This page contains a collection of notes I have written over the years, usually in the context of courses I've taken. 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. As with my teaching material you are welcome to use any of the material on this website, provided you reach out to let me know you're using it.
These are organized in rough chronological order.
Columbia Business School - First Year of the Doctoral Program in Decisions, Risk and Operations
- Stochastic processes
- 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](/files/notes/cu/cheat_deterministic.pdf] 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.
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.
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).
University of Cambridge - Part IB Natural Sciences
- Physics A: notes on quantum mechanics and partial notes on waves.
- Physics B: notes on thermodynamics and partial notes on dynamics (which cover some trickier points).
- Mathematics: notes on complex analysis, fourier transforms, Green's functions, linear algebra, ordinary differential equations, Poisson's equation, Sturm-Liouville theory, variational methods and vector calculus.
- Additional suggested readings around IB courses.
University of Cambridge - Part IA Natural Sciences
- Physics: notes on mechanics, special relativity, waves, circuits, and thermodynamics (which, I believe, is no longer part of the IA syllabus).
- Spectroscopy and orbitals - notes on orbitals, miscellaneous spectroscopy, IR spectroscopy (with useful summary) and NMR spectroscopy (with useful summary).
- Organic chemistry - notes on acidity, basicity and kPa and nucleophilic substitution and mind maps on carbonyl chemistry, miscellaneous organic chemistry and organometallic chemistry.
- Inorganic chemistry - mind map on periodicity.
- Physics chemistry - notes on thermodynamics, and a mind map on kinetics.
- Mathematics: notes on calculus, continuity and differentiability, coordinate systems, differential equations, functions of several variables, limits, power series, probability, series, vectors and miscellaneous stuff.
- Biology of Cells
- Macromolecules - notes on macromolecules and mind map on membranes.
- Chemistry of life - notes on general principles of metabolism (with mind map), amino acid biosynthesis (with mind map), fatty acid synthesis and degradation, glycolysis and gluconeogenesis (with mind map), oxidative phosphorylation, the Calvin cycle and the oxidative pentose phosphate pathway , the Citric Acid cycle, metabolic control and integration of metabolism (with mind map) and the light reactions of photosynthesis.
- Hunting the Gene - mind map.
- Genes in action - mind maps on DNA replication, RNA replication, transcription in prokaryotes and eukaryotes, introduction to translation, translation in prokaryotes and eukaryotes, control of gene expression in bacteria and eukaryotes, DNA damage and repair and post-transcriptional RNA modification.
- The Genetic Revolution - mind maps on chromosomes, gene and genome sequencing, the genetics of human disease, the molecular basis of human disease, genomes and gene cloning and functional genomics.
- Cell proliferation - mind maps on on how cells cycle, patterns of proliferation, viral reproduction and other bits.
- Cell signaling - mind map
- Development - mind map
- Notes on practicals - michaelmas, microscopy, MolStruc, bacterial practicals, and genetic practicals