I algebraic geometry

Part time:

• 
  prof. P. Domański
  
• 
  prof. M. Mastyło
  
• 
  prof. A. Pełczynski
  
• 
  prof. P. Wojtaszczyk
  
• 
  prof. W. Żelazko
  
• 
  prof. R. Latała
  

Foreign:

• 
  N. Tomczak- Jaegerman
  
• 
  S. Szarek
  

Research mission and objectives and programme in relation to the state-of-the-art:

Main areas of the research are:

1. 
   Geometry of Banach spaces, function spaces and applications
   
2. 
   Probability and probabilistic methods in modern geometry and analysis. Free probability.
   
3. 
   Linear and nonlinear approximation and their algorithms.
   
4. 
   Banach algebras, algebraic structure of operators and applications to PDE.
   

The topics mentioned above provide powerful tools for attacking many theoretical and real life problems the team ventures to involve in, e.g. :

• 
  concentration inequalities and random constructions of various geometrical objects are fundamental in modern numerical analysis; just to mention compressed sensing, random algerithms in graph search, identifying sparse functions etc.
  
• 
  approximation theory, multiscale decompositions and greedy algorithms provide indispencible tools for numerical solutions of PDE's and up to date image processing
  
• 
  Banach algebras and modern function spaces provide a framework for investigation of existence and properties of solutions of differential and integral equations
  
• 
  free probability and random matrix theory turn out to the natural language for large sections of modern physics.
  

Major research achievements:

Members of the group have been invited to deliver lectures at the International Congresses of Mathematicians, ICM (A. Pełczyński, plenary 1983) and four sectional (Latala, Figiel, Tomczak-Jaegerman, Szarek). Out of 40 articles in the "Handbook of the Geometry of Banach Spaces" (North Holland, 2001) 4 have been authored by members of the group (~ 10%).

E.g. MathSciNet Math.Rev shows P. Wojtaszczyk 68 papers and two books cited 520 times by 461 authors and T. Figiel 50/443/385. Famous and extremaly useful are two books by Wojtaszczyk:

[Woj1] Banach spaces for analysts. Cambridge University Press, Cambridge, 1991. xiv+382 pp,

[Woj2] A mathematical introduction to wavelets. Cambridge University Press, Cambridge, 1997. xii+261 having 194 and 96 citations (not counting Polish translations).

Prof. Wojtaszczyk is interested in mathematical problems of sparse representation. In recent years the idea of sparsity is getting more and more important in various areas. Essentially it is a belief that many natural phenomena which seem to depend on many factors can be represented as dependent on a much smaller number of judiciously chosen parameters. This general philosophy

give rise to  a variety of different mathematical questions. There exists an extensive mathematical literature devoted to mathematical aspect of this general philosophy. As of now his research deals with three mathematical manifestation of this philosophy:

1. 
   Compressed sensing and its generalisations e.g. sparse functions. Mathematically it uses tools and ideas from probability theory (Latala, Adamczak), geometry of Banach spaces (Mankiewicz), theoretical computer science, combinatorics, approximation theory (A. Kamont, Z. Ciesielski), information based complexity, etc.  Its results and ideas are used in signal processing, machine learning, electronic microscopes, astronomy, etc.
   
2. 
   Greedy algorithms. Mathematically it uses tools and ideas from functional analysis, geometry of Banach and Hilbert spaces,  theory of n-widths, convexity theory, probability theory. It finds applications in mathematical statistics, numerical solutions of PDE's with parameters, machine learning, etc.
   

1. 
   P. Bechler, R. DeVore, A. Kamont, G. Petrova, P. Wojtaszczyk, Greedy wavelet projections are bounded on BV, Trans. Amer. Math. Soc. 359(2007)619-635, IF=1,094; citations
    WoS=4, MSN=3
   
2. 
   F. Albiac, P. Wojtaszczyk, Characterization of 1-greedy bases, J.Approx. Theory 138 (2006)65-86, IF=1,040; cytowania WoS=2, MSN=0
   
3. 
   P.Wojtaszczyk, Greediness of the Haar system in rearrangement invariant spaces, w Banach Center Publications 72, Approximation and Probability, Warszawa 2006, WoS does not record this paper citations MSN=5
   
