## Northfield Undergraduate Mathematics Symposium 2013

The Northfield Undergraduate Mathematics Symposium is an annual event sponsored jointly by Carleton and St. Olaf. In 2013 ten students spoke at the symposium on research they did around the country over the summer. The pictures at the right show the speakers in action, and below are the titles and abstracts of their talks, and in some cases links to their slides.

### Volumes of Hyperbolic Knot Complements

**Martin Bobb (Carleton)**

A knot is an embedding of a circle in a sphere. Many knots, including 2-bridge knots, have complements with a hyperbolic structure determined by the knot. We explore the hyperbolic volumes of knot complements for 2-bridge links obtained by Dehn fillings. We build on work by Purcell to more accurately explore how the geometry of universal covers and gluing operations affect hyperbolic volumes.

Hyperbolic Knot Complement Slides (Please note: these do not display correctly in Mac Preview.)

### An Analysis of the Trojan Y-Chromosome Method of Invasive Species Management

**Jared Brown (St. Olaf)**

Management of invasive species towards the goal of preserving native biodiversity and preventing economic damage has traditionally been one of the most challenging problems faced by modern ecological scientists. The introduction of modified members of the invasive species, carrying extra trojan y-chromosomes, may o er a much less harmful, and thus less expensive technique for controlling or eliminating wild populations of unde- sired, sexually reproducing organisms. This paper (talk) presents both deterministic and Stochastic models of the reaction of the wild population to such trojan introduction. Results upon arbitrary species parameters support the potential validity of this technique, and give insight into more environmentally specific interactions.

Invasive Species Management Slides

### Digraphs, Zero Forcing, and Maximum Nullity

**Cora Brown (Carleton) and Nathanael Cox (St. Olaf)**

A simple digraph, Γ, is a set of vertices and arcs whose elements are ordered pairs of vertices. The zero forcing number for a digraph Γ is the minimum number of blue vertices needed to force all the vertices of Γ to become blue by the color change rule. This rule states that for Γ with all vertices colored blue or white, a blue vertex,v, can force an adjacent white neighbor, w, to become blue if w is the only white out-neighbor of v. The maximum nullity of a digraph, M(Γ), is the maximum nullity of any of these matrices described by Γ. We will present results on maximum nullity, zero forcing number, and other properties of digraphs including techniques for finding the minimum rank of digraphs, results for oriented graphs (digraphs that allow no doubly directed arcs), and results for directed graphs in general.

Digraphs, Zero Forcing, and Maximum Nullity Slides

### Finite Dynamical Systems: A Probabilistic Approach

**Joey Dickens (St. Olaf) **

Since the behavior of large finite dynamical systems is difficult to observe and characterize, we approach these number theoretic objects from a probabilistic point of view. By doing so, we develop expectations about large random finite dynamical systems. We find and prove several explicit and asymptotic formulas describing the growth of the set of periodic elements as set size becomes arbitrarily large. We conclude with several additional conjectures concerning the asymptotic behavior of the set of periodic elements when we have a d-to-one function.

### Non-Mass Action Modeling for the Binding of Phosphorylated Gli1 with SUFU

**Taisa Kushner (St. Olaf)**

Our goal in this research project was to construct a mathematical model to accurately simulate Gli1-Erk2- SUFU interactions that were observed biologically. Gli1 is a protein associated with the Hedgehog (Hh) signaling pathway, which is involved in embryonic development and stem cell differentiation. Overexpression of the Gli1 protein has been linked to many cancers, most notably glioblastoma multiforme (GBM), the most common and aggressive brain tumor. To create our model, we proposed the novel incorporation of Holling Type-II interactions from ecology into a biochemical model constructed using differential equations and a multiple-time- scale system. We compare this with other models, such as the mass-action protein interaction model and a Gli1-dimerization model and show that these are insufficient for explaining the observed dynamics.

