From modeling microscopic genes and proteins to tracing the progression of an epidemic through a country, mathematics plays an important role in combating disease. For example, the basic model used to analyze the dynamics of infectious disease is a system of differential equations. A new field called “data mining”, involving statistics and pattern recognition, helps locate significant information in the vast amounts of data collected from studies of diseases in populations. Mathematics also plays a key role in connecting changes in the human genome to specific diseases. Mathematics has helped recent fights against foot-and-mouth disease in the United Kingdom and against Chagas disease—a disease affecting millions of people in Latin America. Epidemiologists studying the foot-and-mouth epidemic used mathematical models to conclude that early efforts were insufficient to stop what would become a calamitous spread of the disease. The government accepted the conclusions and took a course of action that, although drastic, did indeed arrest the outbreak. In Latin America, mathematicians computationally tested several courses of action against Chagas disease and found a surprisingly simple yet highly effective step (keeping dogs out of the bedroom) to greatly reduce the infection rate. These examples share three important characteristics: a mathematical model of the disease, modern computers to do calculations required by the model, and researchers with the insight to design the former so as to take advantage of the power of the latter.
The collective motion of many groups of animals can be stunning. Flocks of birds and schools of fish are able to remain cohesive, find food, and avoid predators without leaders and without awareness of all but a few other members in their groups. Research using vector analysis and statistics has led to the discovery of simple principles, such as members maintaining a minimum distance between neighbors while still aligning with them, which help explain shapes such as the one pictured. Although collective motion by groups of animals is often beautiful, it can be costly as well: Destructive locusts affect ten percent of the world’s population. Many other animals exhibit group dynamics; some organisms involved are small while their groups are huge, so researchers’ models have to account for distances on vastly different scales. The resulting equations then must be solved numerically, because of the incredible number of animals represented. Conclusions from this research will help manage destructive insects, such as locusts, as well as help speed the movement of people—ants rarely get stuck in traffic.
Mathematics and music have long been closely associated. Now a recent mathematical breakthrough uses topology (a generalization of geometry) to represent musical chords as points in a space called an orbifold, which twists and folds back on itself—much like a Möbius strip does. This representation makes sense musically in that sounds that are far apart in one sense yet similar in another, such as two notes that are an octave apart, are identified in the space. This latest insight provides a way to analyze any type of music. In the case of Western music, pleasing chords lie near the center of the orbifolds and pleasing melodies are paths that link nearby chords. Yet despite the new connection between music and coordinate geometry, music is still more than a connect-the-dots exercise, just as mathematics is more than addition and multiplication.
Waiting in line while boarding a plane isn’t just irritating, it’s also costly: The extra time on the ground amounts to millions of dollars each year in lost revenue for the airlines. Research into different boarding procedures uses mathematics such as Lorentzian geometry and random matrix theory to demonstrate that open seating is a quick way to board while back-to-front boarding is extremely slow. In fact, mathematical models show that even people boarding at random get to their assigned seats faster than when boarding back-to-front. Figuring out your own strategy for boarding a plane is hard enough, but modeling the general problem—which depends on many variables such as distance between rows, amount of carry-on baggage, and passengers’ waistlines—is substantially more complex. So researchers were pleased when they discovered that their theoretical analysis confirmed simulations conducted by some airlines. An added bonus to the research is that the mathematics used in the boarding problem is similar to that used to improve a disk drive’s data input and output requests. One clear difference: Data doesn’t try to carry on an extra bit.
The interplay of water, light, and music in some modern fountains is magical to behold, and mathematics is part of that magic. Geometry is used in the overall design, mathematical modeling simulates the fluid-particle interactions, and powerful algorithms drive the software that coordinates thousands of valves and lights through the numerous sequences in a typical show. The ability to make water act so precisely results from the use of laminar flow streams where all particles move in parallel and at the same speed. A complex mathematical analysis of fluid dynamics makes it possible for water to perform feats such as climbing stairs or behaving like individual marbles. The result is both wondrous and efficient: A four-foot column of water wouldn't fill a normal drinking glass.