4. 
   R. DeVore, G. Petrova, P. Wojtaszczyk, Anisotropic smoothness spaces via level sets, Commumications in Pure and Appl. Math, 61:9(2008) 1264-97 IF=3,873; citations WoS=1, MSN=0
   
5. 
   P. Wojtaszczyk, Stability and instance optimality for Gaussian measurements in compressed sensing, Foundations of Computational Mathematics, 10(2010) 1-13, IF=2,745; citations
   WoS=4, MSN=5
   
6. 
   R. DeVore, G. Petrova, P. Wojtaszczyk, Instance optimality in probability with an $\ell_1$-minimization decoder, Applied and Computational Harmonic Analysis, 27 (2009) 275-288, IF=3,176; citations WoS=5, MSN=2
   
7. 
   R. DeVore, G. Petrova, P. Wojtaszczyk, Approximation of functions of few variables in high dimensions, Constructive Approximation 33:1(2011) 125-143, IF=1,743; citations
   WoS=1 MSN=1
   
8. 
   P. Bechler, P. Wojtaszczyk, Error estimates for orthogonal matching pursuit and random dictionaries, Constructive Approximation 33:2(2011) 273-288, IF=1,743, citations
   WoS=0, MSN=1
   
9. 
   P.Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods, SIAM J. Math. Anal. 43:3(2011) 1457-72, IF=1,744; citations WoS=1, MSN=0
   
10. 
    P. Wojtaszczyk, Complexity of approximation of functions of few variables in high dimensions, Journal of Complexity, 27:2(2011) 141-150, IF=0,986; citations WoS=0, MSN=0
    

Global citations  WoS=392 (without autocitations 248) MSN=605 index h=12.
Remark 1.  IF means 5-year impact factor of the journal, WoS Web of Science, MSN MathSciNet.
Remark 2. Web of Science does not count citations of books. According to MathSciNet may books had 213 and 100 citations.

Prof. Yu Tomilov got in 2009 Sioerpinski prize for mathematicians below 40 of the Polish Academy of Sciences.

Mankiewicz run Warsaw IMPAN node of the FP6 RTN network PhD (Phenomena in High Dimension).

Zemanek and Tomilov run at IMPAN the FP6 ToK programme TODEQ 2006-2010.

Present international collaboration:

Universities and research centers in:

1. 
   France; Boredeaux, Marne la Valle, Paris (mainly VI), Toulouse.
   
2. 
   Germany; Aachen, Bonn, Jena, Kiel, Oldenmburg, Trier, Wuppertal.
   
3. 
   Austria; Linz, Viena d) Spain; Valencia
   
4. 
   USA; College Station (Texas A&M), Cleveland (CWRU), Columbia (Missouri).
   
5. 
   Canada; Edmonton, g) Israel, Beer-sheva, Haifa, Jerusalem, Tel Aviv. h) Armenia, Erewan.
   
6. 
   UK, Cambridge, Oxford
   

Riemann hypothesis constitutes one of the seven Millennium problems (with one mln USD prize assigned to each of them) chosen by the Clay Mathematical Institute as the most difficult problems with which mathematicians were fighting at the turn of the second millennium. It attracted enormous attention of mathematicians (and also physicists) all over the world and led to dramatic development in several areas of complex analysis, number theory and algebra.

There is an approach to the study of Riemann hypothesis by harmonic analysis techniques (with some complex analysis machinery added at several crucial points) proposed by Nyman and Beurling. Namely, they showed that the Riemann hypothesis is equivalent to a special property of a linear span in L2(0,1) of a ceratin fixed and explicitly given function. Unfortunately, this approach has not been received an attention it deserves before and is being actively developed now, e.g. by Prof. N. Nikolski.

At the same time, the IM PAN Functional Analysis Group (e.g Pelczynski, Wojtaszczyk, Kwapien) has a strong expertise in the methods applied by Nikolski. Thus, a collaboration between Nikolski and the group might be extremely fruitful. Moreover, this can be strengthened by expertise of a world class specialist in number theory, Prof. A. Schinzel, and possible synergies might lead to breakthroughs in several fields.

