Mathematics Events at the 2006 AAAS Meeting

Approximately 6500 people attended the annual meeting of the American Association for the Advancement of Science (AAAS) February 16-20. The meeting was held at the America's Center Convention Center and Renaissance Grand Hotel in St. Louis, MO. Below are descriptions of some of the mathematics-related events at the meeting.

AMS Exhibit

Beyond Pi: Grand Challenges in the Mathematical Sciences

Six symposia on Saturday February 18 and Sunday February 19 had mathematical themes.

Million Dollar Mathematics: Challenge Problems in the 21st Century

• "From the Hilbert Problems to the Millennium Problems," James Carlson (Clay Mathematics Institute)
• "Primes and Zeros: The Million Dollar Mystery," Brian Conrey (American Institute of Mathematics)
• "A Bird's Eye View of the P = NP Question," Steven Rudich (Carnegie-Mellon University)

In 2000, the Clay Mathematics Institute announced seven one-million dollar (U.S.) prizes for the solution of seven outstanding problems in mathematics. None of the problems have been solved but the prizes have generated considerable interest. Carlson gave a historical background to some of the problems, going back to Greek mathematics, and noted that the Riemann Hypothesis is the one outstanding problem identified by David Hilbert in 1900 and by the Clay Institute one hundred years later. The Riemann Hypothesis, which deals with the distribution of prime numbers, was the subject of Conrey's talk. He said that the last eight years have been remarkable for theorems about prime numbers, even though the Riemann Hypothesis is still unproved. Conrey talked about much of the history of the Riemann Hypothesis and connections it has to random matrix theory and physics. Rudich spoke on the P = NP question, which is part of complexity theory, and gave some of the history of complexity theory and of algorithms.

NUMB3RS and the Challenge of Changing Public Perception of Mathematics

• Cheryl Heuton and Nicolas Falacci (executive producers and creators of NUMB3RS)
• Gary Lorden (California Institute of Technology)
• David Krumholtz (co-star of NUMB3RS)

Tony Chan (UCLA) moderated this panel and gave an introduction, discussing the popularity of the show and his role in consulting on the pilot episode. NUMB3RS is the most watched show on Friday nights, with 14 million viewers, and was ranked #15 among all television shows at the time of the panel. Heuton told how the show came about: She and Falacci wanted to do a show that showed the different kinds of thinking of a mathematician and an FBI agent. When Heuton and Falacci went to CBS to sell the show to the network, they were not optimistic, but five minutes into their presentation the CBS representatives said that they were sold on the idea. The mathematician, Charlie (played by Krumholtz), is loosely based on physicist Richard Feynman. When the show was being tested for audience reaction, focus groups said that they would watch the show because "We love the math." Lorden, mathematical consultant to the show, is impressed that the people connected to the show care enough to make the math character human, and not a caricature. He said the math in the show must be correct and relevant to the story line. His hope is that by using real mathematics in the show, a 14-year old girl will see that mathematics and be inspired to learn more about it.

Everyone on the panel delighted the audience but Krumholtz was their favorite. He said that it was quite a responsibility to play the part and although he did not do well in algebra in school, he has now become a deductive reasoner. He has drawn inspiration for the character from Tony, Gary, and the atmosphere at Cal Tech (the show takes place at the fictional Cal Sci, which is based on Cal Tech). Krumholtz meets a wide variety of people who are intrigued by the mathematics on the show. He says his role has given him a rare opportunity as an actor to do something that inspires people in an educational manner.

