Minimum knowledge required for admission to the Master’s Degree in Mathematics
Concepts of limit and continuity for real functions of one or more real variables, and more generally for functions between normed or topological spaces.
Ordinary differential equations and solution methods.
Taylor polynomial for a real function of one or more variables.
Line integrals and surface integrals.
Numerical series and the main criteria for convergence and divergence. Power series in real and complex settings; elementary notions of holomorphic functions of a complex variable.
Basic elements of functions of a complex variable, analytic functions.
Basic knowledge of Lebesgue measure and integral in R and R^n.
Basic knowledge of the most important function spaces: C^n, C^∞, L^p, L^∞ on open sets of R or R^n.
Abstract Hilbert space.
Fundamentals of classical mechanics, in particular: Energy conservation law. Qualitative analysis of one-dimensional motion. Equilibrium of mechanical systems. Principle of Virtual Work. Two-body problem. The rigid body.
Elements of analytical mechanics (Lagrangian and Hamiltonian), in particular: Euler-Lagrange equations. Hamilton equations. Canonical transformations. Integrable systems. Hamilton-Jacobi method.
Basics of thermodynamics, in particular: first and second law, entropy and its probabilistic interpretation. Diffusive motion and random walk.
Equivalence relations and order relations.
Divisibility among integers: Euclidean algorithm and Bézout's theorem. Prime numbers and the fundamental theorem of arithmetic.
Polynomials with real coefficients: division of polynomials and irreducible polynomials; polynomials with complex coefficients and the fundamental theorem of algebra.
Elements of group theory; normal subgroups and quotients; homomorphisms.
Elements of ring theory; principal ideal domains; Euclidean rings; homomorphisms.
Elements of field theory: numerical fields. Characteristic of a field.
Vector spaces and linear maps; matrices and linear systems; eigenvalues, eigenvectors, diagonalization of endomorphisms.
