Course 2019-2020 a.y.

• Basics on differential equations, separation of variables, linear equations, linear systems. Partial differentiation, free and constrained optimization, gradient methods.

• Vector spaces, Banach spaces, Hilbert spaces. Separable spaces, Fourier series: $ell^2$ and $L^2_T$.

• Continuity, convexity, compactness. Fréchet-derivatives. Fixed points, contractions.

• Classical problems in calculus of variations, critical points. Maxima and minima, necessary/sufficient conditions. Convexity.
  Control theory, bang-bang principle. Hamiltonians, the Pontryagin maximum principle.

• Dynamic programming. The Hamilton-Jacobi-Bellman equation.

• Carry out a formal mathematical proof.
• Recognize the abstract mathematical structures that underline modern theories.
• Master vector spaces techniques.
• Solve optimization problems from calculus of variations.
• Set up and solve control theory problems.
• Solve dynamic optimization problems.

• Apply to economics and to the social sciences the techniques of contemporary mathematics.
• Work out both the quantitative and the qualitative perspectives.
• Solve dynamic optimization problems which are key in Economic Theory.
• Master topological arguments which are important in Game Theory and Microeconomics.

• Face-to-face lectures
• Exercises (exercises, database, software etc.)

Every one/two weeks there is a problem session where mathematical problems concerning the topics taught in class are discussed and solved.

Written exam.

Lecture notes.