Baldur Hedinsson, PhD student at Boston University and 2009 AMS-AAAS Media Fellow, and Adriana Salerno, assistant professor at Bates College and 2007 AMS-AAAS Media Fellow, summarize some of the invited addresses and other talks presented at the 2010 Joint Mathematics Meetings.
* Chaos and symmetry in partially hyperbolic systems * Modeling of geothermal flow
* Day 1 slideshow
* Sage: Creating a viable free open source alternative to Magma, Maple and Mathematica for Number Theory * Reasonable effectiveness: Trigonometry, ancient astronomy, and the birth of applied mathematics
* Day 2 slideshow
* Challenges in computational medicine and biology * Mathematical methods in origami design * Mathematical analysis of synchronization of pacemaker cells in mammals
* The theater of the mathematically absurd * Is the resurgence of syphilis a backward bifurcation?
In 1885, Wilkinson begins, King Oscar of Sweden posed a challenge to mathematicians: a contest to find a solution to the three-body problem (the two-body problem, regardless of what mathematicians and their spouses may think, is basically trivial). The winner, Poincaré, was unable to find a solution, and instead discovered chaos. Steve Smale rediscovered these ideas 70 years later, and that is the beginning of hyperbolic dynamical systems. Wilkinson proceeded to explain the original three-body problem and Poincaré’s solution and its repercussions. She used great analogies for hyperbolicity of a system, my favorite being the idea of kneading Danish pastry dough (basically, some parts of the dough are stretched while others are compressed). Most interestingly, not all systems are hyperbolic, “or even close,” Wilkinson said. So the central question becomes: Can we weaken hyperbolicity and retain chaotic features? This is where the idea of partially hyperbolic dynamical systems comes in. Wilkinson admitted she wrote the title of the talk before writing the talk itself, and so concluded the exposition by explaining how the idea of symmetry arises in understanding these systems.
Ostermann, a post-graduate researcher, presented a model of geothermal flow as a part of the minisymposium. Geothermal power is a sustainable energy source since the heat extracted is miniscule compared to the Earth's heat content. Depleting geothermal energy locally, however, is a risk. Increased understanding of geothermal flow is clearly beneficial for managing energy extraction, exploring new reservoirs and reducing risk of local exhaustion. Ostermann displayed a three-dimensional model of geothermal flow. The model uses time-dependent partial differential equations (transient advection-diffusion-equations) that take into account the heat flow both due to temperature differences and due to the motion of magma (high-temperature fluid substance) underneath the Earth's surface. Using analytical methods, Ostermann made the case for existence and uniqueness of a solution to the system of equations in the model and showed how to approximate the solution numerically.
Stein released his first version of Sage about five years ago. He was inspired by the fact that he wanted software that he could change and adapt as much as possible and by his desire to create a free option to Magma, Maple and Mathematica, which are widely used and quite expensive. Sage was initially created for number-theoretic purposes, but as evidenced by the diverse and large audience in this talk, it has definitely crossed over into all sorts of mathematical disciplines. (In fact, at one of the Project NExT sessions I learned that it is a really useful tool for multivariable calculus since it creates 3D graphs that one can see with 3D glasses!) Winner of the Trophees du Libre in November of 2007, Sage is now sponsored by multiple institutions and corporations, like Google, Microsoft, the National Science Foundation, and the Department of Defense. The software is constantly being improved and developed by a community of about 200 programmers and mathematicians, featuring a large library, wide distribution, and interfaces with almost all existing packages. The glue that holds everything together is a programming language called Python. Sage also has a powerful web-based graphical interface, and so one can either download the program or work with Sage directly from the web. Stein concluded the talk with a very illustrative demo in which he showed the wide range of number-theoretic computations one can do in the current version of Sage. In the last month, Sage had 22,000 online users. Find out more at the Sage website.
Funny quote: When asked if Sage could do everything GAP (Groups, Algorithms, and Programming--a system for computational discrete algebra) could do, Stein responded: "Think of Sage as GAP++."
Van Brummelen, who is the historian of mathematics at Quest, gave a talk on the history of trigonometry and how demand for accurate trigonometric calculations brought applied mathematics into being. Van Brummelen traced trigonometry's beginnings to ancient Greece where the study of the planets moving across the sky naturally gave rise to triangles on a sphere and the use of trigonometric functions. Chord was the first such function, described by Hipparchus: given a circle and an arc on the circle, the chord is the line that subtends the arc. Soon more familiar functions such as sine and cosine followed. The Greeks managed to describe the planetary movements using trigonometry, but lacked the mathematical tools to make predictive computations. This changed when the Babylonian number system became known across the Mediterranean. Tables listing values of trigonometric functions came to be valuable to anyone interested in astronomy. Hipparchus was again the first to make such a table by tabulating chord values for a series of angles. Tables for different functions became more comprehensive and accurate as mathematicians hired to carry out calculations (known as calculators) learned better numerical techniques. Armed with the descriptive models of the Greek and an efficient place value number system from Babylon, scientists in Greece, and later in India and the medieval Islamic Empire had unparalleled capability to describe phenomena observed in nature.
