This fall, about 80 mathematicians have arrived at the Mathematical Sciences Research Institute (MSRI) to study two kinds of geometry that have flourished dramatically in the last 30 years.

One of them, called contact and symplectic geometry, has deep roots in physics. "Contact geometry is a language for geometric optics, in the same sense that symplectic geometry is a geometric language for mechanics," said Yakov Eliashberg, a co-organizer of the MSRI program. The other featured subject, called tropical geometry, redefines basic concepts like lines and curves (and even addition) in a way that seems bizarre at first, but has unexpected applications to computer algebra and even to biology.

MSRI, located on a scenic ridge high above the campus of the University of California, Berkeley, is one of the seven research institutes in the United States dedicated solely to math. The Institute is funded is funded in part by the Simons Foundation. MSRI has no permanent faculty, but instead rotates new members in every year. To keep abreast of new developments in mathematics, MSRI chooses research programs for each year only a few years in advance.

Video: MSRI director Robert Bryant

Both contact and symplectic geometry derive from an idea called 'phase space', first developed to study the stability of planetary orbits. A particle (such as a planet) moving in three dimensions has three coordinates of position and three coordinates of momentum (or velocity), which can be lumped together into six coordinates in phase space. Although a six-dimensional world is more difficult to visualize than our three-dimensional one, the equations of motion become simpler and more symmetric there. In particular, they leave invariant a mathematical structure called a symplectic form.  As the points in phase space move according to the equations of motion, the symplectic form does not change.  In fact, the fixed symplectic structure serves to convert the principle of least action for the system into the equations of motion themselves.

The 1980s witnessed the emergence of powerful new methods in symplectic and contact geometry, which made it possible to answer questions about the overall topology (or shape) of the phase space. The results applied not only to phase spaces that arise from physical problems, but to any curved space (or manifold, in mathematical parlance) that carries a symplectic form. In this way, a new subject called symplectic topology was born.

This year's Abel Prize winner, Mikhail Gromov, introduced a key ingredient into symplectic topology, called 'pseudoholomorphic curves', which are defined using an auxiliary compatible Riemannian metric.  These are two-dimensional surfaces on which the symplectic form has a particularly simple interpretation — it just measures the area of the surface in the auxiliary metric. You can visualize them as rubber membranes that stretch taut around any bottleneck in the symplectic space. There are not many of these surfaces, and their areas as well as their intersections convey subtle information about the topology of the space that turns out not to depend on the auxiliary metric, but only on the symplectic structure. (Their confusing name comes from the tradition in algebraic geometry of calling something a curve when it has one dimension over the complex numbers, even though this means it has two dimensions over the real numbers.  In fact, these real surfaces do behave much like complex curves, hence their name and their utility.)

In the 1990s and 2000s, Clifford Taubes, Andreas Floer, Helmut Hofer and others used and generalized Gromov's work to study questions in symplectic geometry, Hamiltonian dynamics and low-dimensional topology. For instance, Taubes recently solved a problem called the Weinstein conjecture, about the existence of closed orbits of certain kinds of vector fields on a contact manifold. A characteristic ingredient in all of the recent work is the application of new invariants called Floer homology theories, which greatly extend the concept of pseudoholomorphic curves. Different Floer homology theories often turn out to give identical information, even though they are computed in very different ways. This phenomenon is still far from understood, and will be the subject of a related but separate program at MSRI in the spring of 2010.

One of the big questions is how these ideas apply to lower-dimensional spaces or to spaces that do not have a symplectic form. "There are lots of hints that symplectic and contact geometry will play a central role in our understanding of four-dimensional manifolds," said Taubes, who will spend the year at MSRI as an Eisenbud Professor, a position funded by the Simons Foundation. "There's just a little tail wiggling in four dimensions, but it's just screaming out that this whole big thing should be there. My goal this year is to figure out how to get more than just this tail into the room."

While the contact and symplectic geometers are chasing tails, another group of 43 MSRI members will work in one of the hottest new fields of mathematics, tropical geometry.

In spite of its name, tropical geometry actually has nothing to do with balmy island paradises.  (The term 'tropical' was chosen to honor one of the pioneers of the subject, Imre Simon, a Brazilian mathematician.)   Calculations in this geometry are done in a system of numbers called the 'tropical semiring', a strange algebraic structure on the real numbers in which multiplication is replaced by addition and addition is replaced by the operation (denoted by ⊕) of taking the minimum of two numbers. For instance, 3 ⊕ 5 = 3, the minimum of 3 and 5. 'Polynomials' defined with these two operations represent certain patterns of lines, rays and line segments.

"One common entry point into tropical geometry is the problem of numerically solving systems of algebraic equations," said Bernd Sturmfels, one of the co-organizers of the tropical geometry program. Centuries ago, solving an equation like x4 = 7 was hard. But then logarithms were invented — and even better, slide rules, which were based on logarithms — and it became a snap. You simply find the logarithm of 7 (using a table or a slide rule), divide it by 4 and then take the antilogarithm. Tropical geometry does exactly this same thing for more complicated systems of several polynomial equations. "It's the geometry of the slide rule," Sturmfels said.

Though Sturmfels said the emphasis of the program is "overwhelmingly on pure math," he and a few other members are working on an exciting application of tropical geometry to biology. Now that genome analysis has enabled biologists to estimate the evolutionary 'distance' between two gene sequences, sophisticated computer programs have been developed to infer the evolutionary tree relating different species. In order to match the gene-sequence data, the lengths of the branches on this phylogenetic tree must satisfy a dizzying array of consistency conditions. Remarkably, those conditions are tropical polynomials, and they are identical in form to the equations that define a very different mathematical structure, the Grassmann manifold. The only difference is that multiplication is replaced by addition, and addition is replaced by the operation called Å. In other words, the tropical Grassmann manifold is the space of phylogenetic trees!