Students select Analysis and Probability I in the fall semester, and then pick a specialization in the spring; either Analysis II or Probability II.

### Analysis and Probability I

I Measure Theory

- Construction of the integral, limits and integration
*L*spaces of functions^{p}- Construction of measures, Lebesgue-Stieltjes product measures
- Examples: ergodicity, Liouville measure, Hausdorff measure

II Elements of Probability

- The coin-tossing or random walk model
- Independent events and independent random variables
- The Khintchin weak law and the Kolmogorov strong law of large numbers
- Notions of convergence of random variables
- The Central Limit Theorem

III Elements of Fourier Analysis

- Fourier transforms of measures, Fourier-Lévy Inversion Formula
- Convergence of distributions and characteristic functions
- Proof of the Central Limit and Lindeberg Theorems
- Fourier transforms on Euclidean spaces
- Fourier series, the Poisson summation formula
- Spectral decompositions of the Laplacian
- The heat equation and heat kernel

IV Brownian Motion

- Brownian motion as a Gaussian process
- Brownian motion as scaling limit of random walks
- Brownian motion as random Fourier series
- Brownian motion and the heat equation
- Elementary properties of Brownian paths

### Analysis II: Partial Differential Equations and Functional Analysis

I First Order Partial Differential Equations

- Cauchy’s Theorem for first order real partial differential equations
- Completely integrable first order equations

II Implicit Function Theorems

- Basic examples of linear and non-linear partial differential equations
- The functional analytic framework, Banach and Hilbert spaces
- Bounded linear operators, spectrum, invertibility
- Implicit function theorems in Banach spaces
- Sketch of subsequent applications to the basic examples

III Second Order Partial Differential Equations

- Qualitative description: elliptic,parabolic, hyperbolic equation
- The Cauchy problem
- Maximum principles
- Sobolev and Schauder spaces
- A priori estimates and Green’s functions
- Riesz-Schauder theory of compact operators
- Detailed treatment of basic examples
- The Laplace and heat equations on compact manifolds
- Applications to de Rham and Hodge theory

IV Selected Topics, chosen from

- Riemann-Roch and index theorems
- Determinants of Laplacians, modular forms
- Integral representations, Hilbert transforms, singular integral operators
- Subelliptic equations
- Nash-Moser implicit function theorems
- Non-linear equations from geometry or physics

### Probability II

Prerequisite: Analysis and Probability I. Can be taken concurrently with Analysis II.

I Rare Events

- Cramér’s Theorem
- Introduction to the Theory of Large Deviations
- The Shannon-Breiman-McMillan Theorem

II Conditional Distributions and Expectations

- Absolute continuity and singularity of measures
- Radon-Nikodým theorem. Conditional distributions
- Conditional expectations as least-square projections
- Notion of conditional independence
- Introduction to Markov Chains. Harmonic functions

III Martingales

- Definitions, basic properties, examples, transforms
- Optional sampling and upcrossings theorems, convergence
- Burkholder-Gundy and Azuma inequalities
- Doob decomposition, square-integrable martingales
- Strong laws of large numbers and central limit theorems

IV Applications

- Optimal stopping
- Branching processes and their limiting behavior. Urn schemes
- Stochastic approximation. Probabilistic analysis of algorithms

V Stochastic Integrals and Stochastic Differential Equations

- Detailed study of Brownian motion
- Martingales in continuous time
- Doob-Meyer decomposition, stopping times
- Integration with respect to continuous martingales, Itô’s rule
- Girsanov’s theorem and its applications
- Introduction to stochastic differential equations. Diffusion processes

VI Elements of Potential Theory

- The Dirichlet problem. Poisson integral formula
- Solution in terms of Brownian motion
- Detailed study of the heat equation; Cauchy and boundary-value problems
- Feynman-Kac theorems, applications