February 23, 2009
The news media uncovered a variety of newsworthy topics at the 2009 Joint Mathematics Meetings, which were held in Washington, D.C., in early January.
Slate writer Chris Wilson investigated the question, "Can a bunch of mathematicians make government more representative?"
Wilson covered a special session on the value and use of political redistricting algorithms. Speakers included Daniel H. Ullman of George Washington University, Richard Pildes of New York University’s Law School, Micah Altman of Harvard University, Michael P. McDonald of George Mason University, Nathaniel Alfred Persily of Columbia University, James M. Snyder of the Massachusetts Institute of Technology, and Christopher P. Chambers and Alan D. Miller of the California Institute of Technology.
In the end, Wilson asked, "Why is an algorithmic solution for congressional redistricting such a pipe dream?"
"In part it's because it is surprisingly hard to define, or at least reduce to a set of rules, what a 'gerrymandered district' is," Wilson concluded. "Writing a formula for drawing districts requires us to define how funny-looking is too funny looking."
Ullman described one school of thought on the question. "The idea is that circles are the best shape for districts," he said. "Unfortunately, they don't tessellate well."
Patrick Berry of Science News headlined two items. In an article titled "Mathematicians Show How Beetles Can Share a Niche," Berry described a new mathematical model demonstrating that two competing species might actually coexist. Placing two species of flour beetle in the same jar of flour needn't always result in one species driving the other to extinction, as ecologists had thought.
"I think it opens some questions about this dogmatic view in ecology," said University of Arizona mathematician Jim Cushing, coauthor of the study. "It reopens the issue of what you consider a niche to be." Cushing and his colleagues presented the paper "Evolutionary Reversals in Competitive Interactions: Experimental Occurrences and Model Explanations Using Darwinian Dynamics" at the Joint Mathematics Meetings.
"What makes this work very, very exciting is the assumption was that if you start with equal conditions, you will always get one species outcompeting the other," Joel Brown of the University of Illinois, Chicago commented in Science News. "They're showing how just a tiny bit of evolution might actually explain the discrepancy."
Berry’s second article, titled "Calculating the Geography of Crime," dealt with the idea of combining crime records, geography, and other data into mathematical tools that could better and more quickly pinpoint serial criminals.
In research by Mike O'Leary of Towson University in Maryland, the idea is that, using information about the layout of a city, such as the location of similar crimes during the past few years, new mathematical tools could improve estimates of where a criminal lives based on where he or she commits crimes.
The "profiling problem in criminology is to determine a search area for an offender based on knowledge of the offender's crime locations," O'Leary said. His mathematical approach, described in the paper "Determining an Optimal Search Area for a Serial Criminal," incorporates "geographic and demographic features that influence the selection of a crime site" using Bayesian methods to generate more accurate predictions.
O'Leary's methods have "added some insights into the mathematics that previously we were struggling with," commented crime analyst and researcher Ned Levine of Houston. "He's really cleaning up the mathematics."
Writer Julie Rehmeyer was also at the meetings and reported on the mathematical art exhibition in her Science News column "When Art and Math Collide."
Roberta Kwok, writing for Nature, came up with this item: "Fighting Cholera by the Numbers."
"A new mathematical model could help to determine the best strategy for controlling outbreaks of cholera," Kwok wrote. The model "calculates which combination of vaccination, sanitation and antibiotic treatment levels will most effectively reduce the number of lives lost and the cost of intervention."
The model was developed by Rachael L.Miller of the University of Tennessee and her colleagues and described in the paper "Optimal intervention strategies for a cholera outbreak."
The model, a set of equations describing the likely spread of cholera, takes into account the infectiousness of the bacterium—it is more likely to cause disease when it is first shed—and the distinction between symptomatic and asymptomatic patients. The model predicts the extent and length of treatment, and suggests that vaccination and sanitation should last for three weeks.
The team, Kwok wrote, "introduced equations to determine the best, most cost-effective intervention schedule."
The idea sounds "fascinating," Rita Colwell of the University of Maryland commented. "It allows [epidemiology] to be far more quantitative and far more locale-specific than it has been in the past," she said.
Still, "there is a lot of work that needs to be done to fine-tune the parameters before it's something we're going to want to use as a practical tool," observed Glenn Morris of the University of Florida, who has worked with the Tennessee team. "But it's a start."
Writing in the Feb. 13 Science, Barry Cipra highlighted presentations on fractal billiards, geometric gerrymandering, and overhanging bricks in his reports from the recently concluded 2009 Joint Mathematics Meetings in Washington, D.C.
