An important strand of investigations deals with the classification of problems. In case of non-computability, Descriptive Set Theory is used. It classifies problems according to the complexity of their logical description. However, it was developed under assumptions that are not shared by structures used for program semantics. Quasi-Polish spaces were introduced to remedy this problem. Recently, examples of spaces not belonging to this class were found to which the theory can nicely be extended as well.
Two measures of descriptive complexity were studied: (1) how complex it is to obtain a set from open sets using Boolean operations; (2) how complex it is to test membership in the set. Both measures are equivalent on countably-based spaces, but not in general, the reason probably being the mismatch between topological and sequential aspects of topological spaces.
Much work has been done on computable problems. First of all, the computability of certain dynamical systems has been studied. A typical example of a "chaotic" attractor arising on systems that have a very complex behaviour and sensitive dependence on initial conditions, which appear in the context of many applications like meteorology, e.g. is the Lorenz attractor. The computability of geometric Lorenz attractors and their physical measured was derived.
Computable problems are classified according to the space and/or time resources necessarily needed for in their computation. The behaviour of dynamical systems was studied in this respect. Attractors can be typically computed in an efficient manner but, for some pathological parameters, the cost of computing can be arbitrarily high. For Lorenz-like attractors the main sources which contribute to their computational complexity have been identified. For other ones, the flow passing near a saddle equilibrium point plays an important role on the overall complexity.
Further highlights in this research are:
A restriction of the famous 3-Body Problem is computable in polynomial time on average. The 3-body problem features prominently in astronomy, e.g. when the motion of a space shuttle under the gravitational pull of, say, the earth and the moon should be computed.
One-soliton and two-soliton solutions to the Korteweg-de Vries partial differential equation for shallow water waves are computable in polynomial-time.
A variant of the (still open) Hilbert’s 16th problem was considered. Hilbert’s 16th problem asks for an upper bound on the number of limit cycles that a polynomial system of degree n has on the plane. Instead, the problem of determining the exact number of periodic orbits (or limit cycles) over the plane was considered. As shown, this problem is uniformly non-computable in general. However, the exact number of periodic orbits can be computed if the polynomial systems are structurally stable and considered over the unit ball.
A natural encoding of the space of divergence-free vector fields on the unit square in the Euclidean plane was constructed and used to show first the mild solution of the Stokes-Dirichlet problem and then a strong local solution to the non-linear incompressible Navier-Stokes initial value problem uniformly computable; thus solving a problem of our-El and Richards.