The Kadison-Singer problem on the (unique) extendability of pure states on von Neumann algebra of diagonal operators to the whole algebra of bounded linear operators is one of the most well-known and challenging problems in operator theory. The depth and beauty of the problem can be illustrated by the fact that it appeared to be equivalent to more than ten fundamental open problems in different areas of research in mathematics and natural sciences including engineering, e.g in operator theory, Banach space theory, harmonic analysis, time-frequency analysis, sicnal processing, Internet coding. One of its equivalent formulations have been studied recently by N. Nikolski and V. Vasyunin who relied on a theory of bases in Banach spaces, and a substantial progress towards a final solution of the problem was achieved.

The IM PAN group (e.g. Pelczynski, Wojtaszczyk, Kwapien) is one of the strongest groups in the world in bases theory, in particular, and in Banach space geometry in general. Thus, it is natural to expect that collaboration between the IM PAN group and Nikolski and Vasyunin might lead to a definite progress (breakthrough) in the study of the Kadison-Singer problem and related open problems in the areas mentioned above.

Nikolski (Bordeaux University, Prof. Emer and St. Petersburg Branch of Steklov Mathematical Institute, Prof.)- one of the few world class specialists in operator theory and its applications to function theory and harmonic analysis. He received the highest award of the French Academy of Sciences - Amper Prize in 2010.

Vasyunin (St. Petersburg Branch of Steklov Mathematical Institute, Prof.) - (former student of Nikolski)- top-rank expert in operator theory, function theory and harmonic analysis.

VI. BIOMATHEMATICS
Team leader: prof. Ryszard Rudnicki
Members:

• 
  dr Radosław Wieczorek
  
• 
  1 Phd student
  

Outside members:
Lublin University of Technology:

• 
  prof. Adam Bobrowski
  
• 
  
  

University of Silesia:

• 
  dr Katarzyna Pichór
  
• 
  dr Marta Tyran-Kamińska
  

MCS University of Lublin:

• 
  dr Urszula Skwara
  

Gdańsk University o Technology:

• 
  dr Agnieszka Bartłomiejczyk.
  

Research mission and objectives:

The rapidly developing techniques of molecular biology and genetics produce large quantities of data, that demand mathematical analysis and modeling. Nowadays, mathematical modeling of biological processes is a central topic in theoretical biology and some biologists find that mathematical models are absolutely essential for research in modern biology. Mathematics provides a broad spectrum of methods to study biological and medical issues. The models use all types of differential equations, probability theory, dynamical systems, discrete mathematics and also very complicated systems which include age, stage or size structures. Using mathematical models one can analyse populations [1] and biological systems at various levels, including cells, genes, and biomolecules. Although first mathematical methods appeared in demography, they have become increasingly important in almost all branches of biology including ecology, epidemiology and infectious diseases, genetics, physiology, immunology and cancer growth. Our aim is to construct models of new biological phenomena which appear in genetics and cells physiology and to provide new mathematical tools to study existing biological models.

Research programme in relation to the state-of-art:

1. 
   Genome evolution. We have presented a model of genome evolution [2] which explained some conjectures of Słonimski concerning the size distribution of paralog families in several microbial genomes. But this model needs modifications and further studies. It would be interesting to consider the model with rates of elementary events (mutation, duplication and gene loss) dependent on the gene location in the genome, genome size or the functional importance of a given gene.
   
2. 
   Gene expression. Modelling of gene expression and gene regulatory networks is the topical subject of the modern biology (The Nobel Prizes in Physiology or Medicine in 2001, 2002, 2006, and 2009 are connected with this subject). It is also a challenging problem of mathematical modelling. Due to a small number of copies of molecular species involved, such as DNA, mRNA and regulatory proteins, gene expression is a stochastic phenomenon. In trying to understand observed distributions of intracellular components, the norm in computational and systems biology is to use algorithms developed initially by D. Gillespie to solve the chemical master equation for specific situations. However, these investigations demand long computer runs, are computationally expensive, and further offer little insight into the possible diversity of behaviours that different gene regulatory networks are capable of. Mathematical modelling from an analytical point of view that we are taking can give precise conditions on the long term behaviour of various dynamics. In the paper [3] we proved some results concerning stability of distribution of mRNA and protein levels and we hope to apply our technique in other cases. Mathematical tools in points a) and b) are mainly stochastic processes and semigroups of stochastic operators. The precise study of mathematical and biological properties of our models needs to use or establish new criteria for stability of such semigroups.
   
