## Advanced Mathematics

## Course Description

This background course on mathematics aims to provide fundamental mathematical knowledge essential for advanced economic analysis. Although open to all master students, it is specifically tailored to those wishing to directly pursue the advanced Y-track of courses. Therefore in content and form, this intensive course is intended to deliver methods beyond refreshing advanced calculus and linear algebra.

The course solely deals with deterministic mathematics. For some theorems formally rigorous proofs are presented in order to make participants more comfortable with - and ideally to provide some intuition for – constructing and understanding of mathematical proofs. Throughout the course proper use of notation will be stressed. Topics presented in class constitute the minimal required program given the above aim, and the maximal feasible program given time. Self study should cover topics skipped in class, as well as the

areas of personal weakness.

The lecture takes place as an intensive crash course in the week before the semester.

The registration in the respective Moodle course is obligatory.

## Course Learning Objectives

Advanced calculus, linear algebra, construction and understanding of mathematical proofs

## Requirements

Students are requested to be enrolled into the Moodle course.

## Course Structure

- Sets, Relations, Preferences
- characterization of and operations on sets
- truth function
- mappings, functions and relations
- preference relations

- Vector Spaces and Linear Algebra
- general vector spaces, linear independence, basis of a vector
- linear mappings between vector spaces, matrix algebra
- basis transformations, eigenvalue - eigenvector decomposition

- Topology and Convex Optimization
- general definition topology, open and closed sets, topological space
- metric, metric space, sequences and convergence in general metric spaces
- norm, normed space and completeness of spaces: Banach and Hilbert spaces
- continuity in general spaces
- compactness and convexity, concavity of sets and functions and relations
- separating hyperplane theorem
- correspondences and fixed point theorems
- existence result of convex optimization problem: Kuhn-Tucker Theorem

- Differential calculus
- differentiability in one and higher dimensions
- Taylor approximation
- optimization problems

## Literature and Sources

Schofield, N. (2004). Mathematical Methods In Economics And Social Choice: Study Edition (Vol. 17). Springer.

De la Fuente, A. (2000). Mathematical methods and models for economists. Cambridge University Press.