Please check here for (non-binding) abstracts for upcoming PDO sessions. Following each session, all available course handouts will be posted here, as well.
August 19, 2009: The Works: Mathematics and the City, presented by Kate Ascher
If you thought New York was a fascinating and complex city to live in, imagine how complicated it can be to try and keep it running. This introductory session will explore some of the systems – historical and current—that keep the city functioning smoothly. We will investigate mail delivery, waterways, power, subways, and many other systems. We will then be introduced to the wide variety of mathematical principles and problems that are behind these systems.
August 20, 2009: Using Queueing Theory to Improve Urban Service Systems, presented by Linda Green
Queueing theory was developed by A.K. Erlang in 1904 to help determine the capacity requirements of the Danish telephone system (see Brockmeyer et al. 1948). It has since been applied to a large range of service industries including banks, airlines, and telephone call centers to help manage delays due to short-term mismatches between supply of capacity and demands for service. In this workshop, Professor Green will describe the use of queueing models in identifying efficient uses of resources in emergency systems such as police patrol, fire, ambulances and hospital emergency rooms. Professor Green will give specific examples of queueing models that have been developed and implemented to identify sources of congestion and reduce delays for emergency care.
August 21, 2009: A Tour of Urban Operations Research: For High Schools, presented by Richard Larson
This day presents a sampling from a course taught at MIT for over 30 years, Urban Operations Research. The text for this topic, published by Prentice Hall, is available free of charge on the web. Much of the material was developed while doing major Operations Research (OR) studies with the City of New York, first with the New York City RAND Institute and then through Professor Larson's private OR consulting firm, now called Structured Decisions Corporation. These NYC efforts continue to this very day.
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September 26, 2009: Mathematics of Epidemics, presented by Frank Hoppensteadt
To paraphrase Sherlock Holmes - humans are random, but in aggregate a mathematical certainty. This observation (and the Law of Large Numbers) underlie many of the successes of mathematics in population biology. One branch of population biology is the study of epidemics and how they propagate through populations of humans (or epizootics in the case of populations of other animals). Modeling disease propagation begins with describing an effective transaction between two individuals, then in small groups, and finally in large populations. There are many branches of epidemiology, but we focus here on infectious diseases, such as viral infections (colds, influenza, chickenpox, measles, mumps, etc.) The models account for randomness, so we begin with a simple two-person population, then move to binomial random processes (coin tossing). Then (here's where the LLN comes in) to differential equations where calculus may be used. This analysis reflects moving from small populations to large ones. The principal mathematical result is the Kermack-McKendrick Threshold theorem, and the principal mathematical method used here is the Monte Carlo method for simulating random processes. Both of these are used by the CDC for projecting disease outbreaks, developing optimal vaccination strategies, estimating potential severity of outbreaks, etc. These models also help bridge the cultural gap between mathematicians, who derive and analyze canonical models (a la Ockham's razor), and others who use ordinary language models.
October 24, 2009: The Census and Appointment, presented by Paul Edelman
Since the founding of the United States, the decennial ritual of taking a census and then apportioning representatives among the states to the House has generated controversy. Four distinct methods have been tried and more have been suggested. It has been the subject of political feuds, mathematical feuds, legal challenge and perorations to God. It is also an excellent example of how mathematics can illuminate the law and how law can motivate mathematics.
In this presentation I will give a crash course in the theory of apportionment. While in one way apportionment is an elementary integer programming problem, the traditional focus has been on the algorithmic and axiomatic properties of the different methods. The relationship between the optimization and the algorithms is unusual and very beautiful. After some discussion of the history of the apportionment of the House, I will show how developments in the law point to new methods of apportionment that have never been employed and the surprising implications they have for the make-up of the House.
Paul H. Edelman holds a joint appointment in the Department of Mathematics and the School of Law. A distinguished mathematician whose scholarship in mathematics has focused on combinatorics, Professor Edelman's work pertaining to the law includes articles on judicial decision making and public choice. Before joining the Vanderbilt faculty, Professor Edelman taught at the University of Minnesota, Carnegie-Mellon University and the University of Pennsylvania.
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November 14, 2009: Mathematics and Bridges, presented by The Salvadori Center
The Salvadori Center uses the urbanscape of buildings, tunnels, and bridges to introduce teachers and students to the wonder, beauty, and logic of architecture and engineering. In this workshop, Salvadori educators will share their methodology for incorporating the built environment into the math classroom. The day will begin with an exploration of built environment concepts, followed by a design challenge that will include the construction of bridges, and will close with a discussion on how to plan and implement lessons such as these in your classroom. Salvadori pedagogy is inquiry-based, interdisciplinary and co-operative, which allows for the exploration of numerous math concepts as well as STEM process skills.
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December 5, 2009: Puzzles for City Slickers by Peter Winkler
City dwellers need to stay on their toes; hundreds of decisions must be made each day, some with very little time to reflect. Mathematical puzzles can be a valuable tool in keeping your students' intuition sharp, and helping them cope effectively with the many small problems of probability and optimization that arise in urban life. Of course, puzzles can also get students excited, and even help them develop a bit more appreciation for the usefulness of mathematics. Some favorite puzzles will be presented, along with ideas for slipping them into your curriculum.
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January 30, 2009: Defending against H1N1 Virus, Smallpox, & Other Diseases presented by Fred Roberts
Our society faces threats from newly emerging diseases such as the H1N1 virus and from diseases such as smallpox or anthrax that might be introduced by bioterrorists. How can mathematics help us identify the best strategies to prevent the spread of disease and respond to outbreaks? Mathematical modeling of infectious disease goes back to Bernoulli's work on smallpox in 1760 and is widely used today by government agencies such as the Centers for Disease Control and Prevention (CDC) and the Department of Homeland Security. We will explore how simple models based on vertex-edge graphs can be used to explore strategies like vaccination and quarantine.
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Course Project: Journal Article or Conference Presentation
The Course Project is the component of the workshop series that bridges teachers’ mathematical experiences in the sessions with the experiences they plan for their students. There are two choices for the Course Project: A Journal Article, or a Conference Presentation
