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2004 Newsletter
 
School of Mathematics
Number 11 University of Minnesota Newsletter Spring 2005
       
Featured Colleagues

In this section we give our readers a glimpse of the work and mathematical lives of three colleagues, Professors Ezra Miller, Richard Moeckel, and George Sell. The editors thank them for giving generously of their time and sharing their perspectives to make this section possible.

 

MCKNIGHT LAND-GRANT PROFESSORSHIP AND
NSF CAREER AWARD FOR EZRA MILLER

Ezra Miller

Professor Miller was awarded the University of Minnesota’s McKnight Land-Grant Professorship for the 2005-07 biennium. This prestigious award is bestowed on a few tenure-track faculty members per year, chosen from across all academic fields on the University’s multiple campuses. Ezra is also a recipient of the NSF CAREER grant (Faculty Early Career Development Grant). These grants are not limited to mathematics and represent a major national scientific recognition. The five-year award will fund Professor Miller’s research program on “Discrete Structures in Continuous Contexts”. We congratulate Ezra for both of these honors.

Ezra joined the department in the Fall of 2002, although he spent his first year on leave at the Mathematical Sciences Research Institute (MSRI) in Berkeley. We asked Ezra for some comments to provide our readers with a perspective on his research and other scientific activities. “Many objects in nature and throughout mathematics,” he responds, “are described by quantities that are allowed to vary continuously. Length, width, height, radius, volume—these are all continuous parameters. In a broad sense, Mathematics is all about relations: describe this in terms of that; decompose those as sums of these; and so on. Thus, the goal can be to describe a continuous object by way of other (perhaps simpler) continuous objects, or, as in my research, to extract the essence of the continuous by way of the discrete.”

One manner in which hidden information can be borne out is by deformation: given an initial object of study, allow yourself to alter it in some controlled way, preserving its essential properties. “After all,” Ezra asks, “how much has really changed about the abstract properties of a rubber band if it stretches a little? The hole through the middle will still be there.” For more complicated curved geometric objects, tugging this way and that can force the object to flatten out. However, the complexity that had been hidden in the curvature has to go somewhere, and this is where the discrete structure enters: the complexity becomes reflected in the object breaking into finitely many pieces, each of which is more tractable than the original. If one understands each of the pieces fully (and this conditional is often the source of much interesting research in its own right), then the objectives are (a) to describe which types of simpler pieces arise, and (b) to determine how the simpler pieces glue together. This technique is amazingly powerful, in that it can reveal enormous amounts about the original curved object: remember that the deformation was chosen so as to preserve the relevant data.

Deformation techniques are of course common throughout mathematics; it is the origin of the specific deformations and the way Ezra and his coauthors apply them that is innovative. The degenerations amount to geometric interpretations of algebra used for symbolic computation.

Another direction of Ezra’s research concerns the notion of convexity. The most basic of convex objects are the polyhedra, which have (unlike, say, a baseball) finitely many flat faces. Examples of polyhedra are the so-called Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Convexity and polyhedra—including their manifestations in dimensions higher than
three—are fundamental to certain aspects of many sciences, including computer graphics, mathematical economics, statistics, and theoretical physics, as well as to many branches of abstract mathematics. Recent research of Ezra’s with Igor Pak (at MIT), which he is currently continuing, discovers novel phenomena that are fundamental to the nature of convex polyhedra. The techniques that they use have immediate applications to algorithmic aspects of geometry, but for more theoretical purposes, their methods provide fundamental insight into the nature of convexity. As with the deformation project above, the basis of their approach is to impose discrete structures.

While much of Ezra’s work falls toward the abstract end of the mathematical spectrum, his theorems often produce concrete results—sometimes almost literally. For the convexity project, he says, “If the input is a four-dimensional polyhedron then the output is a three-dimensional jagged crystalline form that I hope to produce in plastic or wax (alas, not concrete) using modern laser technology. Such results realize the artistic aspects of mathematics in a tangible aesthetic sense; they are the kinds of things one puts on display in the front hall (of one’s home or one’s department).”

Ezra recently co-authored a book with Bernd Sturmfels (UC Berkeley), entitled Combinatorial Commutative Algebra, which has now appeared as Volume 227 in the Springer—Verlag GTM series (Graduate Texts in Mathematics). As it says on the back cover, the book provides a self-contained introduction, with an emphasis on combinatorial techniques for multigraded polynomial rings, semigroup algebras, and determinantal rings. Geometric topics treated in the 18 chapters include toric varieties, flag varieties, quiver loci, and Hilbert schemes. The book has over 100 figures, and one of them even made it onto the front cover; in fact, Ezra proudly remarks “It’s the first GTM featuring cover art!” In this, the electronic age, Ezra managed to write the book while printing it just one time, a few months before it was finished. The book, and most of his papers, too, were written with coauthors in remote locations. “As long as one has control over such issues as who’s editing which of the various chapters or files at any given time, electronic collaboration can be quite efficient. Of course, it also helps to like LaTeX.” On the other hand, there is no avoiding the painful process of compiling the index, which took him three solid weeks this fall.

