Ph.D. in Mathematics

Research topics

Algebra: Group Theory, special aspects of infinite or finite groups: finite or infinite groups with some combinatorial property, finite p-groups with some property, generalizations of solvable and nilpotent groups. Graphs connected with groups. Ordered Groups. Module theory techniques applied to groups.

Complementary Mathematics: The main subjects of our researches are the following: "Ars Analytica" (Algebra, Infinitesimal Calculus, Analytic geometry) from XVI-th to XIX-th century; also we publish manuscripts and "commercium epistolicum" (scientific letters) among mathematicians. We provide historical information, mathematical explanations, catalogues and reproduction by VR (Virtual Reality) regarding antique mathematical plaster models and instruments for use in theaching topics in mathematics and exposition. Foundation of geometry, point-free geometry, formal logic and cognitive processes.

Geometry: Topologies on function spaces and hyperspaces. Special topologies on homeomorphism groups. Action on hyperspaces. Construction of point-free geometries by topological methods. Geometry of nonlinear PDEs and Secondary Calculus with applications to theoretical physics, rational mechanics and mechanics of continua. Lie groups and Lie algebras. Differential calculus over commutative algebras. Geometry of solution singularities. Differential invariants of geometrical structures. Cohomological methods in symplectic and contact geometries. Geometric theory of impulsive motion. Group cristalls. Einstein equations (exact solutions). Geometrical theory of nonholonomic mechanics.

Mathematical Analysis: Functional spaces with and without weights, extensions and imbeddings. Second-order elliptic equations with discontinuous (eventually singular) coefficients in unbounded or non-smooth domains; a-priori estimates; maximum principles; regularity; existence and uniqueness theorems. Qualitative theory of viscosity solutions and regularity for quasilinear and fully nonlinear elliptic equations. Blow-up, semilinear and quasilinear elliptic equations with singularities. Applied functional analysis and PDE: evolution equations, one-parameter semigroups with applications to delay equations, population equations and control theory, non autonomous differential equations, spectral theory, elliptic and parabolic differential operators with unbounded coefficients. Calculus of variations. Integral representations of functionals. Homogenization of elliptic, parabolic and hyperbolic equations in domains with interface. Finitely additive and non-additive functions taking values into uniform spaces.

Mathematical Logic: Algebraic semantics for many valued logic. Lukasiewicz's Logic and its algebraic models. Category Theory and Universal Algebra issues in many valued logic. Non boolean probabilities. Fuzzy Logic.

Applied Mathematical Logic: Soft Computing, Fuzzy Logic, Fuzzy Control, Many valued logical gates.

E-learning and online teaching: Theory of oriented graphs and applied ontologies to the modelling of knowledge. Didactic models for the learning. The inductive experiential learning and its applications. The creation of personalized and adaptive learning experiences. The modeling of knowledge and the tools of Web 2.0. Didactic methods for collaborative learning. Authoring and aggregation of advanced didactic contents. Generation of personalized learning experiences from a combination of skill.

Mathematical Analysis: Conservation laws. Weak solutions. Admissibility conditions. Solution of Riemann Problems. Shock and rarefaction waves. Functions with Bounded Variation, Existence of solutions and wave-front tracking algorithm. Total variation estimates. Uniqueness and continuous dependence. Applications to urban traffic networks, telecommunication networks, supply networks.

Numerical Analysis: Numerical methods and mathematical software for evolutionary problems. Numerical treatment of functional equations: Ordinary Differential Equations and Stochastic Differential Equations, Volterra Integral and Integro-Differential Equations. Parallel computation. Numerical stability. Efficient numerical methods for scientific computation.

Operations Research: Multi-flows queueing systems. Queueing systems with retrials, the server searching for customers, impatient service. Queueing systems with MAP (Markovian Arrival Process) and BMAP (Batch Marovian Arrival Process) flows. Queueing networks. G-networks. Optimization problems. Transport and logistic problems. Maximal fluxes on graphs.

Probability and Statistics: Stochastic models and applications. Computational methods in statistics. Neuronal models based on stochastic processes. Nonhomogeneous diffusion processes. Random motions. Queueing systems. Statistical methods in reliability theory.