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Mathematical Analysis, Modelling, and Applications

The activity in mathematical analysis is mainly focussed on ordinary and partial differential equations, on dynamical systems, on the calculus of variations, and on control theory. Connections of these topics with differential geometry are also developed.The activity in mathematical modelling is oriented to subjects for which the main technical tools come from mathematical analysis. The present themes are multiscale analysis, mechanics of materials, micromagnetics, modelling of biological systems, and problems related to control theory.The applications of mathematics developed in this course are related to the numerical analysis of partial differential equations and of control problems. This activity is organized in collaboration with MathLab for the study of problems coming from the real world, from industrial applications, and from complex systems.

Stochastic Control

Outline: We provide a short introduction to stochastic control. After recalling basic facts on random variables and Brownian motion, we illustrate Ito integral and calculus. Then Stochastic Differential Equations and controlled SDEs are treated. The course is ended by a brief sketch of Malliavin Calculus and existence of distributions for solution to SDEs.

Riemannian and Sub-Riemannian Geometry

  • Differential Geometry.
  • Riemannian Geometry:
    • Riemannian metrics.
    • Geodesics.
    • Curvature.
  • Sub-Riemannian Geometry:
    • Sub-Riemannian metrics and geodesics.
    • Pontryagin Maximum Principle.
    • Nilpotent Approximation.
    • Hamilton, Lagrange and Legendre.
    • Examples:
      • Heisenberg.
      • Martinet.
      • Quantum Systems.

Optimal Synthesis and Applications to Quantum Mechanics

  • Introduction
    • Pontryagin Maximum Principle.
    • Abnormal extremals and Singular Trajectories.
    • What is a solution to an Optimal Control Problem?
    • Definition of Optimal Synthesis.
    • Comparison with the concept of feedback.
  • Bidimensional minimum time problems.
  • The Pontryagin Maximum Principle on Lie groups.
    • Trivialization of the cotangent bundle.
    • PMP on Lie groups.
    • Invariants
    • The K+P Problem.
    • Example: SL(2) (wave fronts, spheres, cut and conjugate loci).

Periodic Solutions in Hamiltonian Systems

The search for periodic solutions in Hamiltonian systems is old and originated in the many body-problem of celestial mechanics. In the last decades it was tackled with success using the variational action functional. Our aim is to collect some old and more recent results, with a special emphazise on variational techniques.

Parabolic Problems and Techniques

One of the most remarkable properties of evolutionary processes described by reaction-diffusion equations is the possibility of the eventual occurence of singularities developing from perfectly smooth data. In this short course for parabolic problems, we will discuss:

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