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 regularly documented in the form of Research Reports.
Below you will find a brief description of current (2007) areas of research.
Ergodic theory, dimension theory, multifractal analysis, multifractal rigidity, thermodynamic formalism, quantitative recurrence; geometric mechanics, Birkhoff setting for classical mechanics and holonomic constraints; Hamiltonian systems, reduction using symmetry and moment-energy methods for the study of stability and bifurcation of relative equilibria; hyperbolic dynamics, nonuniform hyperbolicity and Lyapunov stability, robustness, topological conjugacies; smooth ergodic theory and hyperbolic measures; infinite dimensional dynamics, hyperbolicity and Morse-Smale structures for dissipative systems defined by delay or evolution differential equations; integrability of polynomial equations of mathematical physics; topological dynamics, kneading theory, topological entropy and other topological invariants, Bowen-Franks groups, and zeta functions.
Geometry and Topology
Algebraic topology, K-theory and homotopy theory; algebraic geometry, complex surface theory and holomorphic vector bundles; differential geometry; symplectic topology, symplectic and related geometries (contact geometry, Kahler geometry, Poisson geometry); noncommutative geometry, geometric and deformation quantization, quantales and groupoid theory; quantum field theory; string theory, low dimensional geometry and topology, mirror symmetry and topological string theory; mathematical physics, integrable systems; conformal field theory, topological quantum field theory, matrix models; general relativity and Riemannian geometry; combinatorics.
Nonlinear Analysis and Differential Equations
Classical mechanics, Aubry-Mather theory and viscosity solutions of Hamilton-Jacobi equations; coagulation systems, asymptotic behavior of solutions for infinite systems of ordinary differential equations modeling coagulation, and relations with the kinetic theory; harmonic analysis, applications to dispersive nonlinear partial differential equations, and integral transforms in Lie groups with applications to geometric quantization; qualitative theory of functional and difference equations, stability, bifurcation, and oscillatory behavior; reaction-diffusion equations, stability and bifurcation of solutions, pattern formation and monotone systems; variational problems and Morse theory, variational methods in Hamiltonian mechanics and elliptic equations, optimization in spaces of functions of bounded variation with volume restrictions.