In addition to his research and writing, Ezra is active in organizing and taking part in conferences and workshops. For example, last summer he gave a series of lectures, based on his book, at the Abdus Salam International Center for Theoretical Physics in Trieste, Italy. The audience consisted of around 100 graduate students and postdocs from economically challenged countries around the world. Ezra also was one of the organizers for the Institute for Advanced Study/Park City Mathematics Institute Summer Program on Geometric Combinatorics this past July, 2004, with participants ranging from high school teachers and undergraduate students to research mathematicians.

This coming June, Ezra will be a lecturer at a summer school in Snowbird, Utah for graduate students, on the topic of D-modules and local cohomology, and is a co-organizer of a seminar session at the “every-10-year” Summer Institute in Algebraic Geometry, which will take place in Seattle, Washington this coming August.

 

RICK MOECKEL: TROPICAL CELESTIAL MECHANICS

The editors are very grateful to Professor Moeckel for taking time off from his busy schedule as Associate Head to offer our readers a perspective on his current research, with postdoc Marshall Hampton, in one of the most storied areas of mathematics—celestial mechanics. Rick received his Ph.D. from the University of Wisconsin in 1980. After a postdoc at ETH, Zurich, he joined the UM faculty in 1981. His research area is dynamical systems theory with emphasis on problems arising in celestial mechanics. Much of his recent work, including the problem described here, draws on techniques from computational algebra.

Tropical Celestial Mechanics

The Newtonian n-body problem has been challenging mathematicians for over 300 years. The familiar picture of planets moving in orderly elliptical orbits around the sun is misleading. In fact when three or more masses move under the influence of their mutual gravitational attraction, the results can be incredibly complicated. But simple motions are possible if the positions of the masses are chosen in a special way. For example, in 1772, Lagrange showed that if three bodies are arranged in an equilateral triangle and given appropriate initial velocities, they will always remain in an equilateral configuration — the triangle just rotates rigidly about the center of mass. When the three masses are of equal size, the existence of such a solution is clear from symmetry considerations, but when the masses have different sizes, the result is far from obvious. For example, if the three masses are the Sun, Jupiter and a small asteroid, the center of mass will be very near the sun and yet the triangle just rotates rigidly around that point (see figure 1a). It turns out that there really are asteroids moving approximately this way.
Figure 1: Relative equilibria. a. Lagrange’s equililateral triangle with the Sun (red), Jupiter (blue)
and an asteroid (black). b. A surprising example with eight equal masses.

Suppose n mass values are given. Will there always be some way to arrange the bodies so that such a rigidly rotating motion is possible? For this to occur, the mutual gravitational attraction on each body due to the other masses must be exactly balanced by the centrifugal force of the rotation — a very delicate balance indeed. It turns out that no matter what masses are specified, such special relative equilibrium configurations always exist. In fact, many different shapes are possible. The relative equilibria are found by solving a complicated system of polynomial equations for the positions of the masses. When n=3 the solutions have been known since Lagrange, but for n ³4 much less is known (see figure 1b for an example with n=8). The possible shapes and even the number of relative equilibria depend on the choice of the masses. In fact, it is a long-standing open problem in celestial mechanics to show that the number of solutions of the relative equilibrium equations is finite for all masses. This question was included by Smale on his list of problems for the 21st century.

In joint work with NSF postdoc Marshall Hampton, we found a computer-assisted proof of finiteness for n=4. The method uses some recent ideas in computational algebraic geometry. The argument goes roughly as follows: because the equations are polynomial, it follows that if the number of solutions is not finite, there must be a curve (or higher-dimensional algebraic variety) of solutions. Locally, a curve of solutions can be described by writing each variable as a series in some parameter, t. In the 1970’s D.N. Bernstein pointed out that the lowest-order powers of t in these series are subject to interesting geometrical restrictions. Every system of equations in k unknowns determines a polytope in k-dimensional space, a variation on the familiar Newton polygon for an algebraic curve in the plane. Figure 2 shows such a polytope for a system of equations in three variables describing relative equilibria of the three-body problem. It turns out that the vector of exponents of the lowest order terms of the series above must be an inward-pointing normal to one of the faces of this polytope. By examining normal vectors from every face of the Newton polytope, one can systematically rule out the existence of non-constant series solutions to a polynomial system and so prove finiteness. The relative equilibrium problem for n=4 involves a system of equations in 6 unknowns. The corresponding polytope in 6 dimensions (not shown) turned out to have 12828 vertices!

