular with a representative of the Heidelberg University Centre.

Conferences and Workshops:

Second half-year period

- A conference Summer school on Parallel Computing in Biomolecular Simulations in Gdańsk Sep. 1- Sep. 3, 2001, organized (and cofinanced with Polish means accompanying CE IMPAN-BC), took place.
- "Biology vs. Mathematics: Mutual Challenges", Bedlewo, 24 - 26 Feb, 2002. 33 participants, in particular from: Milan, Madrid, Heidelberg, Freiburg, Leipzig, Bratislava, Duisburg, Besancon, Brighton.
- "18
^{th}European Workshop Computational Geometry" 10-12 April, 2002. About

Third half year period

50 participants.

The first part of the report contains invited lectures:

- "Bio-Geometric Modelling" H. Edelsbrunner, Duke University,
- "Units in Group Rings of Crystallographic Groups" Z. Marciniak, Warsaw University,
- "Evolution of apparent contours" G. Vegter, University of Gronningen".

The second part contributions lectures.

d) "Visual Analysis Image Based Computing and Applications", Warsaw, 21-23

- May, 2002. 25 participants, in particular from: Duisburg, Innsbruck, Heidelberg, Freiburg, Bonn, Bratislava, Lubeck, Duisburg, Leipzig, Erlangen.
- 2
^{nd}Fall Workshop “Complex Processes: Modelling, Simulation and Optimisation”, - modelling within microscopic scale
- modelling within macroscopic scale
- modelling on mesoscale
- modelling on macroscopic scale
- optimisation
- Workshop “
*Probabilistic Problems in Atmospheric and Water Sciences*”, Będlewo near Poznań, 16-18 December 2002.

Fourth half – year period

e) Workshop “Navier-Stokes motion in complex domains*”*, Warsaw, 28 September –1 October, 2002.

see: http://www.icm.edu.pl/wydarzenia/konferencje/navier_stokes.php

Organizer: Prof. Wojciech Zajączkowski.

The aim of the workshop was to discuss a relation between theoretical questions concerning regularity and applications which could be described by the Navier-Stokes equations. The main topics were: regularity results for Navier-Stokes equations, applications of Navier-Stokes equation for solving some industry-originating problems, motion in pipes or complex sets of pipes, non-Newtonian fluids. Approximately 20 participants from several countries attended the workshop.

Będlewo near Poznań, 18-20 October 2002.

see: http://www.icm.edu.pl/2fall

Organizer: Prof. Marek Niezgódka.

The 2^{nd} Fall Workshop was intended to update status information on the projects

conducted by college fellows and to provide new stimuli on most up-to-date challenges in modelling of real-world complex systems. Its underlying role was to foster bridging across scientific disciplines, from mathematics to technology. The contributions have been divided into sessions on:

Almost 40 participants from Poland and Germany attended the workshop.

Fifth half –year period

http://www.icm.edu.pl/dzialalnosc_n/excellence/probabilistic.php

*Organizers*: Prof. Krzysztof Haman, Prof. Jerzy Zabczyk, Dr. Bogumił Jakubiak

The aim of the meeting was to create an opportunity to improve understanding of the role played by probabilistic methods in solving problems that originate from the areas such as atmospheric analysis and forecasting, water management, hydraulic forecasting, ocean and climate models. It was the intention of the organizers to invite mathematicians working in the area of stochastic processes and statistics as well as researchers dealing with different problems related to geosciences. This gave an opportunity for formulating the most urgent problems of geosciences on one side and to present the tools offered by the modern probabilistic methods, on the other.

The meeting was divided into two parts. The first part consisted of several lectures giving an extended introduction to statistical methods and approaches (*Probabilistic problems in atmospheric sciences, Probabilistic approach to water management, Introduction to statistics, Introduction to stochastic processes, On the mathematical understanding of stochastic resonance*). The second part of the meeting was devoted to more specialized talks and reports from ongoing projects.

More than 50 people attended the workshop.

** Summary.** This package has almost accomplished its plans, with several personnel visits taking place now and planned for the next autumne. Cooperation with German (Heidelberg, Freiburg) and Slovakian mathematicians (K. Mikula group in Bratislava) developed. It organized several workshops (more than planned, due to a combination with other financing sources). The programme of this package has been accomplished in a close cooperation with ICM (Interdisciplinary Center of Mathematical Modelling of Warsaw University), giving a chance of developing at IMPAN applications of mathematics (in particular numerical methods, PDE's, large scale modelling, complex systems. We are creating now at IMPAN a new center of applications of mathematics.