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The list below is the set of possible reading courses which can be offered by the Phd Program. Each academic year, on the basis of the didactic requirements of the incoming PhD students, the Council will select the ones to be activated.
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
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].
Refs:
- [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
Program:
- 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
The course should be an introduction to the topic of large deviations and its connection with statistical physics. It will cover the following subjects: