Advanced Mathematics - Topology and Convex Optimization

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Advanced Mathematics - Topology and Convex Optimization

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.

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Courselet Content

5 components


  • Knowledge of Calculus I & II

General Overview


Course Structure

  1. Sets, Relations, Preferences
    • characterization of and operations on sets
    • truth function
    • mappings, functions and relations
    • preference relations
  2. 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
  3. 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
  4. 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.

Courses that include this CL

Last Updated 16th January 2023
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Meet the instructors !

About the Instructor

Kainat Khowaja is currently a research associate and PhD student at Humboldt University of Berlin with International Research Training Group IRTG1792 “High dimensional non stationary time series analysis”.

Along with a double masters degree in Mathematics and high international exposure through various exchanges. she has a rich background in research, scientific presentations and teaching. Since 2015, she has worked as a teaching assistant for 4+ courses and completed various research projects related to statistics, econometrics, machine learning and finance. She has also co-taught four graduate level courses related to smart data analysis and mathematical statistics at her current institute.

Presently, she is working on the methodology to construct uniform confidence bands around non-parametric estimates from generalized random forests with assistance of multiplier bootstrap. Her work extends on the scope in which mathematical properties can be utilized to bridge the gap between theoretical understanding of random forests and their practical performance. Her research interests include interpretability of and variable selection through random forests and she would like to continue her research in the same area.