Non-Mass Action Modeling Slides

### Cayley Graphs and the Cayley-Isomorphism Property

**Greg Michel (Carleton)**

For a finite group *G* and a subset *S* of *G*, the Cayley graph *Cay(G,S)* is the graph whose vertex set is *G* such that two vertices *x* and *y* are adjacent if *x ^{-1}y* is in

*S*. A Cayley graph

*Cay(G,S)*is called a CI-graph if for any

*Cay(G,T)*graphically isomorphic to

*Cay(G,S)*, there is a corresponding group automorphism

*σ*of

*G*with

*σ(S)=T*. A finite group

*G*is called a CI-group if every Cayley graph of

*G*is a CI-graph. We show that

*G*is not CI if it admits a non-CI subgroup or if it admits two non-isomorphic subgroups of the same order. We show that this completely classifies the effect of subgroups on non-CI Abelian groups. That is, if an Abelian group is non-CI and its subgroups do not meet the conditions above, then any non-CI graph will be connected.

Cayley Graphs and the Cayley-Isomorphism Property Slides

### Difference Set Transfers

**Dylan Peifer (Carleton)**

Given a finite group *G* of order *v*, a subset *D* of *G* is called a *v,k,**λ*-difference set if *|D| = k* and the set *{d _{i} d_{j}^{-1} | d_{i} ,d_{j}*

*ε D}*contains λ copies of each nonidentity element of

*G*. The fundamental question in the study of difference sets is determining which groups contain difference sets and which do not, and a related question involves finding all difference sets in a group or set of groups. Though exhaustive search can easily determine all difference sets in groups of order 16, there are still many patterns to be found in what we term

*difference set transfers*- where a difference set in one group of order 16 can be transferred to a difference set in a different group of order 16 using power-commutator presentations of these groups. In this talk we will examine and prove many of the difference set transfers found in groups of order 16 and apply transfers to groups of order 64 and 144, where exhaustive search is infeasible and other standard methods for finding difference sets fail.

Difference Set Transfers Slides

**Evaluating an Adaptive Clinical Trial with Quantitative Endpoints, Sample Size Re-estimation, Sequential Monitoring for Efficacy, and Monitoring for Futility**

**Harrison Reeder (Carleton)**

Clinical trials are an essential component to modern evidence-based medicine, and biostatisticians are continually developing new trial designs to maximize their safety, efficiency, and value. This research focuses on exploring the characteristics of a particular Phase II trial design with three key properties: adaptive sample size recalculation, interim monitoring for efficacy, and monitoring for futility. Using a simulation study, we evaluate the performance of the design and compare the value of trials with some or all of these characteristics. Our comparison measures include the accuracy of the designs' estimation of treatment effect, the error rates of the trials, as well as the robustness of the designs to inaccurate assumptions about the treatment. Our research also compares the merit of three different interim monitoring schemes in this design, observing that O'Brien-Fleming boundaries are most suitable. Our overall findings conclude that compared to a simple design with interim monitoring for efficacy, the complete design is an improvement; our design shows comparable performance under most conditions and improved overall performance under conditions where initial design estimates are inaccurate.

Adaptive Clinical Trial Slides

**Nos****é****-Hoover Thermostats**

**Lora Weiss (St. Olaf)**

The equilibrium statistical properties of molecular systems is important to applied subjects such as biology, chemistry, computational physics and materials science. These equilibrium statistical properties are obtained as phase space integrals that depend on q as the position of the system, p as the momentum of the system, and have H(q,p) as the total energy of the system. In 1984 S. Nosé introduced a thermostat to mimic the effect of a heat bath on a mechanical system. W. Hoover simplified this model and showed that even for the simple harmonic oscillator the system can exhibit complicated dynamics. In this presentation, we attempt to find exact solutions of the Nosé-Hoover thermostat; we look for periodic solutions to make a conjecture about the existence of invariant tori; we determine orbit averages along the solution curves; we analyze various numerical methods to solve the system; and we look at the existence of a first integral for the system, building on the work of Legoll, Luskin, and Moeckel.