The new knowledge had application in many sciences other then astronomy, such as in geography. An example of this is how followers of Islam all over the Islamic Empire found the direction of Mecca (the qibla) and kept exact time used for the five daily prayers. This even led to a new profession, as mosques hired astronomical timekeepers: The mathematics of the heavens had moved down to Earth and into the light of day. The application of the newly developed tools (Van Brummelen referred to as "Instruments of Mass Calculation") reached its height with Ptolemy's planetary model. This geometrical model explains the movement of the planets using combinations of circles. The model was a great fit with the available data and made precise predictions about future motions of celestial objects. Unfortunately the model turned out to be completely wrong. With the audience bereft of speech Van Brummelen used this time in his talk to speculate on the unreasonable effectiveness of mathematics in the natural sciences, an observation publicized by physicist Eugene Wigner in 1960. Van Brummelen finished the talk with a reflection on mathematics being the language of the correct universe and the language of the false universe. He then conveyed the importance of emphasizing the beauty of mathematics on its own--not only the value it provides with models that fit data from the physical world, but also the elegance found in the math itself.
Geman gave a thought-provoking talk exemplifying the complex effects genes have on human health and the obstacles mathematicians face decoding how gene products interact to cause disease. Ending on a positive note, Geman illustrated how a very simple math model can help detect diseases earlier. Belief in genetic determinism and the search for individual disease-genes ended when researchers realized how intricately gene products (proteins and RNA) interact. Unraveling this interaction is crucial to understanding how genetic information affects human health. (Though genes don’t physically interact I will refer to the interaction of gene products as gene-interactions for the remainder of this piece.) The genetic code is the set of rules by which information encoded in DNA sequences is translated into proteins by living cells. Decoding this information means understanding what different parts of the genetic code stand for. To grasp the importance of gene-interactions, imagine you have a car to build. The would be car-genome stores information about how each car-part is made and gene-interactions are instructions how to build the car using all the parts. Trying to make sense of gene-interactions in addition to understanding the genetic code is a lot harder task than the latter one on its own. Gene-interactions are a very important piece of the puzzle. Having a sequenced genome, but no information about gene interactions, is akin to having all the parts to build a car but no instructions on how it comes together as one piece. Not knowing where to put the gas pedal has hindered medical genetics from racing into the future.
Recognizing the importance of gene-interactions, geneticists are eager for better decoding methods and descriptive models. Geman suggested viewing genes and gene-interactions as a large system of interacting particles or a network. Various algorithms in machine learning can recognize complex patterns in data sets of interacting particles and such algorithms are used in face-detection and handwriting recognition software, just to name two. Geman gave examples of applying algorithms as mentioned above to unravel the effect genes and gene-interactions have on human disease. Given genetic data from two groups of people: patients suffering from a disease and healthy individuals, the goal is for the algorithms to detect differences in the genetic code and gene-interactions between the two groups. This is often referred to detecting the difference in network structure. But there are two large problems. First, the network structure is very complicated and the information about the structure is coded in a very long string of DNA. To recover the network structure the algorithms have access to genetic data, typically only around 100 individuals. With so few participants (data points) it is close to impossible for machine learning algorithms to reconstruct the network structure without any extra information. There are not enough participants in the study to find consistent differences between the two groups. Another obstacle is cultural. Geneticists want accurate predictions of whether a person has a disease and models that translate into biological understanding. Algorithms that provide the most accurate predictions in machine learning are not based on biological processes and are often so complex they are referred to as “black boxes.” This creates a communication problem since it is hard to explain how the algorithm works and physicians tend to distrust results from a black box they don’t understand.
Geman ended with an uplifting example, showing how very simple algorithms can help with disease detection. He illustrated how comparing the expression rate of just two genes and seeing which is expressed more strongly can work as a disease detector. With genetic data from a study as described above, a simple algorithm searches for a pair of genes where the expression-rate is reversed between the healthy and the sick group. That is, the algorithm tries to find two genes such that if an individual is healthy gene 1 will be expressed more than gene 2, and vice versa if an individual is sick, gene-2 will be expressed more than gene-1. The method does not explain what causes the change in gene expression, but finds a criterion that can be used for testing the general public to see whether they fall into the sick or the healthy group.
Many people by now are aware of Lang's oeuvre, but how many of us knew of the amount of mathematics behind the art, and similarly of the number of applications origami design has in the sciences? The earliest known well-developed origami dates back to about 1734. Since then, with the help of Yoshizawa who created a common language to understand it, origami has evolved into a complex and rich art form, with pieces made out of one sheet with no cuts that take eight hours to fold! What changed from the origins to the current state of origami is that mathematics became a big part of this new "common language." For example, Lang has developed software called ReferenceFinder, which uses well-known results in origami to solve more folding problems. So mathematics was used to understand already existing ways of designing origami, and sometimes origami was used to solve problems in mathematics (like trisecting an angle). But the real breakthrough for origami was that the mathematics could now be used to find new ways of designing origami. This is what Lang jokingly calls the "Era of intelligent design." Besides ReferenceFinder, Lang mentioned that there is other software, like TreeMaker that uses the "mathematical laws of origami" to find algorithms for origami design. Finally, Lang mentioned the wide array of applications of origami to the sciences, from creating telescopes, solar sails, paper airplanes, lasers, stents, airbags, and heart implants, to studying optics, folding membranes, and protein folding. In general, he mentioned, the philosophy is "small for the journey, large for the destination." See some of Lang's designs and download the software.
