Renewing U.S. Mathematics A Plan for the 1990s (1990) / Chapter Skim
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B Recent Research Accomplishments and Related Opportunities
B Recent Research Accomplishments and Related Opportunities
Pages 87-134
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The unification and cross-fertilization of areas within core mathematics, increaser! reaching out to applications (which often uncovers unusual and unexpected uses of mathematics)
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Biostatistics and Epidemiology SYNOPSIS OF TOPICS In the first section, "Recent Advances in Partial Differential Equations'" the items discussed are formation of shocks in non-linear waves, recent advances in elliptic equations, free boundary problems, and finally some remarkable advances in exactly solvable partial differen88
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discusses the key role played by computational fluid dynamics in global circulation models that are used in the analysis of climate change on a worldwide scale. Section 7, "Chaotic Dynamics," shows how ideas of Poincare on aperiodic orbits for ordinary differential equations, complemented by ideas from topology, differential geometry, number theory, measure theory, and ergodic theory, plus the ability of modern computing facilities to compute trajectories, have led to a body of core mathematics that has many interesting and important applications.
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The availability of powerful computers is stimulating research in core mathematics. Section 13, "Computer Visualization as a Mathematical Tool," indicates how computer graphics can be used as a tool in minimal surface theory and other areas of geometry to enhance understanding and provide experimental evidence stimulating conjectures.
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The next two sections discuss recent advances in algorithms for numerical optimization; Section 24 is devoted to the new and very important interior point methods for linear programming, which provide an alternative to the classic simplex methods and are beginning to have a significant practical impact in the design of telecommunications networks and the solution of large-scale logistics planning and scheduling problems. Section 25 discusses yet another approachstochastic linear programming, a technique that allows one to include non-deterministic elements in the formulation and solution of a problem.
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1. RECENT ADVANCES IN PARTIAL DIFFERENTIAL EQUATIONS An important trend of the last 15 years has been the great progress made in understanding nonlinear partial differential equations (PDEs)
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This result is a basic step forward in the analysis of partial differential equations. Important progress has been made also on some singular elliptic equations, namely those having "critical nonlinearity." If the- nonlinear term in such an equation were made slightly weaker, then the equation could be regarded as a small perturbation of a linear problem, but at the critical nonlinearity this becomes impossible.
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The simpler partial differential equations obtained when this coefficient vanishes are called Euler equations. These are accurate enough for studying the movement and accumulation of vortices.
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One recent success in the alliance between large-scale computing and modern mathematical theory is the discovery of a new mechanism of nonlinear instability for supersonic vortex sheets. Recent large-scale simulations have demonstrated that all supersonic vortex sheets exhibit nonlinear instability, belying the predictions of stability made in the 1950s and 1960s.
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This flow pattern would be difficult or impossible to describe and adjust without dependable mathematical models coupled with computer simulations. For very-high-altitude high-speed conditions, numerical simulations are also being used for vehicle design.
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Leonardo da Vinci first described the vortices that form behind heart valves and that enable the valves to avoid backflow by closing while the flow is still in the forward direction. Leonhard Euler first wrote down the partial differential equations for blood flow in arteries.
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This raises the prospect of additional applications such as flow through the aortic valve, the mechanical consequences of localized damage to the heart wall, interactions of the right and left ventricle, flow patterns of blood in the embryonic and fetal heart, the fluid dynamics of congenital heart disease, and the design of ventricular-assist devices or even total artificial hearts. A general-purpose three-dimensional fiber-fluid code has already been developed that solves the equations of motion of a viscous incompressible fluid coupled to an immersed system of elastic or contractile fibers, using the vector architecture of the Cray.
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In MRI a large magnet and surrounding coil measure the resonating magnetic field inside the patient, due to an unknown density of hydrogen atoms, which act like little spinning magnets. The mathematics used to reconstruct the hydrogen density uses the inverse Fourier transform applied to the measured signal.
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Three-dimensional general circulation models provide a means of simulating climate on time scales relevant to studies of the greenhouse effect. These models, which numerically solve a nonlinear system of partial differential equations, are being used to compute differences between a climate forced by increases in greenhouse gases and a control or current climate.
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In 1963, a detailed numerical examination of a specific system of differential equations from meteorology revealed unexpected chaotic trajectories. This work not only pointed out the presence of chaotic trajectories in a specific non-Hamiltonian system but also suggested new directions of research in the theory of dynamical systems.
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Mathematicians are now beginning to create new numerical methods for computing the stable and unstable manifolds of which Poincare spoke. In a related vein, identification of "inertial manifolds" for partial differential equations is a promising route in the quest to reduce the essential dynamics of an infinite-dimensional dynamical system to that of an appropriate finite-dimensional one.
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In broad terms, this affects signal and image processing and fast numerical algorithms for calculations of integral operators (not necessarily of convolution type)
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Another facet of number theory that has seen an enormous amount of activity is arithmetic algebraic geometry. This has figured prominently in work on the arithmetic Riemann-Roch theorem, in the unifying conjectures connecting Diophantine problems to Nevanlinna the104
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APPENDIX B ory and to differential geometry, and in recent results giving a new and very "geometric" proof of Mordell's conjecture over function fields a proof that may translate to an analogous new proof in the number field context. A third significant development of the last five years consists of the conjectures and results that bring some old Diophantine questions closer to the heart of issues perceived to be central to immediate progress in arithmetic.
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It came as a complete surprise, both because dimension-4 behavior is so different from the previously known behavior of other dimensions, and because of the remarkable source of the discovery. In fact, the key invariants used to distinguish the exotic 4-manifold from ordinary Euclidean space have their origin in the study of the Yang-Mills equations, originally introduced in particle physics.
