# Researcher studies connection between geometry and physics

Finding the largest number of marbles that can be packed into a given container is a very hard classic problem originally posed by Kepler.INDIANAPOLIS -- Since the marbles retain their spherical shape, the task hinges on minimizing the odd-shaped interstices between them. Remove the rigidity condition, asking only to preserve the volume of the marbles, and the problem becomes trivial. We have in effect replaced the marbles with Play-Doh balls, letting them squeeze through the crevices as we try to pack more.

Olguta Buse, an associate professor in the Department of Mathematical Sciences, is currently sponsored by a National Science Foundation research grant to further her fundamental results on symplectic packings. In this version of the problem, the marbles are not completely rigid, but still must preserve a “symplectic structure”. This

condition is a balanced compromise between the hard and soft regimes described above. It is not an arbitrary condition either, because symplectic geometry is the exact type of mathematics underlying classical mechanics, and hence is at the very foundation of physical sciences.

The behavior of phenomena as diverse as spinning tops, magnetism, the propagation of water waves, and the gravitational field can be described and understood to a large extent in terms of symplectic geometry. A good grasp on the behavior of the geometry of symplectic spaces, as in Buse’s work, is becoming indispensable in comprehending and manipulating large scale complex systems, like particle accelerators and space missions as they follow their orbits through the solar system.