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Georgetown University Department of Economics Ph. D. Program in Economics MATH CAMP Summer 2013: August 12-23 Time: 9:00am-12:00pm and 2:00pm-5:00pm Classrooms: please see detailed schedule Instructors: Andy WHITTEN [email protected] Jordan MARCUSSE [email protected] Course goals: To help students review mathematics necessary for the first year courses To help students identify any remaining gaps in their mathematics preparation To introduce advanced mathematics topics used in first year courses This is a course to refresh and deepen your existing mathematical knowledge. It is not possible to learn the mathematics necessary for the first year in two weeks. Please take a look at this syllabus prior to math camp. If the topics on this syllabus are unfamiliar, take some time to prepare before arriving (for example, by reading up on unfamiliar topics in the texts below). The time spent in this course will be much more useful if you come in with some experience with the topics, ready to ask questions. Please feel free to email Andy with any questions you have in the time leading up to the course. Topics: WEEK 1 1. Basic Logic and Proofs - review of math symbols - examples of types of proofs 2. Real Analysis / Calculus - metric space, norms, sequences, convergence, topology - functions, continuity, differentiation - convexity, concavity, quasi-convexity, -concavity - intermediate value theorem, inverse function theorem - integration, integration by parts, Leibnitz rule, FTOC - Taylor expansion 3. Optimization - existence/uniqueness of maximum - unconstrained optimization - constrained optimization, Lagrangians, KKT conditions - implicit function theorem 4. Correspondences - definition, examples - hemicontinuity, examples 5. Miscellaneous - envelope theorem - hyperplanes - theorem of the maximum - fixed point theorems WEEK 2: 1. Matrix Algebra - notation, matrix multiplication - vector spaces, projection, eigenvectors, eigenvalues - linear independence, rank, determinant, definiteness 2. Differential Equations - existence/uniqueness of solutions, stability - eigenvalues 3. Neoclassical Growth Model - finite /infinite horizons, dynamic optimization - Lagrangians, Euler equation, TVC - examples 4. Set Theory - definitions, containment, intersections, unions 5. Probability Theory - sigma algebras - measures, definition of probability function Recommended Texts: Note: You are not required to purchase any/all of these books for math camp; however, it is highly recommended that you purchase at least one book on mathematics methods beyond the required texts for the semester. How to Prove It: A Structured Approach – Daniel Velleman This book teaches how to construct rigorous proofs. Previous first year students say that it helped them prepare for the style of proof that will be done in first semester courses. A First Course in Optimization Theory - R.K Sundaram This is a good, shorter reference, especially for math related to microeconomics. It covers most of the important topics with good examples and explanations. The short length requires, though, that it leave out some topics and many proofs. Mathematics for Economists - Simon and Blume Good book to use as a reference or to catch up if you feel behind the curve mathematically. Covers calculus, optimization, and linear algebra well. Has some elementary treatment of real analysis, as well as differential equations. Mathematical Methods and Models for Economists - Angel de la Fuente More advanced book. Recommended by a previous math camp instructor, but we have little experience with it. Probability Theory with Economic Applications – Efe OK https://files.nyu.edu/eo1/public/books.html Approaches probability theory from a real analysis approach. The material is presented in relatively easy terms, and is intended for PhD-level economics students. Until the book gets published, it is available for free in its entirety on Professor Ok’s website. Recursive Methods for Economic Dynamics – Stokey, Lucas, and Prescott An important book for understanding dynamic programming, the math behind modern macroeconomics. (This one will likely be required for your first-year courses anyway.) Appendix of Microeconomic Theory – Mas Colell, Whinston, and Green Useful math reference in the back of the first-year required micro book. Best used as a quick-reference guide, complementing a more complete math book.
