EARLE RAYMOND HEDRICK LECTURE SERIES
THE MATHEMATICS OF DYNAMIC RANDOM NETWORKS
Jennifer Tour Chayes, Microsoft
During the past decade, dynamic random networks have become increasingly important in communication and information technology. Vast, self-engineered networks, like the Internet, the World Wide Web, and Instant Messaging Networks, have facilitated the flow of information, and served as media for social and economic interaction. I will present simple mathematical models that allow us to explain many observed properties of these networks, e.g., the scale-free nature of their degree distribution, the ease of information transmission (including transmission of viruses), and the first-to-market advantage for early nodes on these networks. I will also present a new general theory of limits of sequences of networks, and discuss what this theory may tell us about dynamically growing networks. See what "Scientific American" is saying: http://www.sciam.com/subscribe.cfm?lsource=friendmail
LECTURE 1: MODELS OF THE INTERNET AND THE WORLD WIDE WEB
Friday, August 3, 10:30 am - 11:20 am
Although the Internet and the World Wide Web have many distinct features, both have a self-organized structure, rather than the engineered architecture of previous networks, such as the phone or transportation systems. As a consequence of this self-organization, the Internet and the World Wide Web have a host of properties which differ from those encountered in engineered structures: a broad "power-law" distribution of connections (so-called "scale-invariance"), short paths between two given points (so-called "small world phenomena" like "six degrees of separation"), strong clustering (leading to so-called "communities and subcultures"), robustness to random errors, but vulnerability to malicious attack, etc. During this lecture, I will first review some of the distinguishing observed features of these networks, and then review the recent models which have been devised to explain these features. The basic models have their origins in graph theory and statistics.
LECTURE 2: MATHEMATICAL BEHAVIOR OF RANDOM SCALE-INVARIANT NETWORKS
Saturday, August 4, 9:30 am - 10:20 am
This lecture will be devoted to a mathematical analysis of some of the standard models of random scale-invariant networks, including models of the Internet, the World Wide Web, and social networks. I will show how these models can be rewritten in terms of a Polya urn representation, which will allow us to prove that the models exhibit some of the observed properties of real-world networks, including scale-invariance and vulnerability to attacks by viruses. Using these models, I will also examine various strategies for containment of viruses and epidemics on technological and social networks.
LECTURE 3: CONVERGENT SEQUENCES OF NETWORKS
Sunday, August 5, 9:30 am - 10:20 am
In the final lecture of this series, I will abstract some of the lessons of the previous lectures. Inspired by dynamically growing networks, I will ask how we can characterize general sequences of graphs in which the number of nodes grows without bound. In particular, I will define various natural notions of convergence for a sequence of graphs, and show that, in the case of dense graphs, many of these notions are equivalent. I will also give a construction for a function representing the limit of a sequence of graphs. I'll review examples of some simple growing network models, and illustrate the corresponding limit functions.
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JOINT MAA-SMB INVITED ADDRESS
ON THE DYNAMICS AND EVOLUTION OF EMERGENT AND RE-EMERGENT DISEASES: FROM TUBERCULOSIS TO SARS TO THE FLU
Carlos Castillo-Chavez, Arizona State University
Friday, August 3, 8:30 am - 9:20 am
The role of mass transportation, immigration, tourism, demographic growth, and bioterrorism are but some of the engines behind disease dynamics and disease evolution. Examples using recent epidemic outbreaks will be used to highlight the role of mathematics in the evaluation of the impact of these epidemic drivers. Mathematics will also be used to highlight the relevance of "boderless" health policy perspectives.
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MAA INVITED ADDRESS
MANAGING NATURAL RESOURCES: MATHEMATICS MEETS POLITICS, GREED, AND THE ARMY CORPS OF ENGINEERS
Louis J. Gross, University of Tennessee
Friday, August 3, 9:30 am - 10:20 am
The availability of satellite-based remote sensing, computers capable of handling large databases, rapid communication networks, and small radio sensors able to transmit details on individual animals has fostered the development of computational ecology. By combining mathematical and computer models of natural systems with geographically-explicit details of the biotic and abiotic components of the environment, we can compare alternative virtual futures to better plan sustainable ecosystems. Opportunities exist for mathematicians to develop and apply models for harvest regulation, control of invasive species, fire management, and disease and pest control. This optimistic view of the potential for computational methodologies to aid in managing natural systems is tempered by the reality that factors other than scientific best practices are involved. I will discuss a range of applications from relatively simple models for invasive plant control to models applied to long-term planning of an immense restoration effort in the Everglades of South Florida.