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The study of symplectic manifolds is called symplectic geometry, and it has been revolutionized in the last few years. A major breakthrough was the use of nonlinear elliptic equations (see Section 1, "Recent Advances in Partial Differential Equations")
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If we restrict a symplectic structure to a surface of constant energy, we get a "contact structure." Along with the recent progress in symplectic geometry has come important work on contact structures. In particular, exotic contact structures have been discovered on Euclidean space, distinguished from the standard contact structure by "overtwisting" on embedded discs.
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These different topics all have been points of view within the new field of noncommutative geometry. Basically, the ideas of differential geometry have been shown to extend to a noncommutative setting.
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Finding this minimal surface is easily expressed as a problem in the calculus of variations and thus reduced to the study of a certain partial differential equation, the minimal surface equation. While the solutions of this equation are not difficult to describe, at least in the small, the global behavior of the solutions is very delicate, and many questions remain open.
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These new examples have invigorated the subject of minimal surfaces in general, and recent progress in the subject has been closely linked to computer graphics. More general calculus-of-variations problems have recently been approached by computer graphics techniques, which are invaluable in formulating and testing conjectures.
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At the same time another infinite dimensional Lie algebra, the Virasoro algebra, entered physics in the dual resonance models and string theory. While the dual models lost much of their interest for physicists in the 1970s, there were important mathematical developments that grew from them: for instance, the development of a formula for the determinant of the
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15. STRING THEORY Some of the most exciting developments in recent mathematics have their origin in the physicists' string theory, the so-called "theory of everything." This development offers a classic example where a physical 113
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The entry of new mathematics into string theory is forced by principles of invariance one must eliminate superfluous parameters from the description of the theory. Three such principles emerge: parameter invariance on the string (the labeling of positions on the string is physically irrelevant)
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Their work was motivated by algebraic geometry and partial differential equations, the latter being classically the main vehicle for exchange between mathematics and physics. Many mathematical problems for future study have been posed by string theory.
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17. SPATIAL STATISTICS The keen interest in the development of theory and methods for spatial statistics is strongly driven by an array of applications including remote sensing, resources estimation, agriculture and forestry, oceanography, ecology, and environmental monitoring.
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Thus the needed research described above has a substantial foundation on which to build. An important recent advance in spatial statistical research is the development and application of flexible spatial Markov lattice models for image processing and the application of powerful techniques such as the Metropolis algorithm for implementation of these models.
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acceptance sampling. Statistical process control consists of methods for assessing the behavior of an engineering process through time, in particular for spotting changes; the major methods in this category are control charts, a topic in need of new thinking.
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A number of interrelated results on graph minors, some in "pure" graph theory and some relevant to the design of algorithms, are among recent achievements. The results open up new avenues for research and suggest a number of problems and opportunities.
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One of the main results about graph minors is that for any number (say ten) , there is an efficient algorithm to solve the ten-paths problem.
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Several very surprising properties can be shown to hold for GEI equilibrium, using tools from differential topology. First, in great contrast to the complete markets model, if there are fewer assets than states, then the GEI equilibria are "generically" inefficient.
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The trick to the correct analysis of the problem was to recognize that the GEI budget constraint can be reexpressed in terms of the span of the monetary payoffs across the S states, and hence in terms of a Grassman manifold constructed from the state space. Arguments from degree theory show that the simultaneous equations defined by GEI equilibrium on this Grassman manifold generically have a solution.
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In fact, dramatic progress has been made in the last five years in uncovering nontrivial and highly efficient methods for parallelizing important problems that seem to be inherently sequential. Examples include algorithms for arithmetic, matrix calculations, polynomial manipulation, differential equations, graph connectivity, pointer jumping, tree contraction and evaluation, graph matching and independent set problems, linear programming, computational geometry, string matching, and dynamic programming.
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An interesting aspect of these tests is that they do not provide witnesses for primality, but this weakness was rectified in work that defined witnesses for primality rather than compositeness, showing that if n is prime, most randomly chosen numbers will bear witness to that fact. There are many other randomized algorithms based on the abundance of witnesses.
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23. THE FAST MULTIPOLE ALGORITHM There are great opportunities for progress in algorithms dealing with problems such as particle beams in plasma physics, underwater acous125
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. The Fast Multipole Method (FMM)
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For linear programming, however, researchers quickly discovered that its improved worst-case running-time bound clid not correlate with better performance in practice.
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The solution involves the use of projective transformations and a logarithmic potential function to guide the search, and yields a running time of O(n35L2~. The theoretical improvement over the Ellipsoid method running time was not the main story, however; more important, this algorithm (along with several of its variants)
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25. STOCHASTIC LINEAR PROGRAMMING Deterministic models for linear programming problems and their solutions, ever since their first appearance 40 years ago, have been keeping pace with the extraordinary advances that have taken place in computing power.
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The process of physically locating clones along the genome should be substantially facilitated by an understanding of the design parameters and sources of variation inherent in the process. Obtaining DNA sequence data is only a first step in modern molecular biology.
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It encompasses such varied subjects as the worldwide geographic variation in disease incidence rates, the setting of radiation exposure standards in the workplace, and the evaluation of vaccine efficacy using randomized field trials. Two distinct study designs, cohort and case-control, are used for much of the research in chronic disease epidemiology.
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Better techniques are needed for assessing the magnitude of measurement errors and to correct for the tendency of such errors to dilute the observed association between exposure and disease. Recent advances in statistical computing and in the statistical theory of quasi-likelihood analysis based on generalized estimating equations promise rapid advances in this field.
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It is related to problems that occur in the field of image processing, where rapid progress has been made. But the lack or poor nature of key types of data makes it much more formidable.
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