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Highlights of the 2005 Joint Mathematics Meetings
The 2005 Joint Mathematics Meetings of the American Mathematical Society (AMS) and Mathematical Association of America (MAA) were held in Atlanta, Georgia, January 5-8. There were nearly 5,000 participants--mathematicians, exhibitors, employers, and students--who attended invited addresses, special sessions, mini-courses, the prize ceremony, exhibits, contributed paper sessions, and poster sessions. The meetings provided many opportunities for attendees to meet with colleagues old and new, at receptions, meetings and informal gatherings.
Below are some of the many highlights:
Some from the audience thought that axioms must be simple and natural; Cohen
agreed, while Martin and Woodin disagreed. "I want
axioms to be what they are," said Martin, "and I don't know that
they'll be simple." Woodin noted that while axioms may be simple, the
way in which they reflect reality may not be simple at all--and that
what's considered "simple" can vary greatly over time.
Origami, Linkages, and Polyhedra: Folding with Algorithms
- an MAA Invited Address by Erik D. Demaine
Demaine began his talk by noting that although many basic problems have been addressed in these areas, many
still remain open. After providing some general definitions and constraints, Demaine introduced the topic of linkages:
Given that bars (also known as edges or links) cannot cross, and bar lengths are preserved, Demaine focused on what we know and
don’t know about the folding and unfolding of open chains of bars, closed chains of bars, and trees, in two-, three-, and
four-dimensions. He also discussed three algorithms, including "energy flow," for unfolding two-dimensional closed chains,
and presented a short video demonstrating the unfolding of a few different two-dimensional shapes. He completed this part of the
talk with a section on applications to protein folding.
Demaine then turned his attention to paper folding and unfolding. After showing a few slides of recent origami art,
he discussed some work being done in the young field of "origami mathematics." For example, the
silhouette problem
: Is every
two-dimensional shape made of straight sides the silhouette of a flat origami? has been answered (the answer is yes,
given a sufficiently large piece of paper) but the question of what is the largest regular tetrahedron (or octahedron,
dodecahedron, or icosahedron) that can be folded from a unit square remains open. After discussing different methods of
paper folding, Demaine next talked about paper cutting, demonstrating--and entertaining the audience--by folding a piece of paper,
making a complete straight cut in it, and displaying the resulting cut-out shapes (including a star and a butterfly).
He followed this with a discussion of map folding and concluded his talk with the subject of polyhedron unfolding,
demonstrating with a video how a folded Latin cross could be made into various three-dimensional shapes, in addition to a cube.
For further information, go to Erik Demaine’s webpage
on folding.
How "Hardness" Relates to "Randomness"
- "The Power and Weakness of
Randomness (when you are short of time)," an AMS Invited Address by Avi Wigderson
In his invited address entitled "The Power and Weakness of Randomness
(when you are short of time)," Avi Wigderson
of the
Institute for Advanced Study touched on some of the deepest and most
mysterious questions in theoretical computer science today. Efficient
algorithms exist to solve many real-world problems on computers. Here
an efficient algorithm is one whose running time increases as a
polynomial function of the size of the problem. By introducing
randomness into the algorithm, one can improve the running time but
one must pay the price of a small possible error in the answer. This
error can be made as small as one likes. A simple example: Suppose
you want to know if a polynomial expression is identically zero. The
running time for simplifying the expression could be an exponential
function of the size of the expression. So choose a point at random,
plug it into the expression, and if you get zero, the expression is
probably
identically zero. You can improve the accuracy by
plugging in more points. Wigderson presented a number of problems in
which such probabilistic algorithms work well. He then posed a
question: Are there problems for which an efficient probabilistic
algorithm can be found, but for which no deterministic algorithm
exists? No one knows for sure. Likewise, no one knows the answer to
the famous P versus NP problem, which asks, Are there problems for
which an exponential-time algorithm exists but no polynomial-time
algorithm exists? One of the great achievements of theoretical
computer science, Wigderson said, is the surprising fact that the
answers to these two questions cannot both be "yes." If there is at
least one problem that is truly hard, then efficient probabilistic
algorithms, for any
problem, can always be replaced by
efficient deterministic ones. This achievement requires a new
understanding of the age-old notion of "randomness."
In this lecture Bjorn Poonen
of UC
Berkeley discussed progress on generalizations of the problem. For
instance, what happens if the coefficients in the equation, and the
coordinates of a solution, are allowed to be in a ring other than the
ring of integers Z
? For some rings the answer is obvious, for
others a bit of work is required, and for still others nothing is
known. One ring for which the question is unresolved is the rational
numbers Q.
