Summer 2012 Projects
Quandles and Generalized Colorings of Knots
Dr. Paul Drube
Knot theory is a branch of topology that studies how one may embed a circle within three-dimensional space. Since knots may become quite complicated, much of modern knot theory involves the development of invariants that can distinguish between different equivalence classes of knots. One basic class of knot invariants are the Fox n-colorings of knots, which count the ways that one may appropriately “decorate” the strands of a knot diagram with elements of a cyclic abelian group. In recent years, the mathematical structure underlying these Fox n-colorings has been greatly generalized utilizing abstract binary operations known as quandles. After a brief introduction to knot theory, this project will explore basic properties about quandles and their associated knot invariants, hopefully developing interesting results about the generalized colorability invariants of specific classes of knots.
Prerequisites: linear algebra; an introductory proof course or basic abstract algebra (group theory) may also be helpful
Vertical transmission in two-sex epidemic models with isolation from reproduction
Dr. Daniel Maxin
Vertical transmission is the transmission of an infection from mother to child at birth. During the academic year 2008-2009, Tim Olson and Adam Shull (students at Valparaiso University) worked on a project with me to study the influence of isolation from reproduction on sexually transmitted infections with vertical transmission. One of the questions addressed in their project was the behavior of the disease in the case of 100% vertical transmission (i.e. all newborns from infected mothers acquire the disease). A surprising result showed that, under several conditions on the vital parameters, the isolation from reproduction may prevent a Susceptible Extinction situation (a case when the disease eliminates the entire healthy population) even when the infection rate is large. For simplicity the model did not separate the individuals by gender (to keep the number of equations low). Our project consists in establishing a two-sex version of this model that follows the dynamics of females and males.  The main objective is to investigate if the result described above holds in the more realistic frame-work of gender structured models and whether the result is independent from various types of mating functions commonly used in the literature. A secondary objective is to improve the original model with features that were not included so far such as: diseases with recovery or temporary isolation from reproduction.
Prerequisites: Differential Equations
Generalized Pattern Avoidance in Trees
Dr. Lara Pudwell
A current hot topic in combinatorics is enumerating structures that avoid certain patterns. In the 2010 VERUM program, Gabriel, Peske, and Tay studied contiguous pattern avoidance in ternary trees. In the 2011 VERUM program, Dairyko, Tyner, and Wynn studied non-contiguous pattern avoidance in binary trees. There are a number of variations on tree-pattern avoidance that remain to be explored by modifying what types of trees or what types of patterns are considered. We will consider a new type of tree pattern that is a hybrid of contiguous patterns and non-contiguous patterns. In particular, counting problems in the 2010 project always resulted in algebraic generating functions while counting problems in the 2011 project always resulted in rational generating functions. We will try to characterize whether hybrid patterns fall under the algebraic or rational paradigm and extend our work from there.
Prerequisites: linear algebra, or another proof based course; a course in combinatorics, discrete math, or elementary graph theory would be helpful.