Columbia MA Math Camp 2023

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Information and Course Material for the 2023 Math Camp of the MA in Economics at Columbia University

View the Project on GitHub andreaciccarone/MA-Columbia-Math-Camp-2023

Columbia MA Math Camp 2023

Welcome! This page hosts the course material of the August 2023 Math Camp for Columbia’s Economics Master program.

Table of Contents

Course Information

Course Description

The course will cover the mathematical tools and concepts required for the first year sequence of the Master 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: real analysis, linear algebra, calculus and optimization. While studying these topics, we will refine proof-writing skills and develop familiarity with mathematical rigor and formality. Emphasis will also be put on problem-solving and application of the tools. While the course is largely self-contained, students are expected to have taken courses in elementary analysis and unidimensional calculus, as well as have some familiarity with concepts in linear algebra.

The class will be taught in a hybrid format from Monday August 14th to Thursday August 31st. Lectures will be held in person (Hamilton 517) every weekday from 9.30am to 12pm EST ; they will simultaneously be available on Zoom as well as recorded for asynchronous 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. These 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, in order to be graded. 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 Material

Course Outline and Lecture Notes

Here is a tentative course outline :

  1. Preliminaries : Mathematical Logic, Sets, Functions, Numbers
    1. Introduction to Mathematical Logic
    2. Sets
    3. Relations
    4. Functions
    5. Numbers
    6. Countability and Cardinality
  2. Real Analysis
    1. Metric Spaces
    2. Basic Topology
    3. Sequences and Convergence
    4. Compactness
    5. Cauchy Sequences and Completeness
    6. Continuity of Functions
  3. Linear Algebra
    1. Vectors and Vector Spaces
    2. Matrices
    3. Systems of Linear Equations
    4. Eigenvalues, Eigenvectors, and Diagonalization
    5. Quadratic Forms

  4. Multivariate Calculus with slides

    1. Derivatives
    2. Mean Value Theorem
    3. Higher order derivatives and Taylor Expansions
    4. Log-Linearization
    5. Implicit and Inverse Function Theorems
    6. (Riemanian) Integration
  5. Convexity
    1. Convex Sets, Separation Theorem, Fixed Point Theorems
    2. Convex and Concave Functions
    3. Quasi-convex and Quasi-concave functions
  6. Optimization with slides
    1. General Setup
    2. Result on the set of Maximizers
    3. Optimization on R^n
    4. Kuhn-Tucker Theorem
    5. A brief introduction to dynamic programming
  7. 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). I’m sure there are typos in the notes. If you spot one, please point it out.

Problem Sets and Exam

Problem sets will be posted here.

  1. Problem Set 1 (Logic, Sets, Analysis) with solutions
    • Date Posted : Monday August 14th
    • Date Due : Monday August 21st
  2. Problem Set 2 (Real Analysis, Linear Algebra) with solutions
    • Date Posted : Monday August 21st
    • Date Due : Tuesday August 28th
  3. [Problem Set 3 (Multivariate Calculus, Convexity, Optimization)] (https://github.com/andreaciccarone/MA-Columbia-Math-Camp-2023/blob/gh-pages/Problem%20Sets/PS3_2023.pdf) with solutions
    • Date Posted : Monday August 28th
    • Date Due : Tuesday September 5th

References and Textbooks

Two very useful short introductions to mathematical proofs :

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.

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 :

Past Exams and Problem Sets

You can find Past Exams and Solutions Here and Past Problem Sets and Solutions here.