Here are the schedules from fall 2019 and winter 2020.

Weeks 1-2: Counting things and using python

• Generating functions: counting partitions, permutations, Catalan objects.
• Discrete probability distributions, expectation, variance, and conditional probability.
• Linear recurrence relations.
• Experimental math: guessing formulae from data.
• Algorithmic complexity, big-oh notation.
• Graph algorithms: shortest paths, counting subgraphs.
• Python: basic flow control, functions, lists and dictionaries.

Homework 1: hw1.ipynb or hw1.html

Homework 2: hw2.ipynb or hw2.html

Weeks 3-4: Structure and analysis of graphs and networks

• Terminology: connectedness, degree, matchings.
• Adjacency matrix: eigenvalues/vectors, cycle counting.
• Spanning trees, Wilson’s algorithm
• Laplacian: Matrix-tree and Cauchy-Binet theorems, discrete harmonic functions, Laplacian eigenmaps.
• Random graphs: Erdős-Rényi model, phase transition, Poisson approximation.
• Labeled graphs: the Ising model, Gibbs distributions, and the Metropolis algorithm.
• Random walks on weighted graphs and the PageRank algorithm.

Homework 3: hw3.ipynb or hw3.html Homework 4: hw4.ipynb or hw4.html

Week 5: Point processes

• Definition; Poisson limits; additivity; motivating examples.
• Thinning and labeling; conditional uniformity; characteristic functions and moments.
• Statistics of spatial point processes: testing for independence.
• Poisson jump processes.
• The Cauchy process.
• The Kolmogorov equation for pure-jump Levy processes.

Homework 5: hw5.ipynb or hw5.html

Weeks 6-7: Markov chains and generators

• Finite-state Markov chains: definitions, classificiation, transition probabilities.
• Methods for simulation: Poissonization.
• Matrix exponentiation of substochastic matrices.
• The approach to stationarity.
• Generators for Markov chains and Poisson jump processes.
• Uses and examples of generators.
• Simple random walk and relatives.

Homework 6: hw6.ipynb or hw6.html

Week 8: Brownian motion and Gaussian processes

• Gaussian processes: definition and characterization; conditional probabilities.
• Brownian motion as a limit of simple random walk via convergence of the graph Laplacian.
• Brownian motion as an integral of white noise.
• Other integrals of white noise and the L2 isometry.

Weeks 9-10: Diffusions and associated PDE

• Itô integration.
• Stochastic differential equations
• Backward Fokker-Planck equation from Itô’s lemma; connection to generators.
• Forward Fokker-Planck equation.
• Equilibrium distributions and approach to equilibrium.
• First passage times of diffusions.

Homework 7: hw7.ipynb or hw7.html