##### Parallel session abstracts

**Stephanie Adamkiewicz**, Valparaiso University and Augustana College **Megan Cornett**, Valparaiso University and Indiana State University

*The Mosaic of an Integer *

The mosaic of an integer n is the array of prime numbers resulting from iterating the Fundamental Theorem of Arithmetic on n and on any resulting composite exponents. Bildhauser, Erickson, Gillman and Tacoma generalized several arithmetic functions and attempted to define divisors on mosaics. We continue their work by investigating alternative definitions of multiplication for mosaics.

**Michael Anselmi**, Indiana University and University of Rochester

*The evolution of mutation rates in a population *

For the sake of simplicity, results in population genetics assume that all individuals in a population harbor the same forward and backward genomic mutation rates. This assumption does not hold in reality. Relaxing this assumption, will a given initial population evolve such that as time tends toward infinity, an equilibrium distribution of mutation rate frequencies arises? The goal of this project is to obtain analytical approximations for such distributions.

**Stephen Bidwell**, Wabash Summer Institute in Mathematics and Tufts University **Matthew Hassell**, Wabash Summer Institute in Mathematics and Binghampton University

*A weighted least-squares method for elliptic problems with degenerate and singular coefficients *

We consider second order elliptic partial differential equations with coefficients that are singular or degenerate at an interior point of the domain. We present a formulation of a novel weighted-norm least-squares finite element method using adaptive mesh refinement based on functional minimization. Using asymptotic analysis, we categorize a family of singular solutions and propose a weighting scheme. Our goal is for this approach to minimize the pollution effect and recovers convergence rates. We provide numerical experiments within this context.

**Aaron Blumenfeld**, Rose-Hulman Institute of Technology and University of Rochester

*Discrete Logarithms on Elliptic Curves*

Elliptic curve cryptography exploits the hardness of the discrete logarithm problem, which is to solve $kB = P$ for $k$. It is assumed to be secure because of the belief that discrete "exponentiation" behaves like a random map -- given a random integer $k$, $kB$ should behave like a random point on the curve. In this talk, we investigate the discrete logarithm on elliptic curves mod $p$ for $p \geq 5$ by constructing functional graphs. In particular, we show that the graphs we produce are binary functional graphs and use theoretical and statistical techniques to analyze them.

**Kaitlyn Brady**, Wabash Summer Institute in Mathematics and Worcester State College **Allison Cullen**, Wabash Summer Institute in Mathematics and University of Detroit Mercy

*Numerical Modeling for Capillary Equations: The Small Angle Case *

Capillary action occurs when surface tension causes a fluid to rise along a wall which it has come into contact with; the Laplace-Young equation is a nonlinear elliptic partial differential equation that is used to model this. We are particularly interested in the situation of a domain with a geometric corner; in the small angle case the solution is unbounded and little is known analytically. Our research takes a computational approach. We design a way to linearize and reformulate the PDE, strategically design a mesh, and use the finite element method and least squares approach to approximate the exact solution. Our formulation is designed to produce a sequence of approximations that converge to the exact solution in the L1 norm.

**Jonathan Cain**, Rose-Hulman Institute of Technology and University of Arizona**Daniel Kamenetsky**, Rose-Hulman Institute of Technology and Hamilton College**Noah Lavine**, Rose-Hulman Institute of Technology and Haverford College

*Bilinear Programming and Protein Structure Alignment*

The three-dimensional structure of a protein largely determines its function, and proteins with similar structures are believed to have similar actions. Protein structure alignment compares the structure of a protein with known function to that of a protein with unknown function. We present a spectral method for protein alignment and discuss the associated bilinear program. In particular, we give conditions for a perfect alignment and heuristics for finding quality solutions.

**Joshua Cain**, Wabash Summer Institute in Mathematics and University of Dayton**Lindsey Mathewson**, Wabash Summer Institute in Mathematics and Carroll University**Amanda Wilkins**, Wabash Summer Institute in Mathematics and Beloit College

*Reduced Cozero-divisor Graphs of Commutative Rings *

Let $R$ be a commutative ring with identity. Afkhami and Khashyarmanesh introduced the cozero-divisor graph of a ring, $\Gamma'(R)$, where $x \in V(\Gamma'(R))$ if and only if $x$ is a nonzero nonunit of $R$, and $x$---$y$ if and only if $x \not\in (y)$ and $y \not\in x$. We introduce the reduced cozero-divisor graph, where the set of vertices is $\oo(R)$, the set of nontrivial principal ideals of $R$, and $(x)$---$(y)$ if and only if $(x) \not\subseteq (y)$ and $(y) \not\subseteq (x)$. We examine the basic graph-theoretic properties of the reduced cozero-divisor graph, its relations to $\Gamma'(R)$, and what properties of $R$ can be deduced from its structure. In particular, we examine how the graph reflects the decomposition of a finite commutative ring into a direct product of fields and local rings.

