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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 analysismechanics of materialsmicromagneticsmodelling 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.

Random polynomial systems, Kahler geometry and the momentum map

Lecture 1: On counting solutions of polynomial systems

  •   Bézout's theorem
  •   Smale's 17-th problem
  •   Shortcomings of Bézout's theorem
  •   Sparse polynomial systems, and the mixed volume

Lecture 2: Differential forms

  •   Multilinear algebra over R
  •   Complex differential forms
  •   Kähler geometry
  •   The coarea formula, using bundles.
  •   Projective space

Lecture 3: Reproducing kernel spaces

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