Poonen discussed some approaches to the question,
such as seeing whether the "no" answer for Z
implies a "no"
answer for Q.
These approaches, if successful, would also disprove a conjecture of Barry Mazur
about the topology of rational points on varieties. Poonen has proven that the answer is "no" for a
ring that is in some sense "close to" Q,
but it is unclear if
the method can be generalized to Q
itself. Poincaré Proof Puzzle - an AMS Invited Address by Bruce A. Kleiner
Has the Poincaré Conjecture been solved? What about the
Thurston Geometrization Conjecture? These questions have been
circulating around the mathematical community ever since
Grigory Perelman
posted a series of remarkable papers on the web two years ago. Perelman is a mathematician in the Laboratory of Mathematical Physics
in the St. Petersburg Branch of the Steklov Institute of Mathematics
in Russia. As mathematicians have continued to struggle to understand
his difficult and challenging papers, the jury is still out on whether
the work is 100 percent correct and complete. In his lecture on
Perelman's work, Bruce Kleiner
of
the University of Michigan shied away from giving a simple "yes" or
"no" answer about whether Perelman has succeeded. But Kleiner did
provide a clear flowchart for how Perelman's proof proceeds and
outlined some of the main ideas involved. Essentially, Perelman has
carried out a program proposed in the early 1980s by Richard Hamilton.
Hamilton suggested that the Thurston Geometrization Conjecture could
be solved using a tool called the Ricci flow. The presence of
singularities is a major complication in Hamilton's approach.
Perelman successfully implemented a surgery process that "clips off"
the singular part of the manifold. Kleiner identified three key
phases in Perelman's proof: 1) show that the singularities have a
standard form; 2) carry out the flow-with-surgery method; and 3) study
the long-time behavior of the flow with surgeries. As experts like
Kleiner have worked through Perelman's proof, all the small problems
that have arisen have been fixed. While these experts would probably
be very surprised if the proof turns out to be wrong, no one is yet
clamoring to proclaim it as correct.
Tubular Transport - "New Methods in Celestial Mechanics and Mission Design," by Jerrold Marsden
(Image concept by Martin Lo, Jet propulsion laboratory; Graphics by Cici Koenig, Caltech.) The entire solar system is threaded with these tubes, Marsden explained, and they can be used to transport spacecraft from one planet to another with very little energy. What makes this method so efficient is the use of the unstable orbits of the dynamical system, which Marsden noted are actually very easy for the spacecraft to travel on. (Marsden confessed that he never understood why in the field of dynamical systems so much attention is paid to the stable orbits, while the unstable orbits are so useful and interesting.) These interplanetary transport tubes have been used in previous space missions, such as the Genesis craft that collected solar wind data, and will be used in the Jupiter Icy Moons orbit, to be launched a decade from now.
Marsden's lecture was presented in the AMS Special Session on Current Events, organized by David Eisenbud, AMS president and director of the Mathematical Sciences Research Institute in Berkeley. Talks in the session and speakers are below:
The Green-Tao Theorem on Primes in Arithmetic Progression: A Dynamical Point of View, Bryna Kra A booklet
(4 MB, pdf)
contains all the talks in the session.
The Notices
Celebrates a Milestone
The Notices of the AMS
held a special reception to mark the 10th
anniversary of its 1995 redesign, which encompassed not only a new
look for the publication but also a new editorial vision. At a
reception held on the final day of the Joint Mathematics Meetings,
Notices
editors, staff, and friends celebrated 10 successful
years of one of the world's most popular magazines for the
professional mathematician. See the January 2005
issue
for articles about the Notices
anniversary.
--- Allyn Jackson, Senior Writer and Deputy Editor of The Notices of the AMS The Graph Theory of Blackwork Embroidery, a presentation by Joshua Holden - AMS Special Session on Mathematics and Mathematics Education in Fiber Arts What does the clothing of a sixteenth century English king have to do with mathematics? In the eyes of a mathematician, special stitching on the king’s collar is a graph theory problem waiting to be solved. Blackwork embroidery, also known as Spanish stitching, involves tracing a path in one direction, then following the same path in reverse so that stitches appearing above the fabric on the first pass appear below the fabric on the second. Thus, the design is identical on either side. The legend of this technique traces back to Catherine of Aragon, a wife of Henry VIII. In his talk, Joshua Holden addressed the question of what type of patterns can be stitched using the blackwork technique. The answer? Blackwork stitching is possible on a directed graph that is reversible and has a path passing through all edges; such conditions are met if and only if the graph is connected. (Some of the works on display during the session are pictured here.)