**Justin Chen**, Purdue University

*Models for Analytic Maps of the Disk to Itself or the Ball to Itself *

Every analytic (complex differentiable) map $\varphi$ of the unit disk into itself (that is not an automorphism of the disk) has exactly one fixed point $a$ in the closed disk for which $|\varphi'(a)|\leq 1$. A \emph{model for $\varphi$} is a map $\sigma$, like change of variables, that takes the disk into a domain $\Omega$ and $\Phi$ an automorphism of $\Omega$ such that $\sigma \circ \varphi = \Phi\circ \sigma$. In the decades following Koenigs' solution, in 1884, of the easiest case (fixed point $a$ in the disk) of this question, it has been discovered that for any $\varphi$ with $\varphi'(a)\neq 0$, there is a model with $\Omega$ either the plane or a half plane and $\Phi(w)=w+1$ or $\Phi(w)=sw$ for some number $s$ with $0<|s|<1$. Such a model leads to solutions of Schr\"oder's functional equation that are explicit as functions of $\sigma$. This question can be generalized to analytic maps $\varphi$ of the unit ball in $C^2$ into itself. Even though Koenigs' construction does not generalize to the several variable case, it has been shown that, for most $\varphi$ that have a fixed point $a$ in the (open) ball, $\varphi$ has a model with $\Omega = C^2$ and $\Phi(w)=Aw$ where $A=\varphi'(a)$. However, when $\|a\|=1$, there is little known about models for $\varphi$. This talk will discuss some of the known cases in one and two variables and discuss progress on questions in two variables.

**Kelly Collins**, Rose-Hulman Institute of Technology and University of Evansville**Andy Weaver**, Rose-Hulman Institute of Technology and Indiana University

*Minimum Energy Solutions for the Shape of a Slice of French Bread *

If you have ever wondered why baked goods obtain specific shapes, you are certainly not alone. The answer to this enquiry can be summed up in two words: energy minimization. A heuristic model for the shape of a slice of French bread, derived from the driving principle of energy minimization, has been formulated by past REU groups. Having inherited this mathematical model, we seek to understand under what conditions it provides minimum energy -- and, therefore, physically possible -- solutions.

**Michael Covello**, Valparaiso University and Loyola University, Chicago**Jill Jessee**, Valparaiso University and Simpson College**Matthew Zimmer**, Valparaiso University and St. John's University

*The impact of non-reproductive groups in two-sex demographic and epidemic logistic models without pair-formation *

Mathematical models in demography need to incorporate population effects in the mortality rates whenever they are used for long term predictions. Logistic two-sex models that include single females, single males and couples have been already analyzed in several recent papers. Those models were extended to an STD epidemic model that included non-reproductive groups. The couple formation/dissolution mechanism together with the influence of non-reproductive groups was proved to be essential in the spread and/or containment of the disease. In this talk we establish a logistic two-sex model without pairs and investigate whether similar or different results hold true in the new framework compared to those obtained in populations with stable couples. This approach is particularly relevant in the context of gender structured animal populations that do not form stable pairs except for reproduction.

**Robert Dovovan**, Wabash Summer Institute in Mathematics and Worcester State College**Paul Milner**, Wabash Summer Institute in Mathematics and University of St. Thomas**Abigail Richard**, Wabash Summer Institute in Mathematics and University of Indianapolis**Tristan Williams**, Wabash Summer Institute in Mathematics and University of Wisconsin at Eau Claire

*An exploration of ideal-divisor graphs *

Zero-divisor graphs have given some interesting insights into the behavior of commutative rings as seen in Anderson and Livingston, Axtell et al, and Cote et al. In his paper ``An Ideal Based Zero-Divisor Graph of a Commutative Ring'' Redmond introduced a generalization of the zero-divisor graph called an ideal-divisor graph. In our research, we expand on Redmond's work to see if additional information about the structure of commutative rings is hidden in ideal-divisor graphs.

