### Topic for 2000: Singularities and knots in real algebraic geometry

The group directed by Alan Durfee and Don O'Shea will investigate the topology and geometry of sets defined as the zeroes of real polynomial equations. More specifically, we will look at:

• Singularities defined by polynomial equations. This follows up work done by REU groups in 1996 and before.
• Perturbations of a critical point of a function f(x,y) which give a function with simpler critical points. How many local maxima, minima and saddles can occur? This follows up work done by REU groups in 1989 and 1992.
• Perturbations of a critical point of a polynomial function which defines a curve in three-space. What knots can occur? This follows up work done by an REU group in 1998.
There are many possibilities here. We will focus on a few depending on the interests and background of the participants. There are many interesting and easily accessible questions in this subject. Some of them may have nice payoffs, though of course one doesn't know until one is finished! However we anticipate that participants will be able to do original work in these areas, as has happened in previous years.

The whole area of real algebraic geometry is a rich field with many interesting results linking the structure of the objects (geometry) with the polynomials (algebra). A typical classical result which we will probably use is

Bezout's Theorem: If C is a curve in the real plane defined as {f(x,y) = 0}, where f is a real polynomial of degree m, and D is a curve in the real plane defined as {g(x,y) = 0}, where g is a real polynomial of degree n, then C and D have at most nm intersections.

This theorem is a special case of the corresponding result for complex curves in the complex projective plane, where they have exactly this number of intersections. A more typical (and classical) result specific to real algebraic geometry is

Harnack's Theorem: If C is a curve in the plane defined as {f(x,y) = 0}, where f is a real polynomial of degree d, then C has at most (1/2)d^2 - (3/2)d + 2 topological components.

For example, if d = 2 (a conic section), then the curve has at most 2 components. (Although we anticipate not using this result, one never knows what will happen!)

Good preparation for this project would be course(s) in abstract algebra, topology and similar subjects. If you're interested in the knot theory aspect, then The Knot Book by Colin Adams is excellent (and fun reading, as well).

(Last revised 1/8/00.)