--- Lisa DeKeukelaere, AMS-AAAS Mass Media Fellow
The Mathematical Sciences Employment Center
The Mathematical Sciences Employment Center is held each January at the Joint Mathematics Meetings. The Center provides a service for Ph.D.-level mathematicians seeking employment and for employers, mainly academic, who wish to conduct brief interviews with candidates. Read an Overview of the Employment Center. --- Diane Boumenot, Manager of Membership & Programs
A Walk on the Industry Side: A Mathematician Takes on the Seismic Exploration Business
by Nicholas Coult -
MAA Session on Mathematics Experiences in Business, Industry, and Government
Seismic exploration is a multi-billion dollar industry whose main activity is searching for oil and gas
beneath the earth’s surface, and Nicholas Coult has put his math skills to work
in helping the industry achieve its goals. In order to locate oil,
machines make sound waves that propagate into the crust, then sensors
measure the reflections of the waves, and computer programs construct images of
the ground beneath our feet. These tests generate large amounts of data, making
compression an important problem. Discrete multiwavelet transforms provide a
good way
to perform the compression, however this method works best for smooth
continuously defined data,
which is not always available. To overcome this problem, Nicholas Coult developed an algorithm for eliminating
missing data regions by extrapolating the
data.
Coult noted that problems presented in industry are often a unique challenge
because they are not fully specified, and boundaries and special conditions
consume much of the work time. In the end, he found educating and communicating
with the client to be among the most important keys to success.
The AMS gave away more than $8000 in two games of Who Wants to be a Mathematician at the Joint Mathematics Meetings in Atlanta. The big winner was David Harris of Vestavia Hills High School (AL) who won $4000. Close to David in winnings was Mitch Costley of Rockdale Magnet School (GA), who won $3000 that day.
The games were held on Thursday, January 6. Four students from the Atlanta area played in the first game. The winner of that game, Mitch Costley, then played in a championship game with the winners of four games played during the fall semester in Alabama and Georgia. Contestants in the two games are listed below. Game One
: Championship Game
: (Note: Dragos Ilas qualified in November at a game held in Georgia, but now lives in Tennessee.)
Several of the schools were represented by enthusiastic rooting sections, including about 30 students from Vestavia Hills High School, who made the trip to root for David. Mitch got all the questions correct to win game one. He used his "Help" on the last question and asked his teacher, Dr. Chuck Garner, who supported Mitch’s reasoning and choice. When it was time for Mitch's bonus question, he was on his own, since he had just used his "Help." The bonus question involved properties of an arithmetic operation, and Mitch answered correctly to earn the $2000 prize. The prizes that each contestant in game one received are below.
Mitch Costley, Maple
9.5 from Maplesoft,
and $2000 from the AMS
Then Mitch competed with the four other game winners for a $2000 first prize and a chance at another bonus question worth $2000. This game was very close, with all five contestants having a chance at first place going in to the last question. David Harris got the last question right, which kept him in first place and gave him a chance at the bonus question for that game. The bonus question in the championship game involved probability and logic. David thought for a long time and although he hadn’t used his "Help" yet, he elected not to use it on his bonus question. After David had registered his answer, the audience--composed of mathematicians attending the Joint Mathematics Meetings and knowledgeable students from the participating schools--expressed a clear preference for one of the choices. It turned out that their choice was David’s as well and it was the right choice! Thus, David (but not the audience) earned an additional $2000. Here are the cash totals from the championship game:
David Harris, $4000
(All of the above had won Maple 9.5 from Maplesoft in their earlier games, and four of the five had won $2000 in those games.) It was a pleasure to observe the contestants correctly answer many hard questions and to listen to their explanations. Although she didn’t win, Livia Zarnescu did give quite a cogent explanation for one of the questions in game two.
Since many of the audience members were mathematicians, it is to be expected
that they would choose the right answer when asked, but what was a bit
unexpected--and gratifying--was how other audience members, mostly high school students, knew the
right answer when they were polled. Congratulations to all.
Using Lagrange multipliers, one can show that work is minimized if the ascending and descending velocities are
equal. However, in order to maintain a good time, even Lance Armstrong would be
unable to achieve the power output necessary for the required uphill velocity. A more fruitful approach is to minimize
exertion, the rate at which power output changes. This model can also be applied to wind
resistance, and Gordon’s upcoming work will address similar strategies for
competitive rowing.
The AMS exhibit included books, Mathematical Reviews and MathSciNet, an online connection to the AMS Bookstore, a meeting place for AMS authors, and the AMS Membership booth, at which people could learn more about the Society and pick up giveaways such as calendars, the Mathematics Awareness Month poster, postcards, and various materials on AMS programs. The Society also hosted a booth for the Mathematics Genealogy Project.