I was still adjusting to a new time zone and fighting jet lag, as were many Joint Meeting attendees, when I heard Menaka Navaratna's talk on modeling biological clocks. Navaratna presented a computational model of pacemaker cells and showed how connectivity within a network of such cells affects the time it takes for the cell population to synchronize. The model is based on findings that indicate the importance of a brain region, known as the hypothalamus, in regulating the biological clock in mammals. Especially important is synchronous firing of a population of neurons within the hypothalamus known as pacemaker cells. When you hop on a plane to attend a math meeting on the other side of the country, these cells are desynchronized and until they regroup and fire in synchrony, you experience jet lag. Analyzing data from a computer model of 16,000 pacemaker cells (as many cells as are thought to reside in the hypothalamus), Navaratna showed how the number of long-distance connections in the network determined the time it takes for the cell population to synchronize.
Experimental data suggests that pacemaker cells are able to synchronize within approximately a week. Navaratna explained he wanted his model to be biologically plausible and synchronize in close to seven days. If the cells in the computer model were only connected to their nearest neighbors, it took the network much longer than a week to synchronize. If each pacemaker cell was connected to every other pacemaker cell (all-to-all connections) the synchronization occurred much faster. Such extensive connections are not biologically realistic since a cell expends a lot of resources making each connection. Looking into what sort of connections gave the correct time to synchrony, Navaratna found that a network with mostly local connections only needed a few longer ones to synchronize considerably faster than the nearest-neighbor connections. Using random connections but mostly making connections with nearest neighbors in addition to a few long distance connections, Navaratna's model consistently synchronized in close to seven days. This suggests the possibility that the pacemaker cells in the hypothalamus make a "small world" network, which have come up in protein interactions, social networks and the connectivity of the internet.
This year's theatrical presentation, done with the help of Adams' Williams College colleagues and students, was again hilarious and involved four mathematically inspired "skits." The first, entitled "Happiness is a Warm Theorem," was about a mathematician's session with a therapist who diagnoses him with mathematitis, an ailment we can all relate to, with symptoms including obsessive-compulsive behavior, addictive personality, and antisocial tendencies. "Immortality" was clearly inspired by the current vampire craze, but instead of Edward finding an eternal companion in Bella, we have a vampire-mathematician who has been Galois, Maclaurin, and Riemann among others, who finds an eternal collaborator in an unsuspecting budding mathematician. The third skit was "Group Therapy," which as we all know is when a group of people solve their problems by associating them to a concept in group theory. This was probably my favorite one, although I did have a soft spot for "The Lord of the Rings: The fellowship of the rings," in which a young postdoc in Nebraska named Frodo Baggins finds himself with the "Ring of Power," a ring which would destroy all of mathematics if it fell into the wrong hands. The geomancer Gandalf tells him to destroy the ring by taking it to Purdue, "where nobody cares about non-commutative algebra." He starts his quest with his trusty friends (all NSF Fellows at some point--hence the Fellowship), and the aid of Strider, who saves them from a Ringwraith by taking them into a calculus discussion section and pretending to be undergraduates.
Zhao presented data showing cyclical resurgence of syphilis in the U.S. and put forward a model based on partial differential equations to study propagation of the disease within a community. Syphilis is a sexually transmitted disease caused by the spirochetal bacterium and can generally be treated with antibiotics, including penicillin. The disease has a number of stages, each having different symptoms. If left untreated, later stages can damage vital organs such as the heart, brain and eyes. Transmission of syphilis is almost entirely through unprotected sexual contact, although there are examples of transmission from mother to child during pregnancy. The number of syphilis cases dropped dramatically shortly after the discovery of penicillin, inspiring hope of complete eradication in the U.S. and other western countries. The goal was almost reached in the 1950s, but the disease has resurged every 8 to 10 years ever since. The number of infections in the U.S. increases for a few years, then decreases again, and the cycle repeats. The cyclical behavior puts into question how effective current medical treatments are in controlling the long-term spread of the disease. Syphilis transmission depends on a number of factors including: effects of spirochetal bacterium on the infected population, human behavior--especially sexual activities, and precautionary measures taken by the community. Understanding how these different aspects influence the number of infections has proved difficult and what causes the resurgence is still under debate. The model Zhao presented is based on previously published models of syphilis but adds the possibility of partial immune protection after treatment and behavioral protection through education. Analysis of Zhao's model indicates that unless a vaccine is developed, the re-occurring resurgence cannot be stopped using current treatment options and prevention strategies. Presently a vaccine is not available, but research is being conducted in developing such a vaccine.
Read more of Adriana Salerno's experiences at the Joint Mathematics Meetings at the JMM 2010 blog.
Coverage of the meeting has appeared in O'Reilly Radar (by Ben Lorica).