Ph.D. courses of previous academic years

Ph.D. courses 2011/12

Lecturer Title Duration
Prof. Andrei Agrachev Nonlinear analysis 3 cycles, October 2011-January 2012
Prof. Stefano Bianchini Hyperbolic PDE's 3 cycles, from January 2012
Prof. Gianni Dal Maso Calculus of Variations 3 cycles, October 4 - February 15
Prof. Antonio DeSimone Introduction to Continuum Mechanics 3 cycles, November - February
Prof. Andrea Malchiodi Variational Methods 3 cycles, from October 10, 2011
Dr. Claudio Altafini Dynamical models in Biology 1 cycle, from October 2011
Prof. Massimiliano Berti Nash-Moser theory for PDEs 4 hours (Minicourse), April 2012
Prof. Ugo Boscain Control theory 1 cycle, April-May 2012
Prof. Giovanni Vidossich Ordinary differential equations 2 cycles, from October 2011
Prof. Massimiliano Morini The classical isoperimetric problem and a non-local variant. 2 cycles, from March 2012*
Dr. Stefano Luzzatto Introduction to Smooth Ergodic Theory 1 cycle, from November 7, 2011
Prof. Vittorio Coti Zelati Critical point theory and Dynamical systems 1 cycle, from April 2012
Prof. Giovanni Bellettini Mean curvature flow and singular perturbations 1 cycle, from April 2012


Ph.D. courses 2010/11

Lecturer Title Duration
Prof. Andrei Agrachev: Topics in sub-Riemannian Geometry 3 cycles, November, 29 - March
Prof. Giovanni Alberti: Geometric Measure Theory short course, October, 25 - November, 11
Prof. Antonio Ambrosetti: Nonlinear Analysis 2 cycles, November, 29 - February
Prof. Stefano Bianchini: Partial Differential Equations (mutuated by the course Analisi Superiore 1 for Laurea Magistrale) 30 September - 18 December, plus some practice lectures in January. 2 cylces from November for the PhD students
Prof. Gianni Dal Maso: BV Functions 3 cycles, November, 23 - March
Prof. Antonio DeSimone: Mechanics of Soft and Biological Matter 2 cycles, November, 23 - March
Prof. Andrea Malchiodi: Analisi Superiore 2 for Laurea Magistrale 2 cycles, October - December
Prof. Giovanni Vidossich: Ordinary Differential Equations 3 cycles, 9 November - March
Dr. Claudio Altafini: Dynamical models in Biology 1 cycle, November - December
Dr. Ugo Boscain: Introduction to sub-Riemannian geometry 1 cycle, November, 10 - November, 22
Dr. Luca Heltai: Introduction to Numerical Analysis 1 cycle, from December
Dr. Maria Giovanna Mora: Gamma-convergence and Applications 2 cycles, from March
Dr. S. Zagatti: Istituzioni di Analisi Superiore A for Laurea Magistrale 2 cycles, 6 October - December

Ph.D. courses 2009/10

Lecturer Title Duration
Prof. A. Agrachev: Topics in sub-Riemannian Geometry 2 cycles, November, 17 - February
Prof. A. Agrachev: Morse Theory 1 cycle, April - May
Prof. S. Bianchini: An introduction to descriptive Set Theory from February, 22
Prof. G. Dal Maso: Elliptic Partial Differential Equations (mutuated by the course Analisi Superiore 1 for Laurea Magistrale) 3 cycles, October - December
Prof. A. DeSimone, Prof. C. Micheletti: Topics in the Mechanics of Biological Systems 3 cycles, November, 10 - March
Prof. A. Malchiodi: Nonlinear Analysis 3 cycles, January, 26 - April
Prof. R. Musina: The 59th Yau problem 1 cycle, from March, 16, to April, 21
Prof. G. Vidossich: Topological Degree and Applications (mutuated by the course Applicazioni dell'Analisi Matematica for Laurea Magistrale) 1 cycle, from March, 4
Dr. U. Boscain: Introduction to sub-Riemannian geometry 1 cycle, November, 11 - December, 09
Dr. M.G. Mora: Mathematical Theory of Elasticity 1 cycle, March
Dr. M. Morini: Calculus of Variations 1 cycle, from February, 3
Dr. A. Remizov: Singular Perturbation and Implicit Differential Equations 1 cycle, from February, 16
Prof. D. Fremlin: Brownian motion and Newtonian capacity 3-7 May, 2010

