Notes

Notes

This page contains a collection of notes I have written over the years, usually in the context of courses I've taken (and sometimes courses I've TAed). Alas, I no longer get to take classes with any degree of regularity, so this page is unlikely to receive too many updates over time.

These notes were often written in a hurry in preparation for exams; you should therefore expect varying quality. They were never systematically proof-read.

Columbia Business School - First Year of the Doctoral Program in Decisions, Risk and Operations


  • Stochastic processes
    • Notes from Prof Assaf Zeevi's "Foundations of Stochastic Modelling".
    • Notes from Prof David Yao's "Stochastic Processes II".
    • A copy of the cheat sheet I used for the stochastic part of my qualifying exams.
  • Optimization
    • Notes based on "Optimization 1", a course I took with Prof Donald Goldfarb in the fall of 2010. The notes also draw heavily on Bestimas and Tsitsiklis' "Introduction to Linear Optimization". These notes cover linear programming, including duality, the simplex algorithm and sensitivity analysis, with a particular focus on geometry. (The notes also include a very short condensed section on network-flow problems).
    • Condensed Notes roughly following two courses I took - "Foundations of Optimization" (thought by Prof Ciamac Moallemi) and "Convex Optimization" (thought by Prof Garud Iyengar). These notes are also heavily based on Boyd and Vandenberghe's book "Convex Optimization" (available online) and Luenberger's "Optimization by Vector Space Methods". The chapter numbers in these notes refer to Boyd and Vandenberghe's text. Rough list of topics covered: convexity of sets and functions, formulation of convex programs (from linear programs to semi-definite programs), duality, applications, Hilbert and Banach spaces, minimum-norm problems in Banach spaces, the Hahn-Banach Theorem.
    • Notes covering more technical material in the "Foundations of Optimization" class above, including duality, KKT conditions, no-convex program and more.
    • A copy of the cheat sheet I used for the deterministic part of my qualifying exams.
  • Some notes on game theory based on "Microeconomic Theory" taken during a course by Prof Paolo Siconolfi.
  • MBA classes I TAed:
  • I also TAed a PhD class on Convex Optimization for Prof Garud Iyengar. I designed a number of review sessions for the class;
    • Review session 1 (convex sets, convex cones, dual cones); pdf
    • Review session 2 (convex functions, convex conjugates, simple application to risk measures); pdf
    • Review session 3 (convex programs in standard form (LP, QP, QCQP, SOCP and SDP) with applications, including sparse and robust optimization; one small part of the proof is omitted); pdf
    • Review session 4 (duality, with applications; including adversarial optimization); pdf
    • Review session 5 (various applications of SDPs, introduction to infinite-dimensional optimization); pdf
    • Review session 7 (more infinite-dimensional optimization; Banach spaces); pdf
    • Midterm review pdf

University of Cambridge - Part III Mathematics/Certificate of Advanced Study in Mathematics/Masters of Mathematics


  • Actuarial Statistics: course notes, based on lectures by Susan Pitts, covering aggregate claims, reinsurance, ruin probabilities, no-claim-discount systems, credibility theory and run-off triangles. Solutions to the 2006 paper.
  • Biostatistics: notes on survival data analysis, based on lectures and handouts by Peter Treasure. Notes on deriving the samples size required for a test of given power when comparing proportions.
  • Mathematics of Operational Research: notes, based on lectures notes by Richard Weber, the book Introduction to Linear Optimization and the book Games, Theory and Applications, by L.C. Thomas. Covering optimization (including primal simplex, dual simplex and integer linear programming), algorithms on graphs, a very short section on complexity theory, game theory (zero-sum games, non-zero-sum games, cooperative games, bargaining, market games and evolutionary games) and regret minimization. An example of using Gomory's Cutting Plane method to solve an integer linear program.
  • Monte Carlo Inference: notes based on lectures by Robert Gramacy, covering random number generation, nonparametric inference (importance sampling, control variates, antithetic variables, the bootstrap, the jacknife, bootstrap tests), bayesian inference (Markov Chain Monte Carlo, including the Gibbs Sampler and the Metropolis Hastings Algorithm, reverse jump MCMC, sequential importance sampling) and classical inference (simulated annealing, expectation maximisation).
  • Statistical Theory and Applied Statistics: notes, based on lectures by Richard Samworth and Susan Pitts, and practicals organized by Susan Pitts. Covering linear models (including ANOVA, geometric interpretation and formal treatment of the variance using Cochran's Theorem), likelihood theory, generalized linear models (logistic regression, Poisson regression, contingency tables), high-dimensional models (Hodge's estimator, the Stein estimator, ridge regression, the LASSO, SCAD, LARS), multiple testing (the Bonferroni correction and the Benjamini-Hochberg procedure) and application of LMs and GLMs in R. Includes solutions to many example sheet problems. Note that the proofs of Slutsky's Theorem and of the asymptotic normality of MLEs are still not complete.
  • In my fourth year at Cambridge, I spent a considerable amount of time running example classes for first-year students. This page contains the materials I created for this purpose

Work at MIT


  • Quantum Mechanics II (8.05): review notes for the final exam.
  • Relativity (8.033): miscellaneous notes on various parts of the class.
  • String Theory for Undergraduates (8.251): miscellaneous partial notes (1, 2, 3) taken during the class, unlikely to be anywhere near as good as the course book by Prof Barton Zweibach who taught it.
  • Statistical Mechanics (8.333): miscellaneous partial notes, which might be useful in that they cover some topics in painful detail.
  • Optimization Methods in Management Science (15.052): exam review notes (1, 2, 3).
  • While at MIT, I was fortunate to work as a Graduate Teaching Assistant for courses 8.01 (Classical Mechanics) and 8.02 (Electromagnetism). Here is the materials I created for these classes
    • Course 8.01 (Classical Mechanics)
    • Course 8.02 (Electromagnetism)
      • Notes from a review session covering self-inductance, Maxwell's Equations, Electromagnetic waves (including the Poynting vector), diffraction and interference. I also went through some practice questions with solutions.
      • A guide to surface/flux integrals for perplexed 8.02 physicists.

University of Cambridge - Part IB Natural Sciences


University of Cambridge - Part IA Natural Sciences