Gia Bloomstrand
Northern Arizona University, AZ

Minimal Surfaces

At each point p on a surface M in Euclidean 3-space, we can compute a normal vector n. Any plane that contains n will intersect the surface in a curve c. For each curve c, we can compute its curvature. As we rotate the plane through the normal n, we will get a set of curves on the surface each of which has a value for its curvature. Let k1 and k2 be the maximum and minimum curvature values at p, respectively. The mean curvature of M at p is H=(k1+k2)/2. Then M is a minimal surface if the mean curvature equals zero at every point. The figures on this page are examples of minimal surfaces. We can use ideas from complex analysis to investigate minimal surfaces. In particular:

(1) we can use the Schwarz-Christoffel formula from complex analysis to derive analytic functions that map onto convex polygonal regions. We can then shear these functions and derive the corresponding minimal surfaces and see how they are related to Jenkins-Serrin minimal surfaces which project to convex  polygons;

(2) we can shear elliptic integrals of the first kind to  get a family of minimal surfaces that range from Scherk's  doubly-periodic to the helicoid. What families of surfaces do  we get, when we shear elliptic integrals of the second and third  kind?

Some nice internet sites are:

M. Weber's "Minimal surfaces--introduction"
H. Karcher and K. Polthier's "Touching Soap Films: an introduction to minimal surfaces"
Tech. University Berlin's java applet that graphs various minimal surfaces

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Mathematical Physics

In this program we introduce one of the principle boundary value problems of analytic function theory, the so-called “Riemann-Hilbert” problem:

Let denote a contour dividing the complex plane into two simply connected domains, and . Also let  be a matrix valued function of the complex variable , and let tend to the matrix identity as tends to infinity. Factor  as the product, where each of these two matrix valued functions has the same asymptotic behavior as (if each of the domains is unbounded), but where is analytic in  and  is analytic in .

This introduction should enable participants to enter a fast growing research industry in pure mathematics and mathematical physics: orthogonal polynomials, integrability/soliton theory, random matrix theory (and, so, quantum gravity), field theory renormalizations, singular integral equations, etc. The mentor will explain in detail new applications of the Riemann-Hilbert problem to the energetics of physical systems lacking time reversal invariance. Various publishable projects in this new area will be proposed to participants in the program, additionally direction will be given for those desiring to pose problems from the above list of popular Riemann-Hilbert applications.    

Some nice websites are:

Evolution of eigenvalues of a matrix undergoing a random Dyson Process:
http://www.crm.umontreal.ca/~physmath/images/Dyson.gif

AMS Notices introduction of the (original formulation of the) Riemann-Hilbert analysis to integrable systems:
http://www.ams.org/notices/200311/fea-its.pdf

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Geometric Optimization

The basic problem in geometric optimization is to minimize length, area, or some other quantity, among curves or surfaces satisfying a given constraint. A well-known example is that a circle has least perimeter among curves enclosing a given area. Some problems that the REU participants could research are: (1) the octahedron problem which asks what is the least area surface spanning the edges of the octahedron. The conjectured solution is a beautiful, piecewise-planar soap film consisting of twelve triangles and six kites; (2) large minimizing networks in the hyperbolic plane; that is finding the shortest path connecting a set of points in the hyperbolic plane; (3) Melzak's conjecture which asks what polyhedron with unit volume in Euclidean 3-space has the shortest edgelength (for example, a cube with volume 1 has total edge length 12; there is a polyhedron that has a shorter edgelength--can you figure it out?).

Some nice internet sites are:

Soap bubbles and isoperimetric problems (http://math.berkeley.edu/~hutching/pub/bubbles.html)

Exploratorium soap bubble page (http://www.exploratorium.edu/ronh/bubbles/bubbles.html)

Some open problems in soap bubble geometry (http://torus.math.uiuc.edu/jms/Papers/foams/soap-prob.pdf)

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