Paradise Lost: The Changing Nature of Mathematical Proof

• "Complex Proofs and the Classification of the Finite Simple Groups," Michael Aschbacher (California Institute of Technology)
• "Perelman's Work on the Poincaré Conjecture," Steven G. Krantz (Washington University in St. Louis)
• "Formal Proof and the Kepler Conjecture for Sphere Packings," Thomas C. Hales (University of Pittsburgh)

Session organizer Keith Devlin (Stanford University) gave an introduction to the session. He told the audience that proving theorems is just one of the many things that mathematicians do but proofs are the ultimate arbiter in mathematics. Devlin gave Euclid's proof of the infinitude of the prime numbers as an example of a rigorous proof, yet it is not a proof in formal logic so that when presenting the proof to others, a person would have to leave out details depending on how much mathematics the listener knew. He said that computer proofs and very long proofs have not changed the definition of proof but have changed the "filling in of missing details." Devlin noted that whenever he presents Euclid's proof, someone in the audience will insist that it is incorrect.

Aschbacher began his talk with a fundamental principle: Mathematics seeks to 1. reduce complexity to a manageable level and 2. find and exploit structure, where no structure is apparent. The classification of the finite simple groups (which are the building blocks of algebraic structures known as groups) is an example of the fundamental principle. The classification involves hundreds of papers and about 10,000 pages. There is no road map to the proof so it's truth is not easy to determine. Gaps in the proof have been found and filled. Krantz spoke on the Poincaré Conjecture, one of the Millennium Problems, particularly on the purported proof of the conjecture by Grisha Perelman. In 2002 and 2003, Perelman posted three papers on the Mathematics ArXiv that he claimed proved the Geometrization Problem which has the Poincaré Conjecture as a consequence. Krantz said that the papers are full of great ideas, but they have not been refereed. Perelman has no apparent intention of submitting the papers to a journal, so they may never be refereed in the way most significant papers are. Perelman posted new drafts in September 2005 but they have not resolved the mystery of whether the conjecture has been proven.

In 1998, Hales and Sam Ferguson submitted a proof of Kepler's Conjecture, which states that packing spheres in the pyramid arrangement often used to stack oranges is the configuration that leaves the least wasted space. Their claimed proof is over 300 pages long and uses 40,000 lines of computer code. More than a dozen referees have studied the proof but the best they can say is that they are "99% certain" of its truth. Hales's Flyspeck Project is now translating the proof into code that can be formally checked by computer. Hales said that he feels this process is more reliable than the standard refereeing process. He conceded that the proof and that of the classification of the finite simple groups are long but pointed out that the U.S. Tax Code fills up more than 60,000 pages.

Astrodynamics, Space Missions, and Chaos

• "New Ways of Traveling in Space," Ed Belbruno (Princeton)
• "A New Theory of Trajectory Design and NASA's Vision," David Folta (NASA Goddard Space Flight Center)
• "Using Chaos to Control Satellite Trajectories," Elbert Macau (Instituto Nacional de Pesquisas Espaciais, Brazil)

Speakers in this session explained how non-linear dynamics and chaos theory are used to design space missions, which now use much less fuel than might have been used otherwise. Non-linear dynamics and chaos theory are used because the gravities of more than one celestial body affect a space vehicle. Consideration of the effects of other planets or moons has led to the discovery of many new trajectories in the solar system. Besides saving fuel, use of the new trajectories lessens risk in a mission in that in the case of an error, a vehicle may not have to be routed to its original orbit but can be brought to a nearer, different orbit.

How Insects Fly

• "Falling Paper, Dragonfly Flight, and Making a Virtual Insect," Jane Wang (Cornell University)
• "Active and Passive Flapping Flight," Steve Childress (New York University)
• "Neural Control of Aerodynamics," Michael Dickinson (California Institute of Technology)

Although all of the creatures discussed in these talks are small, their flights are very complicated to model. Mathematics is needed to simulate the motions and test hypotheses about flight. Wang began the session with a description of how a dragonfly flies. The dragonfly has two pairs of wings which it flaps in a rowing-like motion. There is a phase shift in wing activity with the amount of the shift depending on what the dragonfly is doing: Hovering has more of a phase shift than forward flight. Wang has also tried to analyze the path of a sheet of paper as it falls, in an attempt to understand flight. Childress modeled an Antarctican mollusk moving through water. He found that forward "flight" is a bifurcation between small Reynolds numbers (parameters associated with fluid flow) and large ones. Dickinson is studying the flights of fruit flies. He presented films of fruit flies hovering and paid special attention to how flies turn and what makes them turn. His research team has built a large model of a fly, called Robofly, which can simulate flight for some time, but eventually becomes unstable and crashes.

