MathMods :: Joint MSc

• Lebesgue Measure and Integration
• L^p Spaces
• Basic of Topological Vector Spaces, Normed and Banach Spaces, Linear Operators and linear functionals.
• Hilbert Spaces
• Weak topology, Weak * topology, weak compactness
• Applications of Baire Category in Functional Analysis: Uniform Boundedness, Open Mapping, Closed Graph, Inverse Mapping.
• Banach and Hilbert adjointness, self-adjointness
• Compact Operators
• Riesz Fredholm spectral theory

• Terence Tao, An introduction to measure theory.. American Mathematical Society, Providence, RI, ISBN: 978-0-8218-6919-2 . 2011.
• Haim Brezis, Functional analysis, Sobolev spaces and partial differential equations.. Universitext. Springer, New York,. 2011. xiv+599 pp. ISBN: 978-0-387-70913-0
• Alberto Bressan, Lecture notes on functional analysis. With applications to linear partial different. Graduate Studies in Mathematics, 143. American Mathematical Society, Providence, RI,. 2013. xii+250 pp. ISBN: 978-0-8218-8771-4
• Lecture notes provided by the teacher.
• Michael Reed, Barry Simon, Methods of modern mathematical physics. I. Functional analysis. Second edition. . Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York,. 1980. xv+400 pp. ISBN: 0-12-585050-6
• Stein, Elias M.; Shakarchi, Rami , Real analysis. Measure theory, integration, and Hilbert spaces. Princeton Lectures in Analysis, III.. Princeton University Press, Princeton, NJ,. 2005. xx+402 pp. ISBN: 0-691-11386-6

Local Theory of nonlinear systems: initial value problem, hyperbolic equilibrium point, Stable Manifold Theorem. Hartman-Grobman Theorem. Stability and Liapunov functions. Saddles, nodes, foci and centers. Nonhyperbolic critical points. Center manifold theory. Global theory of nonlinear systems: limit set, attractor, limit cycle, Poincaré map, stable manifold theorem for periodic orbits, Poincaré-Bendixson theory. Mathematical background: Fundaments of perturbation analysis. The Multiple Scale Method. Basic concepts of bifurcation analysis: Bifurcation points, Linear codimension of a bifurcation, Imperfections, Fundamental path, Center Manifold Theory. Basic mechanisms of multiple bifurcations: divergence, Hopf, nonresonant or resonant double-Hopf, Divergence-Hopf, Double-zero bifurcation

Lawrence Perko, Differential equations and dynamical systems, Springer-Verlag, 2001

Linear first order PDE's. Method of characteristics. The Burgers' equation. Shocks and rarefaction waves. Riemann problem for scalar conservation laws. Partial differential equations of second order. Well posed problems, IBV problems. The heat equation. Derivation, maximum principle, fundamental solution. The Fourier transform method. Laplace's and Poisson's equations. Maximum principle, fundamental solution and Green's functions. The wave equation. One dimensional equation, fundamental solution and D'Alembert formula. Fundamental solution in three dimension and strong Huygens' principle, Kirchoff's formula.


Introduction to linear systems. Structural properties: controllability and observability. Introduction to Lyapunov stability theory. Introduction to control systems. The Laplace transform. The Nyquist criterium. Frequency based methods. The steady state and the transient responses. Basic control actions. The locus root method.

Greetings and introductions. Expressing likes and dislikes. Talking about daily activities. Understanding and using everyday expressions as well as basic phrases related to daily needs (buying something, asking for directions, ordering a meal). Interacting in a very simple way about known topics (family, nationality, home, studies).
Italian gestures. Italian geography. Introduction to the most important Italian cities. Italian food.

Nuovo Espresso 1, by Luciana Ziglio and Giovanna Rizzo, published by Alma Edizioni, 2014, ISBN: 978-8861823181