Research Archives

2011 Research

# Participating Faculty in Summer Research 2011

**Preston Aldrich, Ph.D.**- Research will involve the study of complex systems through the use of networks. Some of the research is computer-based, though no prior knowledge of programming languages is required. Methods involve the acquisition of open source data (such as through genomics databases), the representation of the data structure as one or more networks (involving nodes and links between nodes), and the analysis of network structure using established software programs and mathematical graph theory (no prior knowledge in these areas is assumed either). Model systems will vary and depend on the specific question being asked, but ideally involves an overlap between student and faculty interests. Two possibilities this summer include (A) network analysis of gene regulation systems in prokaryotes and eukaryotes and/or (B) collaboration with a co-advisor from the math department (Dr. Comar) involving the application of epidemiological models to problems involving invasive species. This latter project would involve some network applications but also would assume some familiarity with calculus.**Timothy Comar, Ph.D.**-**1. Hextile Knot Mosaics:**A hextile knot mosaic is an embedding of a knot diagram in a regular hexagonal grid. We are interested in investigating the following questions: Minimality Issues: There are several different notions of minimality for hextile knot mosaics. These include the minimum number of tiles needed for a hextile knot mosaic for a particular knot, the smallest region (of a couple of standard types) in which a hextile knot mosaic can be embedded for a knot, and hextile knot mosaics that cannot be contracted via planar isotopy. We are interested in determining the quantities for knots of small crossing number and knots in particular families. We are also interested in determining relationships between these notions of minimality and classical knot invariants. Hextile Moves: We are interested in classifying combinatorial moves between different hextile knot mosaics of a particular knot. These moves either preserve the underlying combinatorial structure of the knot diagram (planar isotopy moves) or moves that change the underlying combinatorial structure of the knot diagram without changing the knot type (Reidmeister move). We first need to classify these basic moves. We then can consider the planar isotopies and different embeddings of a particular knot diagram within a particular region in a regular hexagonal grid.**2. Mathematical Biology:**I am interested in studying basic ecological and epidemiological models using differential equations and studying gene regulatory networks. One current project involves the study of integrated pest management models using impulsive differential equations. Another project involves the modeling of gene regulatory with both Boolean (qualitative, discrete) models and differential equation (continuous, quantitative) models. The work requires a combination of computer simulation, modeling, and mathematical techniques. Experience Needed: Preferred: Math 221/208 Biocalculus II or Math 260 Differential Equations.**3. Regular Stick Knots:**An alpha- regular stick conformation of a knot is an embedding of a knot in space consisting of straight edges such that each has the same length and each pair of adjacent edges meets at an angle of alpha. Such conformations can be used for modeling certain types of molecules. The alpha-regular stick number of a knot is the minimal number of edges required to embed in the knot in space as a an alpha-regular stick conformation. Previously, we have determined this number for a couple of knots. (See Comar's homepage for pictures!) We are interested in determining this number for other knots and for families of knots for particular values of alpha. We would also like to find relationships between this quantity and other knot invariants. Experience needed: Mathematical maturity of at least the Calculus I/Biocalculus I level.**Anthony DeLegge, Ph.D.**- An Epidemic Model with a Multi-Stage Vaccination. Last year, one of the biggest news stories was the H1N1 flu pandemic. Besides the fear about how quickly it was spreading and the potential for it to become a very lethal disease, there was also a fear that, when a vaccine for it was discovered, it would be a two-step vaccination. That is, it would require two separate shots in order to be sufficiently immunized. Because this would effectively cut a nation's vaccine resources in half, it becomes important to decide the best strategy to immunize the population: Is it best to get as many people started on the vaccination schedule as possible, knowing some won't finish (ex. they forget to get the second one), or to only work with those who are guaranteed to finish the program? This project's goal is to build an epidemic model to answer this question. We'll initially start with a two-stage vaccine and then, if time permits, look at an n-stage vaccine.**Jeremy Nadolski, Ph.D.**- Detecting multivariate outliers when presented with a singular covariance matrix.**Peter H. Nelson, Ph.D.**- As part of an NSF grant I'm developing new biophysics teaching materials. Topics include: oxygen, water, glucose, ion and drug transport; ion channel gating (neuroscience); motor proteins; DNA and RNA dynamics etc... I'm looking for research students to determine the current state of knowledge and to find the numerical data required for biophysical models. Right now I'm looking for students to help me investigate water transport through aquaporins. In the process, we learn some basic physiology, including osmosis and homeostasis of erythrocytes (red blood cells) etc... from a biophysics perspective! There are many other topics available, for more information visit the project web page http://circle4.com/biophysics- or contact Dr. Nelson directly.**Robin Rylaarsdam, Ph.D.**- McCune-Albright Syndrome is a genetic disease caused by a mutation that permanently activates the Gs alpha protein in cells. Previous work in the laboratory identified a secondary mutation in the Gs gene that can reverse the permanent activation. Summer students will systematically introduce other amino acid substitutions into this secondary site to investigate the structural requirements for inactivating the McCune-Albright mutation. The work will involve site-directed mutagenesis, mammalian cell culture and transfection, western blotting, ELISA assays, and microscopy.**LeeAnn Smith, Ph.D.**- Evaluation of microarray datasets using BRB arraytools and Tigr suite software.**Kari Stone, Ph.D.**- Synthesis of Transition Metal Complexes with Redox-active Ligands for Catalysis."**Monica Tischler, Ph.D.**- Molecular characterization of bacteria from environmental sources.**Ellen Zilliak, Ph.D.**-**Solving Sudoku**- Sudoku is a logic-based, combinatorial number-placement puzzle. The game has become relatively popular in recent years. It began as a puzzle that was published in newspapers and currently Sudoku software is very popular on PCs, websites, and mobile phones. There now exists a computer program which will produce the puzzles quickly. However there is still an unsolved question associated with this puzzle, what is the number of the fewest givens that render a unique solution to the puzzle. Even though this is an unsolved problem there has been some experimentation it has been conjectured that the number without a symmetry constraint is 17. I want to look at a smaller version of the Sudoku puzzle with a 4x4 grid with 2x2 regions. In this situation it might be easier to model using graph theory the solution to this problem. This would then give us a technique that might be used to solve this larger problem.**T****he Algebra of Rewriting**- In mathematics, one method of defining a group is by a presentation. Every group has a presentation. A presentation is often the most compact way of describing the structure of the group. However there are also some difficulties that arise when working with groups in this form. One of the problems is called the word problem which is an algorithmic problem of deciding whether two words represent the same element. I want to study the word problem on group extensions. To solve this problem in a group extension G we will need to know how to solve the word problem in a normal subgroup N and a quotient group G/N. This information along with information about how these groups stick together to make G is enough to solve the word problem in this case. Unfortunately this information about how these groups stick together can be difficult to obtain. In general if G is a group of order n, then there is a set of n3 equations to solve this problem. However, these are not linear equations, as the size of the group increases this problem becomes difficult to solve. Therefore we will instead restrict ourselves to a smaller class of presentations to those which have a confluent rewriting system. In this case we should be able to reduce the number of equations needed to determine how to glue the two groups together to form G.