Project 1:: Geometry of Hyperbolic Percolation Clusters
On any infinite connected graph G we can play a game called "percolation" whereby for each edge we remove it with probability 1-p. Let p_c in [0,1] be critical connection parameter, namely the infimum of the values of p where there almost surely remains at least one infinite connected component of the graph. Percolation has been heavily studied in some settings such as lattice graphs in R^n, in part perhaps because of its connection to problems in physics and other areas.
We are interested in studying "expected" geometry of connected components under percolation, especially near a threshold value like p=p_c, on graphs that exhibit hyperbolic behavior such as infinite trees. (We expect to have no infinite components at p=p_c.) We are seeking to understand metric questions like the expected growth rates of balls in infinite components at supercritical values of p, the expected escape rates of certain paths and ways to quantify “shape.” One motivation for this is the search for algebraic invariants in the case that G is the Cayley graph of a group for a given generating set.
An introductory knowledge of probability would be very helpful, but can be learned. Some knowledge of group theory could be an asset as well.
Project 2:: Identifiability of models for phylogeneticsFaculty Mentor:
A common model for estimating a species tree from DNA data is the General Time Reversible Markov Model with rate variation modeled by a Gamma distribution with invariant sites. Recently, this model was proved to be identifiable using pairwise comparisons with a few exceptions. Identifiability means that given an infinite amount of data from this model, one can recover the model parameters uniquely. Software does not actually use the continuous Gamma distribution to model rate variation. Instead, it uses a discretized version of the Gamma distribution. The calculus that worked in the continuous case fails in the discrete case. This REU project will explore by computer simulations, analysis, and techniques developed in a recent paper for a different tree reconstruction model the identifiability of this model using discrete rates.
Project 3:: Searching for lonely pseudo-AnosovsFaculty Mentor:
Fundamental to the field of low-dimensional topology are the ways that a surface can be re-organized in a continuous fashion, the so-called self-homeomorphisms of the surface. The most prevalent self-homeomorphisms are the so called pseudo-Anosov self-homeomorphisms, those that if repeated over and over reorganize the subsets of the surface in such a way that nothing is recognizable. Amazingly, every pseudo-Anosov discovered so far comes with infinitely many `partners', pseudo-Anosovs which preserve the same notion of linearity. In this project we will be searching for pseudo-Anosovs that have no partners.
A penchant for visualization and computation will be needed, as will be knowledge of linear algebra and group theory. Experience with coding and topology will be very helpful.
Project 4:: Knotted Spheres in Four-Dimensional SpaceFaculty Mentor:
Most people know that a string can be tied in a circular knot in many different ways. In fact, this idea been rigorously studied mathematically for over 100 years. Many beautiful theorems have been proven in the area of knot theory; yet, perhaps surprisingly, there remain many more interesting and elusive questions to be answered. On the other hand, very few people have even considered the possibility of taking, instead of a string, a 2-dimensional sphere, and trying to tie it in a knot! Of course, this is very hard to visualize, because it is only possible to do this if one puts the sphere in 4-dimensional space.
It may seem very difficult to study such knottings of sphere, and it is, in fact! However, we can make it easier on ourselves by looking for connections between normal 1-dimensional string knots and these complicated 2-dimensional sphere knots. There are many known connections between these types of objects that have been discovered and studied over the years. In this project, we'll explore some of these connections. In the process, we will discover how knots, surfaces, and manifolds can be built using simple pieces, and we will see how studying simple objects like tangles and disks can give us deep insight into the mysterious world of knots and manifolds.
Knowledge of basic modern abstract algebra is preferred. Some familiarity with topology at the intuitive level would be helpful, but not required. An interest in drawing pictures of knots, disks, and tori while using lots of colored chalk, is a must!
Project 5:: Rubber Bands EnergiesFaculty Mentor:
Given two rubber band networks, when is one "looser" than the other? Intuitively, this means that however the two networks are stretched, the energy of the first is less than the energy of the second. In some cases, good criteria are known, but not in general.
In this project, we will investigate this problem experimentally and theoretically. From the experimental point of view, good programming experience is necessary, preferably in Haskell.
From the theoretical point of view, a good background in analysis (especially complex analysis) or topology is very helpful.
Project 6:: Curved PaperFaculty Mentor:
Have you ever wondered what really happens when you turn the page of a book? Classical theorems from the theory of surfaces tell us that a flat sheet of paper can be bent just into pieces of generalized cylinders and cones. Can we make a book out of paper that is already curved? What would the spines look like? This question was raised in the 19th century, and mathematicians even developed a theory of scrolls. Today, the theory of pleated surfaces in hyperbolic geometry offers tools to approach the same question. We will explore both theories to see what light they shed on each other.
Basic understanding of the theory of surfaces: First and second fundamental form, principal curvatures, theorem egregium. 2-dimensional hyperbolic geometry. Experience with Mathematica would be helpful.