Prime graphs of solvable groups

This project is at the intersection of finite group theory and graph theory and will be supervised by Thomas Keller. There are several graphs associated with finite groups which are currently of interest in the field. The subject of this project are graphs related to the element orders of a finite group. The best known such graph is the prime graph Γ(G) of the group G and is defined as follows. The vertices are the prime numbers dividing the order of the group, and two such primes are linked by an edge if and only if their product divides the order of an element of the group. For solvable groups it is known that Γ(G) has at most two connected components, and the structure of the group in case of two components is well understood. Moreover, for solvable groups a purely graph theoretical description of the prime graphs is known. This has recently been extended to larger classes of groups. There are two main types of questions which are commonly studied in this context. First, how do restrictions on the graph influence the group structure? For example, groups whose prime graph is a tree have been classified. Second, how do restrictions on the group influence the prime graph? In this project, Keller will lead students to study some interesting questions of this type of various levels of difficulty. Of particular interest would be to obtain classifications of the prime graphs of some more classes of groups.

Depending on the ability and interest of the students, they can choose whether they prefer to work mostly on the group theoretical end or on the graph theory side of things. It is assumed that students have had at least one semester of Modern Algebra or Abstract Algebra. Explanations and background of additional topics, such as solvable groups and Frobenius groups, will be provided. The students will also have an opportunity to learn the computer algebra system GAP, which is useful in examining specific examples of groups that may be useful in making or confirming conjectures.

A Model-based Test for Treatment Comparisons based on Interval-censored Survival Data

   Interval-censored (IC) data occurs frequently in cohort studies, in which the survival time of an individual is only known to belong to an interval, (L, R]. To compare the survival distributions of several treatment groups based on IC data, various nonparametric test procedures have been developed including the (unweighted or weighted) generalized log-rank tests. Alternatively, regression approaches can be used to make such comparisons when covariates indicate treatments under some regression model (e.g., Cox proportional hazards model). In this project, we will be developing a new test procedure for the IC data under other regression models such as proportional odds model. Assuming no difference among groups, a test statistic will be proposed and estimated using the nonparametric maximum likelihood estimate of the common survival function. We will investigate the distributional properties of the test statistic and conduct a simulation study to evaluate the performance of the proposed test.

Chip firing on signed graphs and beyond

Chip firing game is a game played on a graph: imagine chips at each node of a vertex and each node can be ‘fired’ to give one chip to each of its neighbors. This simple game has surprising depth and has become an important part of the study of combinatorics and its related field over the last decade. It has beautiful connections to algebraic geometry, group theory, representation theory, and various other fields of mathematics, and also to physics and biology as well. Our main aim is to investigate and generalize the various known results for chip firing on a graph, to a bigger class of objects called signed graphs and beyond using the template provided by Kilvans-Guzman. Signed graphs are graphs where each edge is assigned a sign. This class of objects is also a deeply studied, fascinating object and has multiple interesting results in linear algebra and matroid theory. We will attempt to use various techniques in graph theory, linear algebra, and commutative algebra to extend the results known for chip firing on graphs to signed graphs and heavily rely on programming languages such as sage or python to experiment on various signed graphs.