Dynamic Optimization and Economic Applications (Recursive Methods), Spring 2003

Abstract:: Deterministic optimization: maximum principle, dynamic programming, calculus of variations, optimal control, dynamic games. Stochastic optimization: stochastic optimal control and dynamic programming, Markov processes, Ito calculus, Markov games. Applications. Dynamical systems: local and global analysis and chaos. The unifying theme of this course is best captured by the title of our main reference book: "Recursive Methods in Economic Dynamics". We start by covering deterministic and stochastic dynamic optimization using dynamic programming analysis. We then study the properties of the resulting dynamic systems. Finally, we will go over a recursive method for repeated games that has proven useful in contract theory and macroeconomics. We shall stress applications and examples of all these techniques throughout the course.

Game Theory with Engineering Applications, Spring 2010

Abstract:: This course is an introduction to the fundamentals of game theory and mechanism design. Motivations are drawn from engineered/networked systems (including distributed control of wireline and wireless communication networks, incentive-compatible/dynamic resource allocation, multi-agent systems, pricing and investment decisions in the Internet), and social models (including social and economic networks). The course emphasizes theoretical foundations, mathematical tools, modeling, and equilibrium notions in different environments.

Abstract:: " Networks are ubiquitous in our modern society. The World Wide Web that links us to and enables information flows with the rest of the world is the most visible example. It is, however, only one of many networks within which we are situated. Our social life is organized around networks of friends and colleagues. These networks determine our information, influence our opinions, and shape our political attitudes. They also link us, often through important but weak ties, to everybody else in the United States and in the world. Economic and financial markets also look much more like networks than anonymous marketplaces. Firms interact with the same suppliers and customers and use Web-like supply chains. Financial linkages, both among banks and between consumers, companies and banks, also form a network over which funds flow and risks are shared. Systemic risk in financial markets often results from the counterparty risks created within this financial network. Food chains, interacting biological systems and the spread and containment of epidemics are some of the other natural and social phenomena that exhibit a marked networked structure. This course will introduce the tools for the study of networks. It will show how certain common principles permeate the functioning of these diverse networks and how the same issues related to robustness, fragility, and interlinkages arise in several different types of networks."

Abstract:: This course is a student-presented seminar in combinatorics, graph theory, and discrete mathematics in general. Instruction and practice in written and oral communication is emphasized, with participants reading and presenting papers from recent mathematics literature and writing a final paper in a related topic.