Ph.D. Courses 2017-2018

 Examination report

Mean Field Limits in Classical and Quantum Mechanics
Francois Golse (Centre de Mathématiques Laurent Schwartz - Ecole Polytechnique)
First meeting: Monday, May 14th, 2008 - at 11:00, room B

COURSE (8 lectures, 16 hours)
Singularities of plurisubharmonic functions and multiplier ideals
Sébastien Boucksom (Centre de Mathématiques Laurent Schwartz - Ecole Polytechnique), INdAM visiting professorship
Lectures will be held in Room B, at 14:30-16:30
Timetable: Tue 5, Fri 8, Wed 13, Fri 15, Tue 19, Fri 22, Tue 26, Thu 28 June 2018.

Second School and Workshop ''Mathematical Challenges in Quantum Mechanics''

Lectures by: R. Frank, B. Schlein, S. Teufel
Rome, February 19-24, 2018

A variational approach to complex Monge-Ampère equations 
Eleonora Di Nezza (IHES)
- January 9th, 2018, Room B, 14:30-16:30
- January 12th, 2018, Room B, 11:00-13:00

Algebre di Operatori
Sergio Doplicher (Dipartimento di Matematica - Sapienza)
Tuesdays, Wednesday and Friday, 12:00-14:00 room B
From January 9, 2018 to February 9, 2018


Geometry of Kähler threefolds

Andreas Höring (Laboratoire de Mathématiques J.A. Dieudonné - Université de Nice)
- November 27th, 2017, Room B, 14:00-16:30 
- November 28th, 2017, Room B, 14:00-16:30


Reading courses activated in the first semester of the academic year 2017-2018:

(1) Mathematics for Artificial Intelligence

Introduction to neural networks: genesis and future perspectives.
Basics of statistical mechanics and statistical inference.
Neural networks for associative memory and pattern recognition.
Hopfield model.
Rosenblatt and Minsky&Papert perceptrons.
Neural networks for statistical learning and feature discovery.
Supervised Boltzmann machines.
Unsupervised Boltzmann machines.   
Bayesian equivalence between Hopfield retrieval and Boltzmann learning.
Multilayered Boltzmann machines and deep learning.

Advanced topics: Numerical tools for machine learning; Non-mean-field neural networks; (Bio-)Logic gates; Maximum entropy approach, Hamilton-Jacobi techniques for mean-field models.

Ref.: A.C.C. Coolen, R. Kuhn, P. Sollich, Theory of Neural Information Processing Systems, Oxford Press.

(2) Stability of Matter in Quantum Mechanics

  • Teacher: M. Correggi
  • When/where: November 30, December 5, 12, and 19, 10:00-12:00 in Room B

Basic topics: Ground state for many-body quantum systems; identical particles: bosons and fermions; Pauli principle; stability of first kind; reduced density matrices; Lieb-Thirring inequalities; electrostatic inequalities; stability of the second kind.

Advanced topics (seminars): Sufficient conditions for existence and uniqueness of the ground state for interacting systems [LL]; variational derivation of bound states [LL]; self-adjointness and boundedness from below of atomic Schroedinger operators [LL]; Lieb-Thirring inequalities [LS]; Coulomb potential and Baxter inquality [LS]; Thomas-Fermi theory of electrons [LS].

- [LL]    E.H. Lieb, M. Loss, Analysis, AMS, 2001.
- [LS]    E.H. Lieb, R. Seiringer, The Stability of Matter in Quantum Mechanics, Cambridge University Press, 2009.

(3) Algebraic Groups

  • Teacher: G. Pezzini
  • When/where: Next lessons on December 6, 13, and 20, 10:00-12:00 in Room B

- linear algebraic groups, definitions, examples;
- general theory of  homogeneous spaces;
- quotients, geometric invariant theory;
- structure of solvable groups, Borel subgroups;
- structure of reductive groups and their representations;
- flag manifolds, symmetric spaces.

Refs (preliminary list):
1. M. Brion, Introduction to actions of algebraic groups (downloadable from his web page).
2. J. Humphreys, Linear Algebraic Groups.

(4) Large Deviations

  • Teacher: A. Faggionato
  • When/where: December 7 and 14, 16:00-18:00 in Room B.

The course should be  an introduction to   the topic of large deviations  and its connection with  statistical physics. It will cover the following subjects:

- Large deviations principle; 
- Cramer's theorem;
- Varadhan's theorem;
- Contraction principle;
- Sanov's theorem for the empirical measure;
- Entropy, pressure, free energy;
- Dobrushin-Lanford-Ruelle variational principle.
F. den Hollander. Large deviations. Fields Institute Monographs, 2000.
F. Rassoul-Agha, T. Seppalainen. Large deviations with an introduction to Gibbs measures. American Mathematical Society, 2015.
O.E. Lanford. Entropy and Equilibrium States in Classical Statistical Mechanics.   Lecture Notes in Physics     20, pp.1-113, Springer, 1973

Reading courses activated in the second semester of the academic year 2017-2018:

(1) Metric spaces with curvature bounded above

  • Teacher: A. Sambusetti
  • When/where: May 2, 17:00 in Room B.

For details, see here


You can also follow courses at the PhD in Mathematics of Rome 2 and 3 Roma:
PhD courses at Roma 2
PhD courses at Roma 3

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