I algebraic geometry

Existing collaboration:

Praha, Paris 12, Hannover, Kiel, Ulm, Essen, Freiburg, Chemnitz, RICAM Institute (Austria), Darmstadt, Sankt-Petersburg, Oxford, Suwon (Korea), Iowa City, Tokyo, Ferrara, Naples.

Future Research Plans:

• 
  Regularity of weak solutions to Navier-Stokes Equations (Ladyzhenskaya-Prodi-Serrin conditions). Existence of global regular solutions under some geometrical and analytical restrictions.
  
• 
  Stability of special solutions to incompressible and compressible NSE. Existence of stationary solutions. Problems of inflow-outflow.
  
• 
  Shape optimization for incompressible and compressible flows.
  
• 
  The free boundary problems for magnetohydrodynamics, which can be treated as a starting point for the pinch and the fusion problems.
  
• 
  Measure-valued, weak and regular solutions for non-newtonian fluids and general parabolic problems. Problems of the flow of non-Newtonian fluids with grom conditions given by an N-function (solutions in Orlicz spaces).
  
• 
  Analysis of the flow of polymers. Combining the equations for macroscopic quantities with microscopic structure of the fluid. Existence of solutions and singular limits to the system with implicit rheology.
  
• 
  Theoretical foundation and numerical methods of inverse problems for differential equations on bounded domains related to problems in current interdisciplinary science.
  

III. GEOMETRY AND TOPOLOGY WITH EMPHASIS ON GEOMETRIC GROUP THEORY
Team leader: prof. Tadeusz Januszkiewicz
Members:

• 
  dr Piotr Przytycki
  
• 
  1 Phd student
  

Outside members:

• 
  prof. Jan Dymara (Wroclaw University)
  
• 
  prof. Jacek Swiatkowski (Wroclaw University)