References are provided __only__ for initial familiarity with the subject.

## Distributions

Sections 11.1, 11.2## Applications of distributions to PDE and weak solutions

Sections 11.3, 11.4## Burgers equation

Section 12.1## Variational calculus of one variable

Sections 10.1, 10.2## Variational problems of several variables

Sections 10.1, 10.2## Classical mechanics: Lagrangian formalism

Section 10.3## Classical mechanics: Hamiltonian formalism

Section 10.3## Physics: Lagrangian & Hamiltonian formalism

Section 10.4## Variational theory of the eigenvalues for PDE

Section 13.1## Asymptotic distribution of eigenvalues for PDE

Section 13.2## Properties of eigenfunctions

Section 13.3## Spectra of PDEs

Section 13.4## Continuous spectrum and scattering

Section 13.5## Basic probability topics

Laws of large numbers, central limit theorems, martingales, Brownian motion, Markov chains and processes, basic ergodic theorems## Applications of probability in other fields

Using the probabilistic method in combinatorics, using random walks or Brownian motion to get results in PDE/complex analysis, probabilistic results in number theory etc.## Specific probabilistic models

Random walks on groups/graphs, random graphs, percolation, random polymers, various interacting particle systems etc.