### Negatively curved surfaces in Euclidean space

**Faculty Mentor:**

Chris Connell**Description:**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.### Recovering Trees via Parsimony

**Faculty Mentor:**

Elizabeth Housworth**Description:**

We will consider the following conjecture in phylogenetics by Victor Albert, modified by Mike Steel, further modified as follows: Consider a long sequence of nodes: A_{1}, A_{2}, ...,A_{n}. The nodes transmit a binary (0 or 1) signal. If A_{i}is 0 then A_{i+1}is 0 with probability (*1-p*) but A_{i+1}switches to 1 with probability*p*. Similarly, if A_{i}is 1 then A_{i+1}is 1 with probability (*1-p*) but it switches to 0 with probability*p*. You should think of*p*as being small but fixed while you should think of*n*, the number of nodes, as growing large.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.### Surfaces tiled by squares

*tentative***Faculty Mentor:**

Christopher Judge**Description:**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.### Harmonic functions and representation theory

**Faculty Mentor:**

Matvei Libine**Description:**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.### Codes and designs

**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.### Braids in mathematical physics

**Faculty Mentor:**

Will Orrick**Description:**

Certain models formulated on two-dimensional lattices that were originally devised to describe phase transitions or to serve as discrete analogs of quantum field theories turn out to have an amazingly rich mathematical structure, with connections to number theory and knot theory. Diagonalizing the transfer matrix associated with a model gives the energy levels of the physical system, which vary with the physical parameters of the system. The energy levels undergo a reorganization as the parameters pass through a critical point. In each phase, the energy levels can be labeled by elements of a braid group. The goal of this project will be to work out in detail the mapping between energy levels in different phases and their braid labelings.**Prerequisites:**

A course or two in linear algebra will be necessary. Some complex analysis may be helpful. We will be carrying out some computations in numerical linear algebra, so programming experience and interest in computational issues are desirable.### Distance formula in the Farey graph

**Faculty Mentor:**

Kevin M. Pilgrim**Description:**

The Farey graph is defined as follows. As vertices, take the rational numbers together with the point at infinity. Join infinity to each integer by an edge. Given p/q and r/s in lowest terms, join them by an edge if ps-rq = +1 or -1. Alternatively: it is graph you get by tiling the hyperbolic plane with ideal triangles. Question: given p/q and r/s, find a formula for the distance d(p/q, r/s) in this graph. A sketch has been given on mathoverflow.net; see http://mathoverflow.net/questions/33002/distance-formula-in-farey-graph . It would be very good to have (i) an airtight complete proof with a nice exposition, and (ii) a computer program, written e.g. in C or in a computer algebra system such as GAP. Certain complex dynamical systems lead to interesting maps from the rationals to themselves; such a computer program would enable us to test conjectures about the behavior of these maps.**Prerequisites:**

A first course in group theory and a little programming experience.### Computation of combinatorial expansion factors

**Faculty Mentor:**

Kevin M. Pilgrim**Description:**

Suppose you have a rectangle R_0 and a subdivision of R_0 into finitely many pieces, each of which is topologically a smaller rectangle and which is identified with R_0. Then you can subdivide each of the smaller pieces into still smaller rectangles, and continue. At the nth stage, consider the number D_n of rectangles in a connected chain joining a pair of opposite sides. Can one compute the growth rate of D_n in explicit examples? Such estimates lead to upper bounds on important numerical invariants of certain dynamical systems.

This is a very open-ended project which can head in many different directions.**Prerequisites:**

A course in linear algebra.### Constant mean curvature surfaces in R^3

**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.