il corso si terrà nel periodo: MARZO-APRILE

The main objective of the course is to provide to an audience of non-specialists (engineers, applied mathematicians physics and chemists) a brief introduction to the theory of differential forms, with particular emphasis on the application to the engineering sciences. Exterior differential forms constitute one of the main tools in contemporary mathematics. Their applications extend from geometry (differential and / or algebraic) topology, complex analysis and global analysis up to high-energy physics, geometric control theory and biophysics. The course will cover the classical aspects of the theory as well as their discrete counterpart: the Discrete Exterior Calculus. Particular attention will be given to the applications to the geometry of manifolds and to the classical electromagnetism : Maxwell field equations, linking numbers, the Gauss integral and the Ampère law. Other more advanced examples will be also covered, such as the Dirac and the Hoft-Polyakov monopoles and Yang-Mills instantons. The final goal will be to give a self-contained explanation of the classical and discrete de Rham theorem which proves the topological invariance of the de-Rham cohomology.

- Differential forms on R^n : differential forms, the exterior derivative, pull-back of a differential form, change of coordinates, the inverse of the Poincarè lemma. Maxwell’s field equations via exterior differential forms.
- Manifolds and integration : differentiable manifolds, maps between manifolds, difeomorphisms, partition of the unity, tangent vectors and differential forms on manifolds, integration of differential forms on manifolds, Stokes theorem, the de Rham cohomology.
- Integration of differential forms on chains : Euclidean simplices, chains and boundaries, integration of a p-form on a p-chain, Stokes theorem on p-chains, periods.
- Applications : winding numbers, degree of a mapping, the Hopf invariant, linking numbers, the Gauss integral and th Ampère law.
- Discrete differential geometry and exterior calculus: Primal Simplicial Complex and Dual Cell complex. Local and global embeddings. Differential forms and exterior derivative. Hodge star operator ant the co-differential. Maps between 1-forms and vectir fields. Divergence and Laplace-Beltrami operator. Contraction and Lie derivative. Discrete Poincarè Lemma.
- Discrete differential geometry and exterior calculus: Primal Simplicial Complex and Dual Cell Discrete Variational Mechanics and DEC