### Bounded surfaces of negative curvature in Euclidean space

Faculty Mentor: Chris Connell

*Description:*In this project we will examine complete negatively curved surfaces living in Euclidean space. These are necessarily noncompact, but it is not known whether or not these can still be bounded in space. Back in the 70's Rozendorn produced an explicit surface which is bounded and negatively curved in a certain sense, but just shy of being twice differentiable (which is necessary for the Gaussian curvature to be defined). In fact, it is smooth and negatively curved except at a discrete set of points. This example was built from building blocks which cannot be made truly negatively curved. Since then, Vaigant produced a more complicated surface with similar overall shape to this building block, and which is truly negatively curved. This project will be focused on seeing if it is possible to use surfaces similar to the building block of Vaigant to make a bounded complete example.*Prerequisites:*Having taken an introductory course in differential geometry would be very helpful, but not absolutely necessary.## 2009 Research projects and faculty mentors

### Counting Involutions in Finite Groups

*Faculty mentor:*Allan Edmonds*Description:*How many elements of order 2 (called "involutions") can a finite group of order n have? Of course, if n is odd, then there are no elements of order 2, by Lagrange's theorem. On the other hand most algebra students have seen the elementary result that if every nontrivial element of a finite group has order 2, then the group is abelian, and therefore isomorphic to (Z/2)^r for some r. In particular the order of the group is a power of 2 in that case. In recent work coming out of an analysis of geometric properties of high dimensional simplices, in particular, their groups of isometries, I needed to know what the maximum number of involutions in a finite group of order n is, where n is necessarily even but not necessarily a power of 2. Although this seems like a natural question, the answer wasn't found in the standard literature. It turns out that I was able to prove that the number of elements of order 2 in a finite group of even order n is at most n/2 + (n_2)/2 - 1, where n_2 denotes the "2-part" of n. Moreover one can characterize the kinds of groups of order n for which this maximum value is achieved. For this project we'll first of all try to understand the idea of the proof of this result. At the same time we will explore possible generalizations. For example, it's clear that if a finite group of order n has the property that n - 2 elements have order 2, then indeed n - 1 elements have order 2 and the group is of the form (Z/2)^r. We'll try to look for the next highest number of elements of order 2 below the maximal number mentioned above. We will at a minimum collect data on the possible numbers of involutions by closely examining the known lists of finite groups of low order. We will expect to use the computer program GAP (at Wikipedia) as part of this study. Indeed we'll be satisfied if we can collect a good amount of data and propose some good conjectures.*Prerequisites:*A solid year of abstract algebra with a good emphasis on group theory, including, for example, the Sylow theorems. A willingness and an ability to download and install GAP (official site) and work with it.*Description:*Rapid sequencing of whole genomes provides the ordering of billions of DNA bases, albeit in short fragments. While several models are used as a basis for assembling these short sequences into larger assemblies (e.g. Lander-Waterman), even these assembled "scaffolds" are much shorter than whole chromosome arms. One approach to ordering the scaffolds along chromosomes is to simply use synteny with a closely related species that has a fully assembled genome, though these methods can only give an optimal ordering with no measures of confidence. For this project, we would like to develop an explicit model of the evolution of synteny that can then be used as the basis for assigning scaffolds to chromosomes along with measures of confidence in the ordering. The project will likely involve both modeling and data handling.*Prerequisites:*A background in probability and statistics is necessary, and some familiarity with stochastic processes is desirable. Some exposure to low-level programming (in any language) is also a plus.## 2009 Research projects and faculty mentors

### Phylogenies

Faculty Mentor: Elizabeth Housworth

*Description:*Phylogenies are trees that describe species evolution. The particular kind of phylogeny involved in this project is a labeled (by species names), rooted, time-ordered, bifurcating tree, with some known constraints (that certain sets of species form a clade, or sub-tree, within the overall phylogeny). Without constraints, generating random trees is a very simple matter. Similarly, if the bifurcations are not time-ordered, generating random trees with constraints is a simple matter. One method of generating random trees with constraints involves looping through three interdependent combinatorial objects which is highly inefficient for large trees. The goal of this project is to develop an efficient Monte Carlo Markov chain algorithm for generating these random trees with constraints.*Prerequisites:*Probability, programming. Some exposure to combinatorics would be helpful.## A Problem of Erdös concerning Lattice Cubes

Faculty Mentor: Nets Katz

*Description:*Erdös posed the following problem. Let A be a subset of {1, 2, ..., N}^3 (that is, a subset of an NxNxN grid). Suppose further that A does not contain the eight corners of a non-trivial box (that is, there do not exist x_1, x_2, y_1, y_2, z_1, z_2 with x_1, x_2 distinct, y_1, y_2 distinct, z_1, z_2 distinct such that {(x_i, y_j, z_k}_{i,j,k=1}^2 is contained in A).Erdös' question is how large A can be.