Cipra's article "Taking a Cue from Infinite Kinkiness" was about research on the motion of idealized billiard balls on tables with a fractal geometry. Fractal billiards, Cipra noted, fits well with an experimental approach to mathematics in which computers explore complex phenomena.
Robert G. Niemeyer of the University of California, Riverside and advisor Michel L. Lapidus investigated how a point-mass cue ball rattles about within a shape whose boundary seems to be all corners. They sought examples of periodic orbits—finite-length trajectories that retrace themselves—on fractal shapes, such as a Koch snowflake.
In a paper titled "Periodic Orbits for Billiards on an Equilateral Triangle," published last year in the American Mathematical Monthly, Andrew Baxter of Rutgers University and Ronald Umble of Millersville University had classified the periodic orbits on an equilateral triangle. Niemeyer and Lapidus described their efforts to find periodic orbits in successive approximations of the Koch snowflake, which are formed from equilateral triangles.
"It's very cool," Victor Hugo Moll of Tulane University said. "It's a new angle on a very classical, well-studied problem."
"There's a lot of interesting behavior when we look at these things analytically," Niemeyer claimed. Eventually, studies of fractal billiards could, for example, provide insights into the ways sound bounces off ceilings and walls in a concert hall or off the seafloor.
Cipra's report "Can Mathematics Map the Way toward Less-Bizarre Elections?" focused on a special session devoted to geometric gerrymandering and the use of political redistricting algorithms. Discussions ranged from the "pie-in-the-sky theoretical to crust-on-the-ground practical" ideas and solutions, Cipra said.
Several presentations addressed the problem of gerrymandering—the shaping of districts to the advantage of parties in power. The problem "is much worse than it used to be," argued Richard Pildes of New York University. Gerrymandering "gives people the sense that they're not really in control of their democracy."
Mathematics, Pildes suggested, "can give you tools for creating processes that are likely to lead people to feel that the process is fair and that the outcome is therefore something to be respected."
"The mathematics of redistricting starts with arithmetic and geometry," Cipra explained. "Ideally, every district in a state would have an equal population and would be, in some sense, both 'contiguous' and 'compact.'"
Alan D. Miller of the California Institute of Technology proposed a method for quantifying the "bizarreness" (or compactness) of geometric shapes, which could be used as one criterion in producing redistricting maps and comparing alternatives. "You can use it to reject districts that are badly shaped," Miller said. He ranked the shapes of current Congressional districts according to shape, ranging from highly compact to convoluted.
Lawyer Sam Hirsch of Washington, D.C., a specialist in election law and voting rights, and Charles R. Hampton of the College of Wooster, in Ohio, offered other remedies. However, Hirsch said, redistricting is "ultimately a political problem."
The complex situation presents "contradictions out the wazoo," commented Kimball Brace, head of Election Data Services in Manassas, Va., and a member of the 2010 Census Advisory Committee.
Cipra's "The Joys of Longer Hangovers" focused on new results in a classical brick-stacking problem. In an invited address, Peter M. Winkler of Dartmouth College described how he and his coworkers found better ways to stack a large number of bricks to create an overhang.
Winkler, Mike Paterson (University of Warwick), Uri Zwick (Tel Aviv University), Yuval Peres (Microsoft Research), and Mikkel Thorup (AT&T Labs-Research) showed that a better way to get an overhang with a large set of bricks is to build what looks like a parabolic brick wall with jagged edges. Unexpectedly, such a stacking boosts the overhang from a multiple of the logarithm of n to a multiple of the cube root of n, where n is the number of bricks.
That "hadn't occurred to people," American Mathematical Monthly (AMM) editor Daniel Velleman of Amherst College said. "It's not obvious it's going to help. And the fact that it helps so much is surprising." Velleman has accepted their findings for publication in the AMM. (For a report on earlier, related findings, see "Overhang.")
Moreover, the mathematicians then translated the problem of stacking bricks into a problem about random walks. "Adding a brick," Cipra reported, "spreads force in essentially the same way that taking a random step spreads the walker's probability of being at a given location equally in each direction."
Still, friction makes a real difference, Winkler said. "And real bricks have a lot of that—not to mention mortar!"
Other JMM highlights can be found at http://www.ams.org/ams/jmm09-highlights.html.
Source: Slate, Jan. 13, 2009; Science News, Jan. 31, 2009, February 13, 2009; Nature, Jan. 9, 2009.