Mathematics

The course comprises topics in the field of Solid Mechanics  and is aimed at the presentation of both theoretical and experimental results.The topics covered include:

• Kinematics of deformable continua
• Eulerian and Lagrangian descriptions of motion
• The balance laws of continuum mechanics
• Conservation of mass
• Balance of linear and angular momentum
• Constitutive Equations
• Fluid dynamics: the Navier Stokes equations
• Solid mechanics: nonlinear and linear elasticity
• Selected topics from the mechanics of biological systems

In this series of talks, we are aimed at constructing the solution formula of the Green's functions for various hyperbolic and hyperbolic-parabolic partial differential equations in a half multi-D space domain. We will use the transform variables to derive a master relationships of the Dirichlet-Neumann data of the PDE, and use it to obtain the full Dirichlet-Neumann data in the transform variables; and obtain the Rayleigh surface.

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.

• 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.

Some examples of shape optimization problems.

• 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).

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.

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: