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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.

Geodesics and diffusion on sub-Riemannian manifolds

  1. Riemannian and sub-Riemannian manifolds
  2. Distance, geodesics
  3. The intrinsic volume
  4. The Laplace Beltrami operator
  5. Computation of the heat kernel on the Heisenberg group and on the Grushin plane
  6. Diffusion on the diagonal and on the cut locus

The Classical Isoperimetric Problem and its Nonlocal Variants

  • Preliminary results on Geometric Measure Theory:
    • Hausdorff measures,
    • tangent measures, 
    • rectifiable sets.
  • The theory of sets of finite perimeter. 
  • The De Giorgi's solution to the Isoperimetric Problem. 
  • Partial regularity theory of Lambda-minimizers of the perimeter functional. 
  • A nonlocal variant of the perimeter: the sharp interface Ohta-Kawasaki energy. 
  • Regularity of  local minimizers. 
  • The second variation of the perimeter and of the Ohta-Kawasaki energy. 

Advanced Topics in Calculus of Variations

Program of the course:

  • Preliminaries on Gamma-convergence.
  • Homogenization: an example in dimension one.
  • Dimension reduction in elasticity: preliminaries on the mathematical theory of elasticity;
    Korn’s inequality; rigorous derivation of a plate theory in linearized elasticity
    rigidity theorem; rigorous derivation of Kirchhoff plate theory.
  • Gradient theory of phase transitions

Crystalline anisotropic mean curvature flow

  • Mean curvature flow of hypersurfaces. Introduction to the problem.
  • First variation of the perimeter.
  • Uniqueness of smooth flows. Short time existence.
  • Examples of singularities: Grayson example.
  • Fattening of the crossing.
  • Anisotropic mean curvature flow.
  • Finsler metrics and their duals.
  • The Wulff shape. Duality mappings.
  • Relations between the Minkowski content (or anisotropic perimeter) and the Hausdorff measure.

Nonlinear Analysis and Dynamical Systems

In the first part of this course, I will present the basic elements of calculus in infinite dimensional spaces, holomorphic functions, and the classical implicit function theorem in Banach spaces.

Nonlinear Analysis and Bifurcation Theory

The goal of the course is to cover the following topics:

  • Basic calculus in Banach spaces
  • Local inversion theorems and implicit function theory 
  • Properties of bifurcation points 
  • Bifurcation from the simple eigenvalue 
  • Bifurcation from multiple eigenvalues
  • Examples and applications 

Reference:

  • Ambrosetti-Prodi, A primer in Nonlinear Analysis

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