Ph.D. courses 2008/09

Lecturer Title Duration
Prof. A. Agrachev: Riemannian and sub-Riemannian Geometry from a Hamiltonian viewpoint 3 cycles, November-March
Prof. S. Bianchini: Optimal Transportation February-March
Prof. G. Dal Maso: Gamma-convergence 3 cycles, November-March
Prof. A. DeSimone: Continuum Mechanics 3 cycles, November-February
Prof. A. Malchiodi: Nonlinear Analysis 3 cycles, January-April
Dott. U. Boscain: Complements to the course of Prof. Agrachev November
Dott. C. Altafini: Introduction to Bioinformatics for the students of the Ph.D. in Functional and Structural Genomics, November-December
Dott. M.G. Mora and Dott. M. Morini: Introduction to Geometric Measure Theory and BV Functions 2 cycles, March-April
Dott. S. Zagatti: Analisi Superiore 1 - Introduzione alle PDEs per la Laurea Specialistica
Prof. G. Vidossich: Analisi Funzionale Lineare per la Laurea Specialistica
Prof. G. Vidossich: Topological Degree 1 cycle

Ph.D. courses 2007/08

Lecturer Title Duration
Prof. A. Agrachev: Geometric Control Theory from 9 november
Prof. A. Ambrosetti: Critical Point Theory 3 cycles
13 november - march
Prof. S. Bianchini: Partial Differential Equations
Prof. G. Dal Maso: Calculus of Variations 3 cycles
8 november - march
Prof. A. De Simone: Topics in the Mechanics of Biological Systems 2 cycles
6 november - february
Prof. A. Malchiodi: Geometric Evolution Equations 3 cycles
14 november - february
Prof. G. Vidossich: Differential Equations and Methods of Nonlinear Analysis
Prof. S. Zagatti: Semigroup Theory and Applications february-april
Dott. U. Boscain: Heisenberg Groups 1 cycle
Dott.ssa M.G. Mora, Dott. M. Morini: Gamma-convergence and Applications 2 cycles
april-june

Academic year 2006/2007

Lecturer Title Duration
Prof. A. Agrachev: Nonlinear Analysis (together with LM)
Prof. A. Malchiodi: Partial Differential Equations (together with LM)
Prof. G. Dal Maso: BV Functions 27 lectures, November to March
Dott. D. Ruiz: Critical Point Theory 9 lectures, January to March
Prof. A. DeSimone: Modeling and Numerical Methodsy 20 lectures, November to Januar
Prof. S. Bianchini: Boltzmann Equation 27 lectures, January to March
Prof. G. Vidossich: Analysis in Ordered Banach Spaces 27 lectures, November to March
Dott. I. Zelenko: Geometric Control 27 lectures, November to March
Dott. U. Boscain: Ordinary Differential Equations 9 lectures, November to December
Dott.ssa M. G. Mora: Sobolev Spaces 9 lectures, November to December
Dott. M. Morini: Calculus of Variations 9 lectures, January to February
Prof. G. Alberti (Pisa): Geometric Measure Theory 10 lectures, April to May
Prof. L. Bertini (Rome): Stochastic Differential Equations 10 lectures, April to May
Prof. M. Berti (Rome): to be announced to be announced
Prof. M. Zhitomirskii: to be announced to be announced

Academic year 2005/2006

Academic year 2004/2005

Academic year 2003/2004

Bebernes, Parabolic Problems and Techniques

Outline: 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:

  1. Local Existence
  2. Global Existence
  3. Blowup
  4. Beyond Blowup
Berti, Periodic Solutions in Hamiltonian Systems

Outline: 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.

  1. An introduction to Hamiltonian systems.
  2. Local existence theory for periodic solutions
    • Conditions at the linearized system (Lyapunov, Poincare, Weinstein-Moser, Fadell-Rabinowitz)
    • Non-linear conditions (Birkhoof-Lewis).
  3. Global results with prescribed energy and prescribed period.
Boscain, Optimal Synthesis and Applications to Quantum Mechanics

Outline:

  1. 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.
  2. Bidimensional minimum time problems.
  3. 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).
  4. Introduction to Quantum Mechanics.
  5. Finite dimensional quantum problems.
    • Elimination of the drift.
    • Reduction to real problems.
    • The choice of the cost.
  6. The key example: 3-level systems.
    • Resonance.
    • Minimizing the energy.
    • Minimizing time with bounded controls.
    • The STIRAP strategy.
Buttazzo, Shape optimization and mass transportation problems.

Outline: Some examples of shape optimization problems

  1. existence and nonexistence results
  2. relaxed formulation of optimization problems
  3. optimization of mass densities
  4. relations with Monge mass transportation problems
  5. some optimization problems in mass transportation
  6. applications to problems in urban planning
Charlot, Riemannian and Sub-Riemannian Geometry

Outline:

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