Camille Jepsen

Steiner Problems

• Closed polygonal surfaces such as the tetrahedron or the cube;
• The Klein bottle of constant Gaussian curvature; 
• The double torus of constant Gaussian curvature; 
• A piecewise flat surface with curved edges, such as the
  cylinder.

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Minimal Surfaces Problems

At each point p on a surface M in R^3, we can compute a normal vector n. Any plane that contains this normal 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 point p, respectively. The mean curvature of M at p is H=(k1+k2)/2. A minimal surface in R^3 is a regular surface for which the mean curvature equals zero at every point. In the theory of minimal surfaces, the Weierstrass representation provides a formula for the local representation of a minimal surface. Images of minimal surfaces can easily be displayed by using computers, and this lends itself nicely to student explorations. One new approach to investigating minimal surfaces is to use results about planar harmonic mappings, f=h+\bar{g}, where h and g are complex analytic functions. Peter Duren, Univ. of Michigan, published a text in 2004 on planar harmonic mappings. There is a nice relationship between harmonic univalent mappings and embedded (i.e., nonself-intersecting) minimal surfaces. We can reformulate the classical Weierstrass-Enneper representation using the harmonic univalent map f=h+\bar{g}.

We can investigate minimal surfaces by using theorems about planar harmonic mappings.

• Various properties of minimal surfaces can be represented by using a system of differential equations of the corresponding planar harmonic mappings. We can then compute the Lie symmetries of this system to investigate these properties of the minimal surface as was done by Bila. This is a computational problem that has been recently simplified by the creation of Maple and Mathematica programs that can compute Lie symmetries, and it allows students to study some differential geometry, abstract algebra, differential equations, and complex analysis.
  
• Jenkins-Serrin minimal surfaces correspond to planar harmonic mappings that are convex polygons. Recently, there has been a paper investigating the minimal surfaces corresponding to planar harmonic mappings that are nonconvex polygons by using Poisson integrals. This topic can also be investigated by using a shearing theorem and the linear combination of harmonic mappings.
  
• In the study of planar harmonic functions, there are some canonical extremal functions such as Koebe functions and right hand-plane functions. If the dilatation of these functions is a perfect square, then they lift to minimal surfaces. Which minimal surfaces do these extremal functions lift to? Are there properties of these minimal surfaces that make the corresponding planar harmonic mapping extremal?

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Graph Theory Problems

Inverse Spectral Problems for Graphs
Structured symmetric matrices arise in many applications in science and engineering and their eigenvalues are of intense interest. The structure of a symmetric matrix is conveniently encoded by means of an associated graph. For an undirected graph G on n vertices, let S(G) be the set of all real symmetric n‐by‐n matrices whose nonzero off‐diagonal entries occur in exactly the positions corresponding to the edges of G. The inverse eigenvalue problem for graphs asks:

Given a graph G on n vertices and n given real numbers (with repetition allowed), is there a matrix in S(G) with eigenvalues equal to the n given numbers. This problem is important in understanding how the eigenvalues of a symmetric matrix are related to its off‐diagonal zero/nonzero structure. However, for arbitrary graphs, the inverse eigenvalue problem is very difficult.

A more modest goal is to ask: What is the maximum multiplicity M(G) of an eigenvalue of a matrix in S(G)? This problem has been intensively studied, and much is known. As one example, undergraduate students recently contributed to a new result in this area: M(G) does not change when an edge in G incident to a vertex of degree 1 or degree 2 is subdivided.

A problem whose level of difficulty lies between the inverse eigenvalue problem and the maximum multiplicity problem for graphs is the inverse inertia problem, which asks: Given a graph G on n vertices and a pair of nonnegative integers (p,q) with p + q between 0 and n, is there a matrix A in S(G) that has p positive eigenvalues and q negative eigenvalues. This problem has been completely solved for trees, but little is known for general graphs.

Some of the questions we will consider are:

1. Suppose e = vw is an edge in a graph G for which deg(v) and deg(w) are greater than or equal to 3 and that H is the graph obtained from G by subdividing e. When is M(H)=M(G)?

2. Find conditions on a graph G in order that M(G) be equal to a graph theoretic parameter Z(G) called the zero forcing number.

3. The inverse inertia problem for connected graphs that are not trees.

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