Columbia MA Math Camp 2021


Information and Course Material for the 2021 Math Camp of the MA in Economics at Columbia University

View the Course Material on GitHub CesarBarilla/2021-Columbia-MA-Math-Camp

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

Course Information

Course Description

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 Material

Course Outline and Lecture Notes

Here is a tentative course outline (each main section is a link to the corresponding lecture notes) :

  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
    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
    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. Probability (TBD, if time permits)
  8. 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.

  1. Problem Set 1 (Logic, Sets, Analysis)
  2. Problem Set 2 (Real Analysis, Linear Algebra)
  3. 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.

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.