Columbia MA Math Camp 2021
Welcome ! This page hosts the course material of the August 2021 Math Camp for Columbia’s Economics Masters program.
Table of Contents
The course will cover the mathematical tools and concepts required for the first year sequence of the Master’s in Economics. The main goal of the course is to prepare for first year classes by reviewing or introducing fundamental concepts in various domains of mathematics – analysis, linear algebra, calculus, probability, optimization. A strong emphasis will be put on proof-writing skills and proper mathematical rigor, as well as problem-solving and application of the tools. Students are expected to have taken courses in elementary analysis and unidimensional calculus, as well as have some familiarity with concepts in probability and linear algebra.
The class will be taught in a hybrid format from Monday August 16th to Thursday September 2nd. Lectures will be held in person (room TBA) every weekday from 9.30am to 12pm EST ; they will simultaneously be available on Zoom as well as recorded for asynchroneous attendance. If possible, students are strongly encouraged to attend the lectures in real time.
The course is largely self-contained. Lecture notes will be posted on the website ; teaching itself will mostly take place on the blackboard but additional notes or slides might be provided. Some additional notes and textbook references are provided below.
Problem sets will be assigned weekly. They are important practice and will be graded for feedback, although no grade will be given for the class. Problem sets will have to be submitted online (modalities to be specified) and will have to be typed – LaTeX is very strongly encouraged as it is an extremely valuable skill that students should acquire as soon as possible. There will be a final exam – the date and modality of the exam will be announced later.
Course Outline and Lecture Notes
Here is a tentative course outline (each main section is a link to the corresponding lecture notes) :
- Preliminaries : Mathematical Logic, Sets, Functions, Numbers
- Introduction to Mathematical Logic
- Countability and Cardinality
- Real Analysis
- Metric Spaces
- Basic Topology
- Sequences and Convergence
- Cauchy Sequences and Completeness
- Continuity of Functions
- Linear Algebra
- Vectors and Vector Spaces
- Systems of Linear Equations
- Eigenvalues, Eigenvectors, and Diagonalization
- Quadratic Forms
- Multivariate Calculus
- Mean Value Theorem
- Higher order derivatives and Taylor Expansions
- Implicit and Inverse Function Theorems
- (Riemanian) Integration
- Convex Sets, Separation Theorem, Fixed Point Theorems
- Convex and Concave Functions
- Quasi-convex and Quasi-concave functions
- General Setup
- Result on the set of Maximizers
- Optimization on $R^n$
- Kuhn-Tucker Theorem
- A brief introduction to dynamic programming
- Probability (TBD, if time permits)
- Correspondences (if time permits)
Lectures notes are susceptible to being continuously updated (be sure to check the date of last update, which is always mentioned at the top of the pdf).
Problem Sets and Exam
Problem sets will be posted here. Below is a tentative schedule.
- Problem Set 1 (Logic, Sets, Analysis)
- Problem Set 2 (Real Analysis, Linear Algebra)
- Problem Set 3 (Multivariate Calculus, Convexity, Optimization)
Final Exam and Solutions
References and Textbooks
Two very useful short introductions to mathematical proofs :
Lecture notes from last year’s math camp are available here.
Below is a list of useful references and textbooks sorted by theme. Within each theme, references are listed in (approximately) increasing complexity. References marked with a (!) are more advanced and are included either for future references or very motivated students.
- General references
- Knut Sydsaeter, Peter Hammond, Arne Strom and Andr'es Carvajal. “Essential mathematics for economic analysis.”, 5th Edition, (2016), Pearson.
- Knut Sydsaeter, Peter Hammond, Atle Seierstad and and Arne Strom. “Further mathematics for economic analysis.”, 2nd Edition, (2008), Pearson.
- Walter Rudin. “Principles of Mathematical Analysis” (1976), International Series in Pure \& Applied Mathematics, McGraw-Hill.
- Ok, Efe A. “Real Analysis with Economic Applications” (2007).
- (!) Walter Rudin, Real and Complex Analysis, Third Edition (1987), McGraw-Hill.
- Linear Algebra
- Rangarajan K. Sundaram, “A First Course in Optimization Theory” (1996), Cambridge University Press.
- Probability and Measure Theory
- Dynamic Programming
The problem sets will have to be typed and students are encouraged to use LaTeX. LaTeX is a powerful tool for seamless and systematic typesetting that produces clean and readable documents. It is arguably the best practical options to typeset mathematical notations and it is the standard tool in the academic world in Economics. For those that are not familiar with LaTeX, here are a few references to get started :
- The website Overleaf is a great practical way to get started with LaTeX. You can create a free account and work on LaTeX documents without having to install anything on your computer, it is all browser based. Furthermore, Overleaf has some useful templates and a very good guide to getting started with LaTeX (which is useful even you choose to use another editor) and many tutorials.
- If you prefer to install a local LaTeX distribution and editor on your laptop, there are several good options that come “pre-packaged” with everything you need. Notable among those are MikTeX for Windows or MacTeX for MacOS, which includes the editor TeXshop and a number of useful packages.
- For the more adventurous, you can download separately a LaTeX distribution and then pair it with any editor you like (VSCode, Sublime Text, Atom,…)
- If you are strongly averse to writing explicit commands, LyX is an alternative LaTeX-based software that wraps it in a more visual “Word-like” environment.
- Another good guide to LaTeX : The Not So Short Guide to LaTeX
- A useful guide for all the math command that you might need in the AMS package : Short Math Guide for LaTeX
- There are a lot of good LaTeX tutorials out there, don’t hesitate to look for them and see if you find one you like. Most importantly, after you grasp the general idea of how LaTeX works, you’ll learn the most by just using it and figuring out how to do what you need to do.
Past Exams and Problem Sets
You can find Past Exams and Solutions Here and Past Problem Sets and Solutions here.