# Research areas and reports

The members of the Center for Mathematical Analysis, Geometry, and Dynamical Systems actively collaborate with other Portuguese and foreign researchers. Research activity is documented in the annual **Research Reports**.

Below you find a brief description of the main areas of current (2013) research.

## Dynamical Systems

Ergodic theory, dimension theory, multifractal analysis and thermodynamic formalism, including additive, almost additive, sub-additive and nonadditive; geometric mechanics, classical mechanics and holonomic constraints; Hamiltonian systems and symmetry reduction; hyperbolic dynamics, nonuniform hyperbolicity and Lyapunov stability, robustness, topological conjugacies and regularity coefficients; smooth ergodic theory, hyperbolic sets, hyperbolic measures and admissibility for nonuniformly hyperbolic dynamics; infinite dimensional dynamics, hyperbolicity and Morse-Smale structures; integrability and nonintegrability of equations of mathematical physics, including equations defined by polynomial vector fields; topological dynamics, including topological entropy and Bowen-Franks groups, Banach algebras and C*-algebras.

## Geometry and Topology

Algebraic topology, algebraic geometry, symplectic topology and geometry, non-commutative geometry, combinatorics and Kähler geometry. Moreover, an important part of research is directed towards Mathematical-Physics including themes such as geometric quantization, integrable systems and algebraic geometry, string theory, matrix models, low dimension geometry and topology, field theories and complex geometry, relativity and Riemannian geometry.

## Nonlinear Analysis and Differential Equations

Variational problems and Morse theory, classical mechanics, Aubry-Mather theory and viscosity solutions, mean-field games, methods of harmonic analysis, spectral theory for differential operators, reaction-diffusion equations, qualitative theory of functional and difference equations, and coagulation systems, modeling of transport phenomena in subsurface environments, boundary value problems of Riemann-Hilbert type, factorization of matrix functions and functional matrix equations.