Ph.D. courses of previous academic years
Ph.D. courses 2011/12
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
- A. Agrachev (SISSA): Geometry of Optimal Control Problems and Hamiltonians Systems (3 cycles).
- A. Ambrosetti (SISSA): Introduction to Nonlinear Differential Equations (2 cycles).
- M. Berti (Univ. of Naples "Federico II"): Hamiltonian PDEs (1 cycle).
- S. Bianchini (SISSA): Optimal Transport Problems (3 cycles).
- G. Dal Maso (SISSA): Introduction to Partial Differential Equations (2 cycles).
- A. DeSimone (SISSA): Topics in the Mechanics of Biological Systems (2 cycles).
- A. Malchiodi (SISSA): Morse Theory (2 cycles).
- Y. Sachkov (IPS-RAS): Control on Lie Groups (1 cycle).
- S. Zagatti (SISSA): Introduction to the Calculus of Variations (3 cycles).
- I. Zelenko (SISSA): Introduction to Dynamical Systems (3 cycles).
Academic year 2004/2005
- A. Agrachev (SISSA): Geometric control theory.
- A. Ambrosetti (SISSA): Critical points theory and nonlinear elliptic equations.
- G. Dal Maso (SISSA): Gamma-convergence.
- A. DeSimone (SISSA): Numerical methods.
- A. Malchiodi (SISSA): Elliptic problems with lack of compactness.
- S. Zagatti (SISSA): Partial differential equations.
- G. Vidossich (SISSA): Ordinary differential equations.
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:
- Local Existence
- Global Existence
- Blowup
- 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.
- An introduction to Hamiltonian systems.
-
Local existence theory for periodic solutions
- Conditions at the linearized system (Lyapunov, Poincare, Weinstein-Moser, Fadell-Rabinowitz)
- Non-linear conditions (Birkhoof-Lewis).
- Global results with prescribed energy and prescribed period.
Boscain, Optimal Synthesis and Applications to Quantum Mechanics
Outline:
- 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).
- Introduction to Quantum Mechanics.
- Finite dimensional quantum problems.
- Elimination of the drift.
- Reduction to real problems.
- The choice of the cost.
- 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
- existence and nonexistence results
- relaxed formulation of optimization problems
- optimization of mass densities
- relations with Monge mass transportation problems
- some optimization problems in mass transportation
- applications to problems in urban planning
Charlot, Riemannian and Sub-Riemannian Geometry
Outline:
- 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.
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.