Mathematics

The Mathematics Ph.D. course spans 24 sessions and deals with the mathematical tools that IESE Ph.D. students will have to use later on during their research, either directly or as part of methodologies, algorithms, or other Ph.D. courses such as Statistics and Econometrics. In particular, it consists of two different blocks with an approximately equal number of sessions: matrix algebra and optimization.

The Matrix Algebra block addresses the definition and properties of vectors, matrices and related operators. The emphasis is on establishing a permanent link between operators of multidimensional mathematical objects (rank, determinant, scalar product, eigenvalues, eigenvectors, null space, range, diagonalization, etc.) and their relation to empirical data analysis. While rigor rules the presentation of concepts, intuition is always used to translate mathematical results into research insights. This block consists of three chapters: 1) Vectors, 2) Matrices, and 3) Eigenvalues and Eigenvectors.

The Optimization block addresses the basics of constrained minimization. The objective of this part of the course is that students are able to become optimization users in the near future. For this reason, equal importance is given to the following four areas:

  1. how to model conceptual problems into mathematical optimization problems

  2. the classification (convexity analysis) of optimization problems into well- and ill-behaved problems

  3. the solution of optimization problems, either analytically (using the Karush-Kuhn-Tucker conditions) or numerically (using the Excel solver)

  4. the interpretation of the solution (along with the concept of shadow prices)

The basics of multivariate differentiation are covered prior to introducing convexity, and the modeling/solving/interpreting stages are worked out using several relevant example problems and business cases. This block consists of two chapters:

  1. Mathematical Preliminaries

  2. The Formulating and Optimization Problem

The evaluation of students is performed with a final exam and intermediate mandatory individual homework after each chapter is finished.