I algebraic geometry

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

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  Fluid mechanics delivers still plenty of challenges for mathematicians. The knowledge of stability of solutions is very poor. The regularity problem of weak solutions for the Navier-Stokes equations in the three-dimensional case is an open problem, one of the most important issues in mathematics ('Millenium problem')
  
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  The regularity theory for general parabolic systems, including non-diagonal ones, and non-Newtonian Navier-Stokes equations are still open problems.
  
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  The rudiments of theory for the free boundary problem are well investigated but there are still many open questions related to physically reasonable systems which cannot be solved using standard techniques.
  
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  The field of inverse and ill-posed problems has certainly been one of the fastest growing areas in applied mathematics.
  
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  Many important new results on the regularity of global solutions to thermoviscoelasticity and Cahn-Hilliard equations have been proved recently.
  
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  The studies on qualitative properties of solutions of hydrodynamics are core interest for mathematicians and there is substantial progress in this field.