I algebraic geometry

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.