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.
- Affine spaces, subspaces, and affine transformations. Inner product, Euclidean vector spaces.
- Quadratic forms and their classification by congruence; conics and quadrics: affine and metric classification.
- Topological spaces and continuous functions, homeomorphisms, product and quotient topology, connected and compact spaces. Real projective spaces.
- Differentiable curves in the plane and in space; surface theory in E3, geodesics.
- Axioms of probability calculus; events, independent events, and conditional probability.
- Random variables and main distributions (binomial, geometric, hypergeometric, Poisson, exponential, gamma, Gaussian, beta).
- Multinomial distribution and multivariate Gaussian distribution. Stochastic independence and uncorrelation.
- Bernoulli scheme. Weak law of large numbers. Generating function. Characteristic function.
- Basic notions of descriptive statistics.
- Basic elements of numerical calculus: finite arithmetic, numerical linear algebra, data approximation.
- Principles of programming.