Mathematics is used to understand how to precisely identify the parts of the brain that correspond to specific functions. Current research involves mapping our three-dimensional brain to two dimensions, similar to translating a globe to a map. Yet because of the many fissures and folds in the surface of the brain, mapping our brains is more complex than converting a globe to a map. Points of the brain that are at different depths can appear close in a conventional image. To develop maps of the brain that distinguish such points, researchers use topology and geometry, including hyperbolic and spherical geometry. Conformal mappings—correspondences between the brain and its flat map that don’t distort angles between points—are especially important to accurate representations of the brain. Just as a map of the earth aids navigation, conformal mappings serve as a guide for researchers in their quest to understand the brain.
Galaxies can be more than 100,000 light years across, consisting of hundreds of billions of celestial bodies, and with a mass more than a trillion times that of our sun. Modeling such huge, complex systems, in which many of the stars have chaotic orbits, requires new computational techniques. Advances in the speed and memory of computers have improved models, as has parallel computing, but advances in algorithms—the way the mathematics of a problem is converted into steps a computer can perform—are indispensable in developing accurate galaxy models. The complexity of simulating the behavior of a galaxy is not limited to the galaxy itself. Since a galaxy is usually part of a cluster or supercluster of galaxies, the external forces exerted by these larger agglomerations on the galaxy must also be accounted for. Thus, models must be accurate across many scales of distance. Instead of numerically solving the equations of the model uniformly across all sectors, researchers employ multi-scale algorithms that do more calculations in sectors determined to be more significant. This kind of technique uses computing power more efficiently, giving us a glimpse of the underlying structure of the universe.
The colored “strings” you see represent air flow around the soccer ball, with the dark blue streams behind the ball signifying a low-pressure wake. Computational fluid dynamics and wind tunnel experiments have shown that there is a transition point between smooth and turbulent flow at around 30 mph, which can dramatically change the path of a kick approaching the net as its speed decreases through the transition point. Players taking free-kicks need not be mathematicians to score, but knowing the results obtained from mathematical facts can help players devise better strategies. The behavior of a ball depends on its surface design as well as on how it’s kicked. Topology, algebra, and geometry are all important to determine suitable shapes, and modeling helps determine desirable ones. The researchers studying soccer ball trajectories incorporate into their mathematical models not only the pattern of a new ball, but also details right down to the seams. Recently there was a radical change from the long-used pentagon-hexagon pattern to the adidas +TeamgeistTM. Yet the overall framework for the design process remains the same: to approximate a sphere, within less than two percent, using two-dimensional panels.1 Daniel Bernoulli (BurrNOOlee) was a Swiss mathematician who did pioneering work in fluid flow.
Experimenting with real human hearts isn’t possible, but experimenting with accurate mathematical models of the human heart has led to a new understanding of its complex processes. Mathematics and the computer can replace years of experimentation in laboratories. For example, understanding resulting from mathematics greatly speeds up the design and implementation of artificial valves. Equations based on Hooke’s Law model the geometry of the heart by representing muscle fibers as closed curves of different elasticities. The Navier-Stokes equations, which describe all fluid flows, model blood flow in and around the heart. The fact that the heart’s shape is constantly changing, however, makes the equations especially hard to solve, and a precise solution to the equation can’t be found. Approximate solutions are generated by computer.
Robots of all shapes and sizes now perform tasks as routine as vacuuming the living room floor and as remarkable as discovering a hydrothermal vent on the ocean floor. Geometry, statistics, graph theory, differential equations, and linear algebra are some of the areas of mathematics that allow navigation and decision making so that robots can function autonomously and do things we either can’t, or would rather not, do. The robot pictured below not only dances but also greets visitors and escorts them to their destinations, providing news and weather updates along the way. Abilities like these require algorithms for vision, pattern recognition, speech recognition, and dealing with uncertainty so that accumulated error doesn’t render the robot ineffective. Most researchers think that we are a long way from creating machines that behave like humans, but improving algorithms will improve the capabilities of robots, which have already served in space, in rescues at disaster areas, and in the operating room, where physicians use robotic arms that allow for more precise, less invasive surgery.