Figure 2: A Newton polytope in 3D.
The study of possible exponents of series solutions of polynomial systems is part of an active research area called tropical algebraic geometry (apparently in honor of one of its Brazilian practitioners). Given a polynomial system, there is an associated tropical variety which contains all the exponent vectors of nonzero series solutions. Our proof amounts to showing that the tropical variety determined by the relative equilibrium equations reduces to a single point (the exponent vector 0 coming from constant solutions).

 


GEORGE SELL: GLOBAL CLIMATE MODELING

Professor George Sell is a leader in the area of Dynamical Systems. A biography of George in honor of his 65th birthday, and authored by Professor Victor Pliss of St. Petersburg State University, was published in the October 2004 issue of the Dynamical Systems Magazine (see http://www.dynamicalsystems.org/ma/tc/toc?issue=11). It will also appear in a special issue of the Journal of Differential Equations dedicated to him. George’s extensive contributions include the Sacker-Sell spectral theory for invariant manifolds and many important results on the theory of reaction-diffusion equations and Navier-Stokes equations. His recent monograph “Dynamics of Evolutionary Equations”, coauthored with Y. You, is “a major contribution to the literature on the dynamics of infinite dimensional problems”. George’s service to the mathematical community includes being the co-founder with Professor Hans Weinberger of the Institute for Mathematics and Its Applications (IMA) and serving as the IMA’s Associate Director for several years; serving as the director of the Army Computing Research Center; serving as Program Director at the National Science Foundation, and being the founder and serving as editor of the Journal of Dynamics and Differential Equations. His many honors include an invited address at the 1982 International Congress of Mathematicians, an Honorary Doctor of Science Degree from St. Petersburg State University (1990) and a conference in his honor organized by the University of Valladolid and held in July 2002 in Medina del Campo, Spain.

Professor Sell recently developed a timely and innovative graduate level course on Global Climate Modeling. He is presenting the course for the first time during the Spring 2005 Semester and he intends to offer it on a regular basis. We thank him for sharing with our readers the following description of his course.

Global Climate Modeling

The key to building good climate models is to have a good understanding of the heat transfer phenomena in the oceans of the Earth. The physical forces driving the oceanic flows include the radiation from the Sun and the changes in the gravity due to the Earth, the Moon, the Sun and the other planets. The longtime dynamics of any climate model are located in the global attractor of the underlying oceanic model.

Since the time scales for many climate models run into the tens, or even hundreds, or thousands of years, it is natural to ask: what would happen to the longtime dynamics of a model if one were to ignore high frequency events, such as the daily rotation of the Earth or the Lunar phases? (One might use a partial time-averaging method to eliminate the physics of these events.) However, it must be noted, for example, that the daily heating and cooling of the Earth’s surface due to the Sun’s radiation is both a major physical force AND it differs widely from its mean value. In short, by adding the daily rotation of the Earth to some time- averaged model, one is introducing a large perturbation into the model. “Large” perturbations can destroy many dynamical properties. That being the case, one needs to address the issue of whether any such time-averaged model can lead to good information about the global climate of the Earth.

In this graduate-level course we describe the basic theory of oceanic flows. The lectures start with the Leray-Hopf theory of solutions of the Navier- Stokes equations (NSE) in 2D and 3D. A description of the MILLION dollar NSE problem will be an early goal in this theory.

The next step is to extend the Leray-Hopf theory to oceanic models. This will include the theory of the NSE on thin 3D physical domains. The (unfunded) 10K dollar NSE problem arises in this context! The latter open problem is to show that the global attractor for the NSE on thin 3D domains (say oceanic domains) has “good” dynamical properties.

In the climate models of interest in this course, we will examine the nonautonomous forcing of the oceanic flows due to the planets and the Sun. In order to study the diversity that is seen in the global climate, one needs to study models with many time scales. The time scales for the El Nino event, for example, differ from the time scales needed for a Kyoto protocol, or the time scales used for the onset of the next Ice Age. One challenging problem arising in all models is the effect on the longtime dynamics due to the daily rotation of the Earth. Among other theories, we will show that there is a rigorous mathematical basis for ignoring the daily rotation of the Earth in many global climate models.

 
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