Tsunamis: Their Hydrodynamics and Impact on People

• "Dynamics of Large Ocean Waves and the Prognosis for Tsunami Propagation Forecasting," Walter Craig (McMaster University)
• "Modeling Deepwater Propagation of Tsunamis with Boussinesq-Type Equations," Jerry Bona (University of Illinois, Chicago)
• "Source Characteristics for the Sumatran Earthquake and Prospects for Tsunami Warnings and Forecasting," Arthur Lerner-Lam (Columbia University)

Craig gave an introduction to the topic, explaining that tsunamis occur very often although serious ones occur roughly every 20 years. Predicting tsunamis is difficult and currently unfeasible because predicting earthquakes in not presently within reach. There is some progress in modeling the propagation of tsunamis over open water, but modeling their behavior on or near shore is again out of our reach. Tsunamis differ from typical waves in that there can be a huge distance between their peaks. Bona discussed dispersions of tsunamis and important constants associated with the waves. He said that an asymmetric Gaussian distribution seems to model tsunami propagation pretty well. Lerner-Lam talked of problems estimating the magnitude of the Sumatran earthquake and of large earthquakes in general. The problem arises because the large seismic waves generated by a big earthquake interfere with one another. He also said that one reason tsunami forecasting is hard is the record for large earthquakes is sparse.

Other Sessions

Arches: Gateways from Science to Culture

This topic was a natural, given the host city but speakers pointed out that the St. Louis Gateway Arch is a catenary and not truly an arch. Although this symposium was not in the mathematics section, there was a good deal of mathematics in the talks. The arch is an extremely stable structure. John Ochsendorf (Massachusetts Institute of Technology) showed simulations of what it takes for an arch to collapse. If the two ends of an arch are pulled apart an arch will remain standing much longer than if the two ends are squeezed. Paul Calter (Vermont Technical College) gave the equation of the St. Louis Arch, y = 68.8(cosh(.01x) -1) , and explained the connection between catenaries (hanging chains) and exponential functions.

Strengthening the Scientific Basis of Biometric Identification and Authentication

Anil K. Jain (Michigan State University) introduced the subject, saying that biometrics is based on who you are (using features such as fingerprints or iris patterns) rather than what you know (a password or PIN), and noting that the 9/11 hijackers had 63 valid driver's licenses at the time of the attack. Biometrics has two stages: enrollment, in which features are extracted, processed, and stored in a template database and authentication, in which features are sampled and compared to those in the database via a matching algorithm. Whether it's face recognition, fingerprints, or some other method, a biometric has to be invariant regardless of lighting conditions, for example, and must be secure. Geof H. Givens (Colorado State University) talked about the many challenges of face recognition. Experimental design and data analysis are just two areas of statistics that offer help to the problem of face recognition.

Measuring Politics

Simon Jackman (Stanford University) explained how he uses Bayesian statistics to quantify politician's positions. He disputed the National Journal's characterization of John Kerry as the most liberal U.S. Senator. The Journal used only 62 of the senator's votes, but Jackman said that if all of Kerry's votes were factored in, his relative ranking would be about 17th. Keith Poole (University of California San Diego) presented graphical data on every U.S. Congress through 2004. He likened plotting political positions on a grid to constructing a map given only the driving distances between cities. Poole said that based on his analysis, the American political system is in a state of flux and is polarizing very fast.

The 2007 AAAS annual meeting is in San Francisco. The AAAS website has more information.

You can also read about mathematics at the 2005, 2004, 2003 and 2002 AAAS meetings.