A review of differential calculus for ODEs and PDEs, UPD (Emeric. Bouin)
Differential calculus:
- C^1 function, optimization, convex function;
- The implicit function theorem, constrained optimization;
- The inverse function theorem;
- Domain and its boundary, submanifold in R^n;
- The divergence theorem;
- Brouwer Theorem;
- Applications to PDE.
Ordinary differential equations:
- Examples: gradient flow, Hamiltonian flow; others;
- Cauchy-Lipschitz theorem;
- Gronwall lemma;
- smooth dependence by perturbations;
- linear stability;
- nonlinear stability and Lyapunov function;
- volume preserving flow;
- Variations calculus and Euler-Lagrange equation.
High dimensional probability, cours avancé Master 1 Mathématiques Fondamentales-PSL, ENS (Djalil Chafai)
Possible contents (will be fixed more precisely in Fall 2021):
- Analysis and geometry of Markov processes
- Functional inequalities and long time behavior
- Concentration of measure and transportation of measure
- Entropy and large deviations
- Random matrices and universality phenomena
- Lindeberg and Stein methods for central limit phenomena
Large Deviations and applications in Physics and Analysis, UPD (Stefano Olla)
Large deviations are at the center of modern probability and statistics. The theory originated form the risk analysis for insurance companies. Today there are applications in almost all domains of applied mathematics.
This course will revise various generalizations of Cramer and Sanov Theorems and will illustrate applications to Statistical Physics and Analysis, in particular
- Statistical Physics: Gibbs probability distributions for mean field models (Curie-Weiss) and for models with local interaction (Ising Model). Local Large deviations and equivalence on ensembles.
- Non-linear PDE: Viscous solutions in Hamilton-Jacobi equations.
- Random perturbations of dynamical systems: Freidlin–Wentzell theory.