There a number of interesting open questions about negatively curved surfaces embedded in R^3.
For instance, we may investigate the possibility of the existence of complete (noncompact) negatively curved surfaces which are embedded in a ball of R^3. This is an attempt to generalize a construction of Rozendorn for nonpositively curved surfaces in the 60's. Other questions of interest involve the relationship between topology and degree of negatively curved algebraic surfaces, and the existence of certain negatively curved "hedgehog" surfaces. These questions have the advantage of being both accessible and visualizable.

Prerequisites:
A solid multivariable calculus course is essential. A course in the differential geometry of curves and surfaces, and a little experience with a computer algebra system such as Mathematica, would be useful.

Now suppose you don't know the order of the nodes, you only see the signal that the nodes transmit. And, in fact, you can see as many signal sets as you would like. Can you recover the order of the nodes (with high probability) simply by minimizing the number of changes that occur in the signals that you see?

If you solve this simplified version of the problem, the real problem is to do the same thing for any tree shape, not just the particular tree shape described above. The above tree is called a comb or caterpillar phylogeny. It is an extreme tree shape. The other extreme shape is the symmetric phylogeny, which is the shape to tackle next. Other phylogenies are, in some sense, a combination of these two extreme shapes. 

Prerequisites:
Previous experience with probability and discrete mathematics might be helpful. Some programming experience might also be beneficial.

One can construct surfaces that approximate any conformal type by gluing together n squares whose sides have length equal to one. In particular, one need only consider gluings where the top of a given square is identified with the bottom of one other, and the left side of a given square is identified with the right side of one other. Labeling the squares 1 through n, we see that the gluing pattern corresponds to a pair of permutations of {1,2,3, ..., n}, one corresponding to the vertical identifications and one corresponding to the horizontal identifications.

How does this pair of permutations reflect the topology and geometry of the constructed surface? For example, given the pair of transpositions, how many connected components does the surface have? What is the genus of the surface? What are the symmetries? If pairs of permutations are multiplied pairwise, how does the geometry and topology change? These are the first of many questions that will be explored.

Prerequisites:
A first course in linear algebra. Some familiarity with permutations and elementary topology would be helpful.

Representation theory studies symmetries and plays an important role in many different areas of mathematics and theoretical physics. You will learn about Lie groups and their representations. Important examples of Lie groups are GL(n) - the group of all invertible nxn matrices, O(n) - the group of all orthogonal nxn matrices and U(n) - the group of all unitary nxn matrices. A representation of a Lie group is its action on a vector space by linear transformations.

It turns out some classical results from analysis can be proved in new ways using representation theory; while not exactly new, published, accessible proofs are scarce. Students working on this project will work towards self-discovery of some of these results.

Prerequisites:
Linear algebra and an algebra course on groups, rings and modules are necessary. It would be useful for students to have had some exposure to harmonic functions, fractional linear transformations (usually taught in a complex analysis course) and tangent vectors, tangent spaces (usually taught in a multivariable calculus course). But these topics can be learned in the process.

Faculty Mentor:
Will Orrick 

Description:
Error-correcting codes, in addition to their many applications, are intimately related to some intriguing structures in mathematics. The famous binary Golay code, for example, is connected with the Leech lattice (which provides an extremely efficient packing of balls in 24 dimensions) and with some of the sporadic finite simple groups. A certain subset of the codewords in the extended Golay code can be used to form a 24 by 24 matrix of -1s and 1s that has orthogonal rows (a Hadamard matrix) and that, as a consequence, has maximal determinant among all {-1,1} matrices. We will explore the connection between codes and maximal-determinant matrices in various dimensions with the goal of classifying such matrices, and possibly discovering new examples.

Prerequisites:
A course or two in linear algebra will be necessary. Some knowledge of group theory, or a desire to learn, would also come in very handy. We may do some calculations using computer algebra packages such as Mathematica or Sage, so interest in computation is desirable. Prior experience with a programming language is a definite plus.

Faculty Mentor:
Bruce Solomon 

Description:
"Constant mean curvature" (CMC) is the condition satisfied by surfaces in space that (locally) minimize surface area while bounding a given volume. The simplest CMC surfaces are spheres, but there is a family of undulating CMC surfaces of revolution named after C. Delaunay, who, in 1841, showed that the profile curves that generate these surfaces are all gotten by rolling an ellipse, parabola, or hyperbola along the axis of revolution.

Very recently, Oscar Perdomo has shown that an entirely different, "helicoidal" family of CMC surfaces called "twizzlers" can also be generated by a rolling construction. Perdomo's construction is totally different than Delaunay's, yet Perdomo shows that the twizzlers and Delaunay surfaces all fit into a single, continuous family. The family is continuous, but the rolling construction goes through a puzzling abrupt phase transition.

The question for this project: Is there some associated "continuous" way of deforming Delaunay's rolling construction into Perdomo's?

We would approach the problem by studying both Delaunay's and Perdomo's construction, perhaps doing some computer graphic visualizations using Mathematica, and trying to find the hidden common thread that I suspect connects the two constructions.

Prerequisites:
Basic multivariable calculus and differential equations. Exposure to a first course in Differential Geometry would be a big plus, but not a prerequisite. Same for experience using Mathematica or the like, especially with regard to graphics.