Mathematical ecology is a growing and active area of interdisciplinary research between mathematics and ecology, using almost every part of mathematics (linear algebra, analysis, differential equations, stochastic processes, numerical simulations, and statistics) to understand and model complex biosystems. This modeling helps establish important parameters and thresholds, such as the area required to sustain a species or how fast an invasive species will spread through a region. Models must be fairly complex to capture how a single species interacts with other species and with its environment. Today’s mathematical ecology researchers are faced with the far more daunting task of simulating several interconnected networks of organisms across different scales of time, size, and space. To do that, researchers resort to some relatively new areas of mathematics, for example non-linear dynamical systems and spatial statistics.
Imagine trying to find the right information quickly in a library where billions of pages are randomly piled in a heap, instead of being in books shelved in order. That's what Web search engines do, millions of times a day. First-generation search engines often found useful pages, but those pages may have been too far down the list to be of any practical use. Current search engines rank pages by using mathematics— probability, graph theory, and linear algebra—so that sites most relevant to a query are listed at the top, where the user can most easily see them. The vast number of pages and links on the Web can be represented as a graph in which the nodes are Webpages and the directed edges are links. Today's search engines determine the relevance of a page to a query by incorporating the importance of pages pointing to and from that page. Thus, when it comes to a search, a page’s links can be just as important as its content. The final ranking comes from techniques in linear algebra and probability that help formulate and solve equations which, according to the founders of one search engine, involve millions of variables and billions of terms. In the future, search engines may use artificial intelligence and information on past searches to discern the actual intent of a query.
The state-of-the-art technology used by researchers to identify active (expressed) genes in cells is the microarray: a “gene chip” imprinted, not with circuits, but with DNA. Active genes of fluorescently tagged cell samples placed on the chip reveal themselves when they bind with their DNA complements on the chip. The amount of data generated by this microscopic activity is enormous: just one row in an array can have 15,000 points. Pattern recognition and image analysis are two fields which use mathematics to help extract important genetic information about several diseases, including Alzheimer’s and Parkinson’s, from microarray data. In the future, microarrays may enable an individualized approach to medicine, in which your doctor could use these chips to diagnose disease and determine the best treatment for your unique genetic profile. In one particular area of medicine, cancer research, the points in each column of an array can be thought of as genetic coordinates of samples from tumors. Yet there are so many coordinates that it is difficult to determine which tumors are similar. Algorithms employ statistics and different measures of distance in higher dimensions to group genetically similar tumors into “clusters” so that experiments can be done on treatments corresponding to the clusters. In one case, microarray technology not only distinguished between two different types of leukemia (verifying in the time it took to hit “Return” what had taken 35 years to discover) but also found different clusters within tumors that had been thought to be similar—resulting in clinical trials to confirm the distinction.
The CBS-TV program NUMB3RS shows modern mathematics and mathematicians at work—both instrumental each week in solving and preventing crimes. Although the series is fictional, many of the show’s episodes are based on true stories. In fact, statistics, combinatorics, and graph theory are just some of the mathematical fields being used today by real-life investigators to solve actual crimes. One of the most impressive instances of mathematics solving a crime was a case in which an algorithm pinpointed a serial offender’s location, based on the sites of previous crimes. When DNA samples cleared all the suspects living in the area, however, the natural conclusion was that the mathematics was unsound. Then a tip led investigators to a deputy who had been above suspicion (because of his job) and who had lived in the target area. He was eventually arrested and sentenced, proving that crime doesn’t pay but checking your assumptions does.
People in a society, neurons in the brain, and pages on the Web, along with their connections, are all examples of networks. Mathematicians study characteristics of networks, such as the number and distribution of connections, to discover what such attributes may reveal about the intrinsic nature of a network. For example, the colors in the picture below indicate how disruptive deleting a node would be to the network, in this case a living cell. The discovery and verification of network properties such as this has significance for applications ranging from the microscopic to the worldwide, including the protection of both computers and humans against viruses. The study of networks spawned the phrase “six degrees of separation”, the theme of a game involving actors’ connections via common film appearances. In an experiment done in the 1960s, over 100 randomly chosen people in the Midwest were found to be connected to a Massachusetts stockbroker (by a friend of a friend of a friend, and so on) in an average of just six steps. That people halfway across the country could be so closely connected was quite a revelation and proved that even a large network could be a “small world”. Today, researchers use parameters from graph theory and probability in analyzing networks to determine whether an elaborate network, be it a power grid or actors connecting to Kevin Bacon, is indeed a small world after all.
