Tropical celestial mechanics
Wednesday, November 15, 2006 - 11:15am - 12:15pm
Some interesting problems in mechanics can be reduced to solving systems of algebraic equations. A good example is finding relative equilibria of the gravitational n-body problem. These are special configurations of the n point masses which can rotate rigidly such that the outward centrifugal forces exactly cancel the gravitational attractions. The algebraic equations are complicated enough that it is a long-standing open problem even to show that the number of solutions is finite. I will describe a solution to this question for n=4 which makes use of some ideas from what is now called tropical algebraic geometry – Puiseux series solutions, initial ideals, etc. The problem is open for larger n.