MRes Optimization 2025-26

This course provides the rigorous mathematical foundations required for advanced work in economic theory. The emphasis is twofold: first, on mastering the core concepts of topology, analysis, linear algebra, convexity, and fixed-point theory; second, on developing the ability to construct clear and rigorous mathematical proofs. The goal is not only to acquire technical tools, but also to train in the style of reasoning and argumentation that underlies modern economic theory.

We proceed systematically. The course begins with the formal language of sets, topology, and metric spaces, followed by compactness, sequences, and continuity. We then develop tools of differentiation and linear algebra in finite dimensions, before turning to convex analysis and the geometry of separating hyperplanes. The course culminates with Brouwer’s and Kakutani’s fixed-point theorems, which are central to existence results in general equilibrium and game theory. Throughout, students will be expected to practice formulating and writing rigorous arguments.


Syllabus


Meeting Times:

Class Dates: 30 September – 20 November 2025

Time: Tuesday and Thursday 11:15-13:15 (Practical session Fridays at 12:30)

Location: Lecture Room C



Course Outline:



Textbook: Rudin, W. (1976). Principles of Mathematical Analysis (3rd ed.). New York: McGraw–Hill.



Lecture Notes, Slides, and Other Resources: Materials will be posted below as the course progresses.


Problem Sets: You are encouraged to work with your classmates in solving them, but you must write up and turn in your own answers. All work is to be handwritten by you and delivered on the link provided by Thursday at 5pm; late submissions will not be accepted.


Exam: An in-person exam will take place on December 5, 2025. This is a closed-book/closed-notes individual test.



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