| Center

CMAF (Centro de Matemática e Aplicações Fundamentais) is a research unit at Faculdade de Ciências da Universidade de Lisboa. It had its beginning in 1975. The Center, devoted to research in several fields of Pure and Applied Mathematics, is hosted at Instituto para a Investigação Interdisciplinar da Universidade deLisboa, Avenida Professor Gama Pinto, 2, 1649-003 Lisboa. The fund management, formerly carried out by Fundação da Universidade de Lisboa up to 2010, is currently run by Fundação da Faculdade de Ciências da Universidade de Lisboa. The team currently consists of 82 (integrated) members. Of these, 67 have a Ph.D, (6 having a contract Ciência 2007 or 2008, 13 having a post-doctoral position and another two with other grants); 15 are students (of which 10 Ph. D. students). In addition we now collaborate with 29 external members with a PhD.
The program Ciência 2007/8 together with post-doctoral grants of Fundação para a Ciência e a Tecnologia have allowed an important renewal of the team.
The Center, which has gone through some changes in recent years, has been structured into 8 groups corresponding to different directions of research. There is a Directing Board of five members: Luís Sanchez (coordinator), João Paulo Dias, José Francisco Rodrigues, Teresa Faria and Orlando Neto.
The unit has an Advisory Board composed by Giuseppe Buttazzo, K. D. Elworthy, J. Mawhin and A. Mcintyre. Recent years decisions on membership and rearrangement of groups partially reflect hints contained in their latest reports. In a move towards better visibility of our activities, our web page http://cmaf.ptmat.fc.ul.pt/ now ads to the information on groups three new items: Geometry, History of Mathematics and Communication of Mathematics.

PROJECTS: A number of FCT funded research projects associated to our unit are currently running, see http://cmaf.ptmat.fc.ul.pt/research_projects.html
In addition, the Mathematical Biology group has a significant participation in two EU-projects: EPIWORK (since 2009), and DENFREE (since 1.1.2012), there leading as WP leader the workpackage on "descriptive and predicitive modells for dengue fever”.
The DENFREE project is part of 3 such projects funded by the EU on the research of dengue fever, the largest presently funded activity on dengue research worldwide.

Further applications to a new projects call are taking place.

| General Objectives

1. Research in Mathematics, going from LOGIC to APPLICATIONS. Some groups are mainly concerned with DIFFERENTIAL EQUATIONS and APPLIED ANALYSIS; applications go from physical, biological and economic models to EPIDEMIOLOGY and involve deterministic, stochastic or STATISTICS methods. Specifically:

- Mathematical Logic and Foundations. O-minimal sheaves. Quasi-analytic classes. Locally definable groups. Elimination of imaginaries in o-minimal structures. Dense pairs. Metric space model theory . Finitistic consistency proofs of Fregean systems. parametrized interpretation of nonstandard arithmetic. A Turing machine as model of a physicist’s tasks.

- Theoretical and numerical analysis of PDE's: Young measure solutions for wave type equations with non-standard conditions. Linearized stability of progressive waves for the Schrodinger-Conservation law coupled system. Ground state and non ground state solutions of some strongly coupled elliptic systems. Nodal solutions for supercritical Laplace and p-Laplace equations.

- Blow-up for doubly nonlinear parabolic equations with nonstandard growth conditions.

- Reaction diffusion systems of elliptic PDES with non local competition.

- Estimates in Statistical Mechanics; smoothing properties of semigroups; Mathematical Physics methods in signal analysis, market geometry and complex systems.

- Variational models of two-phase transition problems, in the presence of surfactants; boundary layer problems, solid body touching effects, porous media flows.

- Canonical decomposition of the resolution graph of a plane curve. Semi-universal deformation of a Legendrian curve. Classification of the Legendrian curves of low modality. D-module theory and twistor D-modules

- Infinitesimal deformations of harmonic maps; geometry of submanifolds.

- Boundary value problems in unbounded intervals. Travelling waves in reaction-diffusion.

- Chaotic dynamics, asymptotically linear systems.

- Nicholson-type systems of FDEs with patch structure and multiple discrete delays -permanence, stability of equilibria, bifurcations.

- Lyapunov Exponents of linear cocycles. Connections between number theory and dynamical systems.

- Ecological and epidemiological models. Tools from dynamical systems and phase transitions in the study of the spread (or not) of diseases in human populations.

- Statistical methods in environmental, earth and health sciences, econometrics and marketing research.

2. Advanced training of PhD and Master students.

3. To assist our most recent group (Mathematical Biology) in maintaining its dimension and strength. Two new members have been added to the team, one with a post-doctoral position and another with a research grant. The group will gain a new Ph.D. member shortly.

4. Research in History of Mathematics, and initiatives in communication of Mathematics.

| Main Achievements during the year of 2011

LOGIC: Existence theorems for definable quotients of locally definable groups, structure theorems for semi-bounded groups and compact domination for semi-bounded groups.

Study of the geometry of sets definable in an expansion of the real field by a family of generalized quasi-analytic algebras and proof that such a family of algebras generates an o-minimal polynomially bounded, model complete structure.

Proof that all interpretable o-minimal groups are definable. An implicit characterization of the class NP.

PDEs: Existence and blow-up of the solutions for a class of nonlinear wave equations with variable exponents. Linearized stability of shocks for a Schrodinger-Burgers system. Convergence of numerical methods for a coupled Schrodinger-KdV system. Ground state and non ground state solutions of some strongly coupled elliptic systems. Morse index theorems for second order elliptic boundary value problems. Multiple solutions in systems of competition-diffusion equations.

GEOMETRY: Limits of tangents of quasi ordinary singularities. The triviality of the limits of tangents is a topological invariant of the quasi ordinary hypersurface. The solution and the De Rham complexes associated to a D-Module underlying a twistor are perverse for a natural t-structure. Geometry of immersions with parallel mean curvature into a homogeneous space.

MATHEMATICAL PHYSICS. The Kac conjecture (about the rate of convergence to equilibrium ) for the hard-spheres model where the rate function depends also on the relative velocity. Stochastic dynamics of infinite particle systems in the continuum. Solutions for Magnetohydrodynamics. Construction of stochastic solutions for the scrape-off equations.

COMPLEX SYSTEMS . Somatic evolution of cancer, the evolution of cooperation, the spread of infectious diseases and risk in climate change agreements.

ODEs and DYNAMICAL SYSTEMS: analytical results about periodic orbits in the restricted three body problem. Formulas for Lyapunov exponents of random linear cocycles. A computable version of the stable manifold theorem.

FUNCTIONAL DIFFERENTIAL EQUATIONS. Aspects of the global dynamics of a Nicholson's blowflies model with patch structure and multiple discrete delays.

APPLICATIONS IN EPIDEMIOLOGY. Persistence in seasonally forced epidemiological models.

We analyzed models with external drivers as seasonality and import under population noise, and could estimate simultaneously some of these influences including all initial conditions.

MATH METHODS in MECHANICS: minimization with constraints and Lagrange multipliers optimization of functionals involving eigenvalues or eigenvectors of elastic structures. Singular perturbations of a non-convex second order functional via Gamma-convergence.

STATISTICS. Assessment of prevalence and control of hypertension and their predictors in 60 Community Pharmacies.
Statistical methodology for water quality monitoring via cluster analysis, spatial statistics and linear models.