3. 
   Chaos in structured population models. Models of this type are described by partial differential equations usually with a nonlocal reproduction term. Till now we have shown chaotic behaviour only for models without replication term [4],[11],[12]. Now it becomes clear for us that we could also prove chaos for more complex structured models. In order to establish chaotic behaviour we use tools from the ergodic theory. We construct invariant mixing measures for flows generated by our equations supported on the whole space. This implies existence of dense trajectories and instability.
   
4. 
   Rate of convergence of individual-based models to transport equations. We have constructed and studied some individual-based models [5] of plankton dynamics and showed their connections with macroscopic ones. The convergence rate of those models is very important for numeric simulations. Mathematically, it requires rather involved tools of stochastic processes with values in the space of some generalized functions [17],[18].
   

Major research achievements:

R. RUDNICKI has published 72 papers, most of them in well known mathematical and biological journals. He has MathSciNet Math. Rev. 157 citations by 109 authors, ISI gives 251 cited references. His scientific activity is connected with three fields of mathematics: differential equations, probability theory and biomathematics. In particular he is interested in the following subjects:

1. 
   ergodic properties of dynamical systems generated by partial differential equations
   
2. 
   asymptotic properties of Markov semigroups and their applications
   
3. 
   population dynamics and modelling of cell cycle.
   

Ad 1. He has given a general construction of invariant measures for dynamical systems generated by partial differential equations [11]. Measures of this type are constructed by means of stochastic processes and stochastic fields and have strong ergodic and analytic properties (positivity on open sets, mixing and so on). From these properties follow additional features of the dynamical systems: chaos in the sense of Auslander and Yorke and the existence of turbulent trajectories in the sense of Bass [12].

Ad 2. Markov operators are linear transformations from the space L1 into itself which preserve the set of densities. They appear in the ergodic theory and in the theory of Markov chains. Semigroups of Markov operators are often generated by partial differential equations and by partial differential equations with some perturbations. Rudnicki's results are applied to transport equations describing diffusion and coagulation-fragmentation processes and structured population models.

Ad 3. The most cited paper with M. C. Mackey [15] (cited 21 times in MSN MR and more than 50 found in the Internet) deals with the problem of asymptotic stability of a nonlinear partial differential delay equation describing cellular replication. This equation is difficult to study and Rudnicki developed completely new analytic and probabilistic methods to establish its asymptotic stability, playing an important role in mathematical models of the cell cycle and in a model of the electrical activity of neurons. With O. Arino [16] Rudnicki introduced a coagulation-fragmentation model of phytoplankton dynamics and showed with R. Wieczorek how to obtain this equation as a limit of some individual-based model.

He is an advisor to the Board of the European Society for Mathematical and Theoretical Biology. He has organized four conferences. Now he is president of the Scientific and Organizing Committees of VIII European Conference on Mathematical and Theoretical Biology}, Kraków, June 28 - July 2, 2011.(we expect about 600 participants).

He was awarded the Hugon Steinhaus Prize of the Polish Mathematical Society in 2010 which is the main prize of the society in applications of mathematics.

PROF. ADAM BOBROWSKI is a well-known specialist in the theory of semigroups of operators and mathematical biology. He has authored and co-authored 36 scientific papers in leading journals including Journal of Functional Analysis, Journal of Evolution Equations, Semigroup Forum, Studia Mathematica, Mathematical Biosciences, Theoretical Population Biology and Journal of Mathematical Biology. He is also the author of the successful book [9]. Along with papers in pure mathematics, he has published extensively on population genetics models involving drift and mutations, and on convergence of operator semigroups involved in various models of mathematical biology, including those of gene expression, gene regulation, and age-structured and space-structured populations.
Current research:

Our current research is connected with the following applications of mathematics in biology and medicine:

1. 
   Mathematical models of the genome evolution [2].
   
2. 
   Physiological processes in cells and in cellular population models [3].
   
3. 
   Chaoticity of structured population models [4].
   
4. 
   Individual-based models, their convergence to coagulation-fragmentation equations and applications to phytoplankton dynamics [5],[6].
   
