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Summer 2015, Summer 2016, Summer 2017
Syllabus and Course Outline. [on canvas]
Goal: The course is meant as an introduction to the mathematical analysis useful for both economics and econometrics. This is basically an introductory course in real analysis targeted at students who are interested in going to grad school in economics or who are interested in a deeper understanding of the mathematical foundations of economics.
The emphasis of this course will be on proving theorems rather than solving problems.
Some topics that we will cover are: logic, set theory, metric spaces, function spaces, optimization theory (sufficient and necessary conditions for optima to exist), and Lebesgue integration.
- Check out this [link] on what mathematical concepts you will need to know for grad school in economics and why.
This class is supposed to prepare you for grad school in economics so you are expected to (1) do a lot of self-study and (2) collaborate with your classmates on solving the problem sets.
- Please read Azuma's guide to understand that these are some skills that you'll need to do well in grad school.
There are no makeup exams. Exam dates are posted a week before classes start, so you have plenty of time to organize your schedule accordingly.
- For summer 2017, exam dates are: Exam 1, June 28 (30% of grade) and Exam 2, August 2 (50% of grade).
Useful books:
"Book of Proof" by Hammack.
"The Way of Analysis" by Strichartz.
"Applied Analysis" by Hunter and Nachtergaele,
"Elementary Real Analysis" by Thomson, Bruckner, and Bruckner,
"Mathematical Economics" by Carter.
LECTURES to EXAM 1
Lecture 1 , 05/17 Logic. Proofs:
Lecture 2,3 , 05/24, 05/31: Set Theory. Binary Relations. Utility Representation Theorems.
- Rubinstein notes on preferences and utility representation
- Gilboa (2010): Questions on Decision Theory
- Herden (1989): On the Existence of Utility Functions
- Beardon et al (2002): The Non-Existence of a Utility Function and the Structure of Non-Representable Preference Relations
Lecture 4 , 06/07: Order of existential quantifiers. Functions. Speed of convergence (big Oh)
The Book by Hammack covers all the topics in Lectures 1 to 4.
Lecture 5 , 06/14: Continuity. Right- and left-continuity. EVT and IVT.
Lecture 6 , 06/21: Review.
Exam 1, 06/28
LECTURES to EXAM 2
Lecture 7 , 07/05: Metric spaces
(i) metrics
(ii) open and closed sets
(iii) Weierstrass Theorem
- closed does not imply open
- Scumbag topologist "opens" a store Comic by Abstruse Goose, #394
- a bit more practice on open-closed sets.
- Simmons (1963) Introduction to Topology and Modern Analysis, Chapter 2
- Carter (2001) Foundations of Mathematical Economics, Section 1.3
- Menselson (1990), Introduction to Topology, Chapter 2
- Chichilnisky on Topology and Economics
- Proof of the Heine-Borel Theorem
Lecture 8 , 07/12: Lipschitz continuity. Convexity. Sufficient versus necessary conditions for optimization.
Lecture 9,10 07/19 and 07/26: Measure Theory. Lebesgue integration
- The Banach-Tarski paradox in video and written forms
- Hunt's notes on measure theory. Skip outer measure in Ch 1. Ch3 and Ch4 will be relevant for the last lecture.
- If there is one chapter that you will want to have with you throughout your grad studies, it will probably be this one.
- List of definitions and concepts related to measure theory.
- More notes on measurable spaces and on measures.
- probability and measure notes 1 and notes 2.
- Probability via measure by Ok (page 3 explains measurable spaces in terms of information sets)
Exam 2, 08/02
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