o Note: Prior knowledge of genetics is NOT necessary for this project. Genome-wide association (GWA) studies are an increasingly popular way to attempt to identify the genetic components of complex human diseases. In short, individual’s genotypes (AA, AB, or BB) are measured at thousands of locations across the genome. The distribution of genotypes for people with the disease of interest is compared to the distribution of genotypes for individuals without the disease of interest using standard statistical methods (e.g. chi-squared tests; trend tests; logistic regression). Strong differences in the distributions suggest that the genomic location is associated with the disease under study. GWA studies have successfully identified genes associated with diabetes, Crohn’s disease and Rheumatoid Arthritis, among many others. This summer we will investigate one or both of the following statistical research questions for GWA studies: 1. Traditionally, genotypes have been “called” for each sample presented for analysis. Calling an individual’s genotype means identifying individuals as “AA”, “AB” or “BB” for each location on the genome. New technology, however, assigns posterior probabilities of genotype assignment (e.g. 92%, 7% and 1% for the AA, AB and BB genotypes, respectively). We will explore the implications of using posterior probabilities instead of called genotypes and explore different statistical methods of using posterior probabilities in analysis. 2. New technology is available to predict an individual’s genotype at particular genomic locations, even when those locations are not measured directly. However, errors in these predicted genotypes can increase both the type I and type II error rates in related tests of association. We will explore how genotype errors are created and, subsequently, document how much type I and type II error rates increase as a result. Students will participate in addressing the research questions using a combination of mathematical proof, computer data simulation and real data analysis. Students will write-up results in journal article form in order to be submitted for publication in a recognized peer-reviewed statistical genetics journal.
Background: Students should have had at least 2 semesters of Calculus and at least one course in probability and/or statistics. Experience computer programming would be helpful though not essential. Prior knowledge of genetics is not necessary.
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Students in my research group will investigate questions at the
interface of algebra and topology. The exact nature of the project may
be more algebraic or more topological depending on interest, but
typically the problems we will explore will be algebraic questions whose
answers shed light on questions in topology.
Recent REU research projects have focused on representation theory. A
representation of a group produces matrices that correspond to the
elements of the group and obey the same relations as the group. Some of
these representations are basic in the sense that any representation of
the group is comprised of these basic representations, known as
irreducible representations. The irreducible representations are thus
analogous to prime numbers: just as any number may be decomposed into
its unique product of primes, any representation of a group may be
decomposed into its irreducible representations. We have developed a
geometric model for a class of groups known as metacyclic groups, and
from the geometric model we are able to obtain many, but not all,
irreducible representations of these groups. It is well known by those
who know it well that the number of irreducible representations of a
group equals the number of conjugacy classes in the group, but the exact
relationship between the conjugacy classes and irreducible
representations is not well understood.
There are several unanswered questions in this area that students might
explore this summer. First, can the model be modified in some way to
account for all irreducible representations? Second, can the model be
extended to other classes of groups? Third, can the model(s) be used
to establish a correspondence between conjugacy classes and irreducible
representations? Fourth, how much structure do the geometric models
incorporate? For example, how much structure in the geometric model is
preserved by Adams operations on the representation ring, for instance?
Background: a semester of linear algebra and a semester of
abstract algebra; additional coursework in algebra and familiarity with
computer algebra systems such as Maple are beneficial, but not necessary.
o This summer we plan to investigate graph pebbling problem. It is relatively new area of combinatorics that studies a game on a connected graph. Suppose that pebbles are configured on vertices of a connected graph G. A pebbling step consists of removing two pebbles from a vertex v and placing one pebble on a neighbor of v. A configuration is called solvable if r-solvable if it is possible to move at least one pebble to vertex r by a sequence of pebbling steps. A configuration is called solvable if it r-solvable for any vertex of G. The smallest integer number t such that any configuration of t pebbles is solvable is called the pebbling number of G. For example, for a complete graph on n vertices pebbling number is n.
More information on the pebbling problem can be found at math.hope.edu/bekmetjev/pebbling.
We will look at open questions and new ideas in this area such as critical pebbling, cover pebbling and optimal pebbling as well as algorithmic aspects of the problem. One of our goals will be finding pebbling solutions for various types of graphs using some specific graph properties, heuristics and random algorithms. We will also consider probabilistic models in pebbling.
Background: Students should have a background in combinatorics and/or graph theory. Some knowledge in programming, probability theory/statistics, familiarity with computer algebra systems (such as MAPLE) is a plus.
o There are many open questions dealing with the zeros and zeros of derivatives of real polynomials and functions of the form P(z)eQ(z) where P and Q are real polynomials. One such questions deals with the number of points of extreme curvature of a polynomial. Another asks about the number and location of non-real zeros of the derivatives of P(z)eQ(z). And finally, one asks about the relationship between the number of non-real zeros of a polynomial and the number of critical points of the logarithmic derivative of the polynomial. The second and third problems listed have roots in Pólya and Gauss respectively, so they are very old. This summer we will be looking at those three questions using a geometrical level curves technique. In particular, we will look at the level sets {z in H+:Im f(z)=0} and {z in H+:Re f(z)=0}.
Background: sophomore level linear algebra, multivariable calculus, some experience
with complex numbers, some familiarity with programming in Maple or Mathematica. Knowledge of real and complex analysis would be helpful, but not essential.
o During the Hope College REU in the summer of 2004, two undergraduates worked on the following problem: Given 3 distinct points in the plane P0, P1, P2, a parameter t in (0,1), and the rule Pk+3 = t Pk + (1-t) Pk+1, determine the length of the resulting spiral. In 2006, a second group of REU students at Hope worked on a generalization of the problem that begins with m points in the plane and classifies which starting configurations lead to spirals in which the lengths of the segments form a geometric series. Their work appears in the current Pi Mu Epsilon Journal. In particular, the students began to use some of the techniques of experimental mathematics as described by Borwein, Bailey, et al. For the current project, we'll generalize the problem to points in R3 and explore the length of the spiral and the location of its limit point. We may also consider some additional questions about the spiral in the plane.
Background: a semester of linear algebra, experience with infinite series and complex numbers, some familiarity with programming or computer algebra systems.
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