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The present project concerns a large list of problems from Statistical Mechanics, Quantum Mechanics and Quantum Field Theory. The mathematical tools involve Functional Analysis, Multiscale Analysis, Probability Theory, Stochastic Processes, Operator Algebras and other fields of Mathematics. Besides the main research group members from the state of São Paulo, our mathematical physics group algo incorportes collaborators from abroad and from other states, as well and postdocs and graduate students. The collection of problems, expanded upon in more detail in the sequeI, includes Aperiodic and disordered systems. Quantum Walls. Stability of quantum systems under external fields. Superfluidity in Bosonic systems. Schrõdinger operators with potentials defined by interval interchange transformations. Spectral Transition. Singular perturbation theory. Directed percolation. Renormalization group and convergence of the Mayer series. Multi-scale analysis applied to Boson condensation in a hierarquical modelo Distribution of Lee-Yang zeros. Lacunary series, sparse potentials and singular spectrum. Noise dynamics effects in stochatiscal models. Non-perturbative methods: the Bogoliubov 8(g) matrix in super-renormalizable theories. Casimir Effect. Quantum Equivalence principle. AIgebraic quantum field theory. Constructive methods, mass spectrum in QCD. Spectral properties in lattice quantum chromodynamics (QCD). Algebraic formalism in quantum field theory in curved space-time. Thermal scattering theory. (AU)

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