5. 
   Stochastic models of ecological interactions [7].
   
6. 
   Application of the probability theory and the theory of semigroups of operators in population models [8-10].
   

Existing and previous research collaboration:

Our team consists of a number of collaborating Polish scientists from different institutions. All of us take part in a regular seminar "Computational Biology" in IM PAS. This seminar plays very important role in development of applied mathematics in Poland. Members of our team have also strict collaboration with Michael C. Mackey from McGill University in Montreal, Anna Marciniak-Czochra from University of Heidelberg, Laurent Pujo-Menjouet from Universite de Lyon, Marek Kimmel from Rice University in Houston and Jacek Banasiak from University of KwaZulu-Natal in Durban (South Africa). We also plan to establish closer contacts with Systems Biology Centre at Warwick University within Interdisciplinary Programme for Cellular Regulation.

Future research plans:

Mathematical biology develops dynamically and it is very difficult to predict what will be interesting and the most challenging in future. Our team (especially the leader) has rather broad mathematical knowledge and experience concerning different applications of mathematics to respond to these challenges. We probably concentrate on the problems connected with genetics. The second direction is to study physiological models. We have a long and large experience with such models, but everything depends on the collaboration with physicians. It would be a great success of mathematics and mathematicians if it help in developing of new methods of the medical treatment.

Research mission, objectives and future careers of our visitors.
The rapidly developing techniques of molecular biology and genetics produce large quantities of data, that demand mathematical analysis and modeling. Nowadays, mathematical modeling of biological processes is a central topic in modern biology and some biologists find that mathematical models are absolutely essential for research in modern biology. Mathematics provides a broad spectrum of methods to study biological and medical issues. The models use all types of differential equations, probability theory, dynamical systems, discrete mathematics and also very complicated systems which include age, stage or size structures.

Using mathematical models one can analyze populations and biological systems at various levels, including cells, genes, and biomolecules.

Although first mathematical methods appeared in demography, they have become increasingly important in almost all branches of biology including ecology,

epidemiology and infectious diseases, genetics, physiology, immunology and cancer growth.

Our aim is to construct models of new biological phenomena which appear in genetics and cells physiology

and to provide new mathematical tools to study existing biological models.

We address our programme concerning biomathematics to people who want to enhance their background in science and prepare them for modern jobs

connected with medicine, pharmacy and biotechnological industry.

The leader of our team in biomathematics team Prof. Ryszard Rudnicki has a long-term scientific and teaching experience in this field.

He was the president of the Scientific and Organizing Committees of VIII European Conference on Mathematical and Theoretical Biology,

Kraków, June 28 - July 2, 2011. This meeting brought almost 1000 participants together from 48 countries.


VII. QUANTUM – NONCOMMUTATIVE GEOMETRY
Team leader: dr hab. Piotr M. Hajac
Members:

• 
  Prof. Paul Frank Baum (visiting professor)
  
• 
  Dr Tomasz Maszczyk (part time)
  
• 
  Jan Rudnik (Ph.D. student)
  
• 
  Dr Adam Skalski (post-doc)
  
• 
  Dr hab. Piotr M. Sołtan (part time)
  
• 
  Dr Bartosz Zieliński (part time)
  

Research mission and objectives:

Noncommutative Geometry is a modern and highly advanced branch of mathematics extending classical geometry to the realm of functional analysis and noncommutative algebra. It is motivated by the need to reconcile the languages of General Relativity (geometry) and Quantum Mechanics (operator algebras) in order to provide mathematical tools for the much desired fundamental physical theory.

Our main goal is the discovery and classification of new structures on quantum spaces by means of analysing symmetry and computing invariants. This should allow the development of new tools to solve problems beyond the reach of currently available methods.