Votes are cast by the full membership in each house of Congress, but much of the important maneuvering occurs in committees. Graph theory and linear algebra are two mathematics subjects that have revealed a level of organization in Congress— groups of committees—above the known levels of subcommittees and committees. The result is based on strong connections between certain committees that can be detected by examining their memberships, but which were virtually unknown until uncovered by mathematical analysis. Mathematics has also been applied to individual congressional voting records. Each legislator’s record is represented in a matrix whose larger dimension is the number of votes cast (which in a House term is approximately 1000). Using eigenvalues and eigenvectors, researchers have shown that the entire collection of votes for a particular Congress can be approximated very well by a two-dimensional space. Thus, for example, in almost all cases the success or failure of a bill can be predicted from information derived from two coordinates. Consequently it turns out that some of the values important in Washington are, in fact, eigenvalues.
The processes that cells perform are as wondrous as their individual mechanisms are mysterious. Molecular biologists and mathematicians are using models to begin to understand operations such as cellular division, movement, and communication (both within the cell and between cells).The analysis of cells requires many diverse branches of mathematics since descriptions of cellular activity involve a combination of continuous models based on differential equations and discrete models using subjects such as graph theory. It may be surprising, but cell functions are depicted with complex wiring diagrams of circuits with signaling pathways, gates, switches, and feedback loops. Researchers translate the diagrams into equations, which are often solved numerically. Solving the equations is only part of a process in which solutions are analyzed, models are refined, and equations are reformulated and re-solved. This may be repeated many times. The aim of this process is an accurate representation of cell behavior, which may allow drugs and treatments to be designed in the same precise way that electronic circuits are today.
Anyone who has used a garden hose knows that knots appear in strange places. Scientists have found that a branch of mathematics called knot theory appears in many familiar places, including in our DNA. Mathematics plays a key role in understanding how DNA functions and replicates itself. Certain enzymes cut a strand of DNA at one point, pass another part of the strand through the gap, and then seal the cut. Knot theory gives insight on how frequently an enzyme has to act, from which one can infer how long the enzyme might take to make a product. This kind of complex manipulation is significant in many cellular processes—including DNA repair and gene regulation—and is the type of problem central to the theory of knots.
Sequencing the human genome was a tremendously significant accomplishment, but now comes the hard part: Understanding the structure and function of proteins. The 100,000 proteins in our bodies initiate, control, or perform every one of our biological functions through shapes (called folds) and communication with other proteins. Misfolded or mistargeted proteins can cause diseases such as cancer, mad cow, and cystic fibrosis. Computational biologists are using geometry, probability, and knot theory to begin to describe the intricate folding of proteins. Once it is known just exactly how a malfunctioning protein goes awry, drugs can be designed that address the problem, thereby restoring affected cells. Proteins assemble and re-assemble in an infinitesimal space and, most often, time span, yet the simulations of their functions are enormous, involving millions of calculations at each of billions of tiny time intervals. Almost every subject in mathematics—including integrals, partial differential equations, linear algebra, and numerical analysis—goes into simulating protein behavior, which,even for the simplest proteins, requires parallel computation to solve. It may seem unusual to concentrate such massive effort on such a small scale, but it is productive: Some strains of HIV had been resistant to treatment, but models of an HIV protein, integrase, revealed a nanoscale-sized trench that researchers can fill with a compound to overcome the resistance.