The AMS also hosted a reception for Mathematical Reviews editors and reviewers, and another for AMS authors. Shown below at the Mathematical Reviews reception are (left to right) Jane Kister (past Executive Editor of Mathematical Reviews), David Eisenbud (AMS President) and Carol Hutchins (Head Librarian at the Courant Institute of Mathematical Sciences Library at New York University).
 --- Annette Emerson, AMS Public Awareness Officer Using Mathematically Rich Activities to Develop K-12 Curricula, Part I - an MAA Special Presentation This presentation on mathematics education began with a hands-on exploration of the Road Coloring Problem, introduced by Gregory Budzban of Southern Illinois University. This activity is used by the Algebra Project, an organization that according to its website has "for 20 years . . . worked to increase the proportion of African-American and other students who succeed in college preparatory mathematics. Classroom activities emphasize the development of mathematics concepts out of physical experiences and social interactions of learners," Given a worksheet and a series of questions of gradually increasing complexity, participants in small groups used paper, masking tape and colored pens to indicate buildings and the possible paths between them. Budzban noted that in this way students are introduced to functions, combinatorics, ordered pairs and matrices in a way that engages them. Robert Moses, founder and president of the Algebra Project, spoke about the work he’s currently doing in Jackson, Mississippi with a group of high school students. These students spend 90 minutes each day in mathematics classes. The current goal of the four-year program is that these students pass the state exams and improve performance on SAT exams, as well as meet college mathematics requirements. He noted the importance of determining how teachers can make students more willing to learn mathematics. David Henderson, Professor of Mathematics at Cornell University, presented the audience with another "mathematically rich activity," one that he has used with students from second grade through a graduate-level differential geometry course. He asked three questions: "What do we mean by saying something is straight? How can you check that a 'straight' line is straight? How can you be certain to draw a straight line?" After audience members discussed these questions in their small groups, Henderson talked about the different meanings of straightness, depending upon the context. Through these questions, he noted that students get to "deep meanings" in geodesics and differential geometry. Next, William McCallum of the University of Arizona, presented an expression for the audience to comment upon: P(1 + r/12)12n. After a few minutes of input from the audience, he noted that the audience members were able to start with this "blob," put it into some context, then see it as a meaningful structure. McCallum asked, "How do we get our students to do this? They see it as a blob." One audience member suggested starting with a specific example for students to work on: "You earn one penny every day. What happens if that doubles? . . . "
The final speaker was Ed Dubinsky from Kent State University. He noted that the work he’s been doing for the past 20 years--teaching programming to students--has helped his students learn mathematics.
Dubinsky noted that we believe that "the ultimate source of mathematical knowledge is direct human experience. . . . Too many of us assume that the mathematics just
'jumps out' at the students. How do we help kids go from 'mathematically rich experiences' to the rich mathematics?"
On Saturday night, the annual AMS Banquet was held in the Marquis Ballroom at the Atlanta Marriott Marquis Hotel. Those in attendance with more than 25 years of AMS membership were recognized by AMS Executive Director John Ewing, who introduced each decade's list by offering a brief quiz on the AMS Presidents who had served during that decade. The highlight of the evening was an address by outgoing AMS President David Eisenbud, who spoke of the family of mathematicians that he has come to appreciate over the course of his career, and especially during his term as President. President Eisenbud requested that two former AMS Presidents who were in attendance that evening--Peter Lax (1979-80) and Andrew Gleason (1981-82)--come to the podium as he presented the new AMS President, James Arthur, with a personalized gavel as a symbol of the change in leadership.
--- Diane Boumenot, Manager of Membership & Programs
Although the AMS and MAA co-sponsored the joint meetings, several other organizations also held meetings, sponsored prizes, and hosted sessions: the Association for Symbolic Logic, the Association for Women in Mathematics, the National Association of Mathematicians, the National Science Foundation, Pi Mu Epsilon, the Rocky Mountain Mathematics Consortium, the Society for Industrial and Applied Mathematics, and the Young Mathematicians Network. See the complete program. We look forward to seeing you at next year's Joint Mathematics Meetings in San Antonio, Texas, January 12 - 15, 2006. Note that the meetings will be held Thursday through Sunday.
 Photos by Diane Boumenot, Mike Breen, Bill Butterworth, Lisa DeKeukelaere, Annette Emerson, Sandy Frost, Allyn Jackson, Michael Reeves, and Norman Richert. See also: Highlights of the 2004 Joint Mathematics Meetings in Phoenix, Arizona.
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