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