**Nathan Dowlin**, Indiana University and Yale University

*A Geometric Approach to Determining Algebraic Properties of CAT(0) Groups *

We attempt to demonstrate that torsion subgroups of CAT(0) groups are always finite. This is equivalent to the translation subgroup having finite index, where the translation subgroup is the set of all elements having infinite order. We will discuss some background information, including the definition of a CAT(0) space, the concept of rough negative curvature, and delta-hyperbolic groups. We will conclude with our current approaches to the problem.

**Matthew Friedricksen**, Rose-Hulman Institute of Technology and St. Olaf College**Brian Larson**, Rose-Hulman Institute of Technology and Wheaton College **Emily McDowell**, Rose-Hulman Institute of Technology and University of Pennsylvania

*Theoretical Structure and Statistics of the Self-Power Map *

We investigate the structure of a cryptographic object called the self-power map, given by $x \mapsto x^x$ mod $p$, for $p$ a prime. As a variation of the Discrete Log Problem, the self-power map is thought to be difficult to solve in the inverse and therefore considered safe for use in some versions of the ElGamal digital signature algorithm. Nonetheless, utilizing functional graphs to represent the map has revealed non-random structural properties, which we describe primarily through number theory and statistics.

**Nathan Gabriel**, Valparaiso Univesity and Rice University **Katie Peske**, Valparaiso University and Concordia College**Sam Tay, **Valparaiso University and Kenyon College

*Pattern avoidance in ternary trees*

In recent years, pattern avoidance has proven to be a useful tool to describe connections between different combinatorial structures. In this talk, we consider the enumeration of ternary trees avoiding a contiguous ternary tree pattern. We use recurrence relations and generating functions to count the number of n-leaf ternary trees avoiding a given pattern. We then look at bijections between sets of trees with the same avoidance sequence. This work extends Rowland's research in binary trees to a much larger class of trees.

**Hamza Ghadyali, **Indiana University and University of Michigan

*A Geometric Approach to Determining Algebraic Properties of CAT(0) Groups *

We attempt to demonstrate that torsion subgroups of CAT(0) groups are always finite. This is equivalent to the translation subgroup having finite index, where the translation subgroup is the set of all elements having infinite order. We will discuss some background information, including the definition of a CAT(0) space, the concept of rough negative curvature, and delta-hyperbolic groups. We will conclude with our current approaches to the problem.

**Monica Grigg**, Rose-Hulman Institute of Technology and Colorado School of Mines email

*g-Lattices for an Unrooted Perfect Phylogeny *

In this talk we look at the Pure Parsimony problem and the Perfect Phylogeny Haplotyping problem. From the Pure Parsimony problem we consider structures of genotypes called g-lattices. These structures either provide solutions or give bounds to the pure parsimony problem. In particular, we investigate which of these structures supports an unrooted perfect phylogeny, a condition that gives a solution additional biological interpretation. By understanding which g-lattices support an unrooted perfect phylogeny, we connect two of the standard biological inference rules used to recreate how genetic diversity propagates across generations.

**Eric Haengel**, Purdue University

*Fractals, complex dynamics, and Newton's method*

Complex Dynamics is the subject which deals with so-called fractal sets in the complex plane. These sets exhibit a very broken and complicated geometry, and one might be surprised to find that it is actually very easy to stumble across them. As an example, Newton's method is a practical algorithm for finding roots of a given function $f(z)$. If one knows that a given point $z_0$ is fairly close to a zero of $f(z)$, then Newton's method will always converge to this zero, and similarly all points in a neighborhood of $z_0$ will also converge to this zero. This gives a notion of an Attractive Basin for each zero of a function, and it turns out that the boundary of this basin is in general a complicated fractal set. In my talk I will discuss Newton's method for polynomials in relation to Complex Dynamics, and describe possible pitfalls in using it as a root-finding algorithm. Expect pretty pictures and explanations of their structure. I will also discuss the work of John Hubbard, Dierk Schleicher, and Scott Sutherland, and how they managed to remedy the situation.