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MAA STUDENT LECTURE
SPLITTING THE RENT: FAIRNESS PROBLEMS, FIXED POINTS, AND FRAGMENTED POLYTOPES
Francis Edward Su, Harvey Mudd College
Friday, August 3, 1:00 pm - 1:50 pm
How do you divide the rent among roommates fairly? My friend's dilemma was a question that mathematics could answer, both elegantly and constructively. We show how it and other "fair division" questions --- the most famous of which is the problem of Steinhaus: how do you cut a cake fairly? --- motivate a host of "combinatorial fixed point theorems" and problems about polytopes. They provide excellent examples of how mathematics can address an old class of problems in new ways, and conversely, how problems in the social sciences can motivate new mathematics--- where topology, geometry, and combinatorics meet social applications, and where research by undergraduates has played a big role.
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MAA INVITED ADDRESS
REVENGE OF THE TWIN PRIME CONJECTURE
Daniel Goldston, San Jose State University
Saturday, August 4, 8:30 am - 9:20 am
Two years ago Pintz, Yildirim, and I proved that there always exist primes that are very close together - very close meaning much closer than the average distance between neighboring primes. Our method also proves that if the primes are well distributed in arithmetic progressions then one can obtain results not too far from the twin prime conjecture. For example, if the Elliott-Halberstam conjecture is true then there are infinitely many pairs of primes with difference 16 or less. With these successes I was hopeful that before too long our method could be pushed to unconditionally show that there are infinitely often pairs of primes closer than some fixed bounded distance, i. e. bounded gaps, a giant step towards the twin prime conjecture. In this talk I will discuss the method and why perhaps further progress towards bounded gaps and the twin prime conjecture is going to be difficult, although I will be delighted to be proved wrong.
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JAMES R. LEITZEL LECTURE
On Being a Mathematical Citizen: The Natural NExT Step
Lynn A. Steen, St. Olaf College
Saturday, August 4, 10:30 am - 11:20 am
As public concerns about education and competitiveness evolve, so too must the responsibilities of collegiate mathematicians, including especially the participants and alumni of Project NExT. No longer can we afford to focus only on our students, our department, our college, or our research. Mathematics at all levels and of all kinds is at the center of major challenges to the nation's education and economy. These issues challenge us all to be good mathematical citizens in this evolving national landscape.
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NAM DAVID BLACKWELL LECTURE
PUZZLING PROBABILITIES FEATURING THE STREET GAME OF CRAPS
Jack Alexander, Miami Dade College
Saturday, August 4, 1:00 pm - 1:50 pm
The study of probability has, for some time now, been quite intriguing to me. Part of this fascination is fueled by the fact that some probability challenges require strategies that employ various aspects of mathematics to obtain a solution. This presentation uses calculus, algebra, geometry, graphing, as well as probability theory. To illustrate this contention, this presentation will give analytic solutions and computer simulations for three probability problems that I find quite interesting. These problems are: Count Buffon's Needle Problem; The Triangle from a Line Segment Problem; and The Street Game of Craps. The Street Game of Craps was detailed in a problem from a book entitled Introduction to the Theory of Statistics, 3rd Edition, 1963. This text was written by Alexander M. Mood, Franklin A. Graybill, and Duane C. Boes. It was edited by David Blackwell and Herbert Solomon. The book was part of a series of probability and statistics texts published by McGraw-Hill.
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PME J. SUTHERLAND FRAME LECTURE
NEGAFIBONACCI NUMBERS AND THE HYPERBOLIC PLANE
Donald E. Knuth, Professor Emeritus of the Art of Computer Programming, Stanford University
Saturday, August 4, 8:00 pm - 8:50 pm
All integers can be represented uniquely as a sum of zero or more "negative" Fibonacci numbers F-1 = 1, F-2 = -1, F-3 = 2, F-4 = -3, provided that no two consecutive elements of this infinite sequence are used. The NegaFibonacci representation leads to an interesting coordinate system for a classic infinite tiling of the hyperbolic plane by triangles, where each triangle has one 90° angle, one 45° angle, and one 36° angle.