Erdös conjectured that for any\epsilon>0 there is a constant C_{\epsilon}>0 and for arbitrarily large N, a set A_N in an N \times N \times N grid, not containing the corners of a box with at least C_{\epsilon} N^{{11 \over 4} - \epsilon} elements. It is not difficult to see that much more than N^{{11 \over 4}} is impossible. But finding Erdös' conjectured examples has proved elusive. The best example known to date has about N^{{8 \over 3}} elements.

This known example is produced using linear algebra over finite fields and has a simple algebraic description. It is not clear whether larger examples with algebraic description are possible. This project will consist of a crash course in finite field theory followed by an attempt either to improve the known exponents by algebraic methods or to determine the limits of these methods.

*Prerequisites:*A really strong background in linear algebra will help.### Computational Methods and Models in Mathematical Biology

Faculty Mentor: Michael Lynch

*Description:*Depending on the student's interests, the project will involve:- 1) the development of statistical / computational methods for estimating (in an unbiased way as possible) levels of DNA sequence variation (and covariation) from whole-genome surveys that are now generating very high-throughput data, but erroneously so;
- 2) developing models for estimating the vulnerability of organisms to cancer with increasing levels of organismal complexity (i.e., more cell divisions); or
- 3) developing models for the evolution of various types of genomic elements in populations of various sizes (which influence the role of chance in evolution).
*Prerequisites:*A background in calculus, differential equations, and some probability theory is needed. Some familiarity with computer coding would be highly desirable.

### Multiscale Analysis and Deriving Principles of Nanobiology from Laws of Molecular Physics

Faculty Mentor: Peter Ortoleva

*Description:*The laws of molecular physics are summarized in the Liouville equation, a partial differential equation in 10^6 to 10^8 variables when describing a virus or other bionanosystem. The phenomena being investigated in our lab (e.g., the interactions of a virus with a host cell or structural transitions of viruses or intracellular organelles) involves processes from 10^-14 to 10 seconds and 10^-10 to 10^-7 meters. These systems are being analyzed via techniques that account for the separation of scales and the existence of collective variables which emerge from the many atom description. In this project, REU students are introduced to the theory of nanosystems and associated multiscale mathematical techniques. They then graduate to a specific example problem such as developing order parameters that describe viral structural transitions, the derivation of equations for their stochastic dynamics, and the development of numerical methods to solve these equations. They will also gain a knowledge of how this type of mathematics can forward the treatment of viral diseases, the design of nanocapsules for delivery of anticancer therapeutic agents, or the forecasting of viral pandemics.*Prerequisites: exposure to linear algebra, vector calculus, ordinary and partial differential equations*### Growth of the Airplane Group

*Faculty mentor:*Kevin Pilgrim*Description:*Suppose G is a finitely generated group and S is a finite generating set. An element g has length n with respect to S if n is the smallest positive integer such that g can be written as a a product s_1s_2...s_n, s_i in S. The growth function associated to G and S is the function phi(n) = #{ g : g has length n} which counts the number of elements of length n. While phi depends on S, the property of phi being asymptotically exponential or not is independent of the generating set used.The airplane group is a three-generator group of automorphisms of a rooted binary tree; it gets its name from a well-known fractal associated to a complex polynomial. It is not known whether this group has exponential growth or not.

*Prerequisites:*A solid background in group theory is required.### Isoperimetric Problems

Faculty Mentor: Matthias Weber

*Description:*The standard isoperimetric problem asks to determine the sets with fixed volume that have minimal surface area, and the solutions to this problem in Euclidean space are well known to be round balls. The same question can be asked in other Riemannian manifolds. Smooth boundaries of solutions are known to be constant mean curvature hypersurfaces. There are still many open problems, even for such simple spaces as 3-dimensional tori or the cube.Here is link to a survey article showing what current research is about:

*Prerequisites:*The project can be scaled to various levels, therefore none of the following are essential. They all will help, however, to communicate more easily with me:Analysis, Curves and Surfaces, Complex Analysis, Topology (both point set and a little algebraic topology including fundamental group and surface classifications), Linear Algebra, basic Riemannian Geometry (Riemannian metric, geodesics, Riemannian volume). Finally, experience with Mathematica might be helpful, too.

From left to right: Sharon Ulery, Jacek Skryzalin, Zachary Norwood, Adam Abegg, Zachary Doenges, John Brown, Ari Nachison, Joseph Thurman, Ziva Meyer, Leah Wolberg, Chengcheng Yang (not shown: Jamil Merali)