**Cassandra Hall,** Rose-Hulman Institute of Technology and University of Michigan

*Sessile Drops Having Periodic Boundary *

If you've ever seen a close-up photograph of a water droplet on a table, you know the form of a basic sessile drop. In that case, we see a nearly-spherical sessile drop coming into contact with its solid substrate on an almost circular region. But suppose we have a sessile drop with its base a region having a very different shape. Can we reconstruct the entirety of the drop, and if so, what does it look like?

**Elizabeth Kammer**, Indiana University and University of Alabama

*Natural Logic with Adjectives *

In natural logic, the goal is to create a system of logic that is as similar to natural language as possible. In this talk, we'll consider first very simple fragments of the form All X are Y and then add No X are Y, where X and Y are nouns. We will look at the rules of logic and completeness of these systems. Next, we will introduce the idea of interesecting adjectives which are adjective whose meaning is separate from the word it is modifying. For example, red cars means those things that are both red and cars. Finally, we will consider the rules and completeness of our earlier systems now with intersecting adjectives added.

**Bridget Kraynik**, Wabash Summer Institute in Mathematics and College of Wooster**Yifei Sun**, Wabash Summer Institute in Mathematics

*Multiscale adaptively weighted least-squares finite element methods for convection-dominated elliptic PDEs *

Our research considers a least-squares finite element method approach to solving convection-dominated elliptic partial differential equations, which are difficult to approximate numerically. This process uses adaptive mesh refinement and also in conjunction uses an iterative approach that adaptively adjusts the least-squares functional norm. In our talk we will present the basics of the algorithm we use and discuss the results we have from our research.

**Hongshan Li**, Purdue University

*Proof of Liouville's Theorem *

Liouville's Theorem states that if $u$ is a harmonic function and is bounded from below on $R^n$, then $u$ is a constant function. In my presentation, I will prove this theorem by Harnack's inequality. Before going into the proof, I will introduce some preliminaries including Laplace operator, harmonic function, and Poisson kernel.

**Komi Messan**, Indiana University and North Carolina Agricultural and Technical University

*Average time until fixation of a mutant allele in a given population *

A mutant gene in a given population will eventually be lost or established. the particular interest of this research is to know the mean time for a mutant gene to become fixed in a population and we will exclude the case when this gene is lost. A population of N individuals will be considered with a forward and backward mutation of u and v respectively per bases. Using a set of nonlinear equation, we will calculate the genotype frequencies which will allows us to find the equilibrium points for the infinite population.

**Hayley Miles-Leighton**, Indiana University and University of California San Diego

*Complex Dynamics and the Smale Mean Value Conjecture *

Let f(z) be a complex valued polynomial and let c be one of its critical points. The Smale Mean Value Conjecture (SMVC) states that |f(c)/c| < 1. In this talk, I will show that this holds for the cubic polynomial by looking at its parameter plane (which is defined by c). However, what if we were to look at where the iterates of each c converge? Are there any correlations between SMVC and convergence? With the aid of pictures, I will show where this holds and where we are still interested in looking into, as well as how we plan to find out what happens in our mystery area.

**Garrett Proffitt, **Indiana University

*Classifying Periodic Tilings with a Single Tile of the Real Line and Realizing the Deformation Spaces through Dual Graphs of Two and Three Square Periodic Tilings of the Plane *

The real line has not been studied much with regards to tiling, offering an opportunity to classify and name the different types of tilings one can obtain. Though the underlying mathematics are simply permutations and modular arithmetic, the pictures and the tiles one can obtain through real line tiles can be quite fascinating. Even more fascinating is the deformation spaces of periodic square tilings in the plane. The topological objects these spaces create can be quite interesting in the many ways they touch and intersect. For example, some of the components include two-dimensional tori, three-dimensional tori, and loops, and the ways these objects link up is quite amusing. The three square case is very complex and intriguing, but even the two square case is enjoyable to study and observe.

**Anika Rounds**, Purdue University

**Linh Truong**, Indiana University and University of Pennsylvania

*Finding Intersections of Schubert Varieties Using Combinatorial Measures *

Given a measure on a lattice with a finite number of branch points, there is an associated intersection problem: Can we find a vector subspace in the intersection of the three Schubert varieties determined by the measure? This problem can be solved explicitly by examining the exit densities of the measure. The goal of this talk will be to formulate the intersection problem associated with a measure, and then to construct a solution using properties of the measure.