Iris recognition may allow us to live in a world without PIN numbers—identifying ourselves just by looking at the ATM. Identification by iris recognition is based on pattern recognition, wavelets and statistics. The first two fields are used to translate the patterns in your iris into a string of 0s and 1s, while statistics establishes that the scanned iris is yours. The iris is a good physical feature to use for identification because of the tremendous variability in iris patterns, even between twins. This variability guarantees that a correct identification is made when the code for a scanned iris matches a stored code in at least two-thirds of the bits. Furthermore, the eye and iris are easy for a scanner to find, due to their shape and placement. Once the iris is located, wavelets are used to translate the pattern of the sampled portion of the iris into two bits. These bits reflect the agreement between that portion of the iris and specific wavelets. The entire iris is encoded in about 2000 bits. Finding a relative match between this bit pattern and one of the thousands of iris codes in the database completes the identification. This comparison is done in parallel, so that the whole process takes place in about the blink of an eye.
Archimedes was one of the most brilliant people ever, on a par with Einstein and Newton. Yet very little of what he wrote still exists because of the passage of time, and because many copies of his works were erased and the cleaned pages were used again. One of those written-over works (called a palimpsest) has resurfaced, and advanced digital imaging techniques using statistics and linear algebra have revealed his previously unknown discoveries in combinatorics and calculus. This leads to a question that would stump even Archimedes: How much further would mathematics and science have progressed had these discoveries not been erased? One of the most dramatic revelations of Archimedes’ work was done using X-ray fluorescence. A painting, forged in the 1940s by one of the book’s former owners, obscured the original text, but X-rays penetrated the painting and highlighted the iron in the ancient ink, revealing a page of Archimedes’ treatise The Method of Mechanical Theorems. The entire process of uncovering this and his other ideas is made possible by modern mathematics and physics, which are built on his discoveries and techniques. This completion of a circle of progress is entirely appropriate since one of Archimedes’ accomplishments that wasn’t lost is his approximation of π.
Sudoku puzzles involve a lot of mathematics. Of course, the puzzles are filled with numbers, but the solution process would be the same regardless of the symbols used. More interesting is the logic behind the solution process, which can provide extra satisfaction upon solving a puzzle (with a lot less erasing). In addition, the puzzles are examples of Latin squares—important in abstract algebra and in statistics, in experimental design. Two Sudoku counting problems are: What is the fewest number of filled-in squares possible for a puzzle, and how many different puzzles are there? There are Sudoku puzzles with 17 numbers that have only one solution, but no one knows if there are puzzles with only 16 numbers that have a unique solution. As for the second question, there are more than five billion different puzzles. For counting purposes, puzzles that can be transformed by processes such as interchanging numbers or the top two rows are not considered different. This result depends on group theory and symmetry, crucial for much of modern physics and chemistry.
Much of modern research—from genome sequencing to digital surveys of outer space—generates tremendous amounts of multi-dimensional data. Unfortunately, visualizing dimensions higher than three is not easy, which makes analyzing and understanding the data difficult. Topology, a branch of mathematics concerned with the properties of geometrical structures, helps make sense of large data sets by providing a way of classifying the shapes of these sets. It’s especially useful for locating groups of similar points called “clusters”, which can, for example, distinguish between distinct types of a given disease, each requiring its own treatment. Topology (specifically algebraic topology) is also important in the operation of wireless sensor networks, which are used in applications as diverse as monitoring automobile traffic and controlling irrigation. Combined with numerical integration, results from algebraic topology provide the complete picture based on strictly local data. The advantage is that such sensor networks, maintained without GPS or other distance measures, are generally much cheaper to operate. So, in the case of irrigation, mathematical discoveries made almost a century before the advent of today’s technology save money while helping us use precious water wisely. In topological terms, just like the Möbius strip: What goes around, comes around.
Fitting just-broken pieces together is hard enough, but imagine how difficult it is to do after thousands of years—and a few civilizations—have passed. Archaeologists faced with hundreds of thousands of pieces at a site have turned to mathematicians to help reassemble the fragments. The pieces are first digitally scanned; then software uses geometry, combinatorics, and statistics to reconstruct ancient artifacts, even when many pieces are missing. Mathematics is also used in other new approaches to archaeology and paleontology: in the precise mapping of buried shipwrecks and the recreation of the movement of dinosaurs. In these cases and others, progress, perhaps paradoxically, actually brings us closer to understanding the past. Whether it’s refining a basic technique like triangulation or applying an involved subject such as partial differential equations, mathematics researchers are breaking new ground to uncover antiquity’s secrets.