I algebraic geometry

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.