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*This page contains a summary of the research and publications that have resulted from the GVSU REU, from 2000 to 2019. Note that there may also be papers currently submitted or forthcoming that are not listed here.*

**For information on past REU participants, please visit our alumni page. **

2019 ~ 2018 ~ 2017 ~ 2016 ~ 2015 ~ 2014 ~ 2013 ~ 2012 ~ 2011 ~ 2010 ~ 2009 ~ 2008 ~ 2007 ~ 2006 ~ 2005 ~ 2004 ~ 2003 ~ 2002 ~ 2000

**Random Sample Voting:**(Caitlyn Patel, Rachel Roca, and Professor Jonathan Hodge): This team studied random sample voting (RSV), a voting method in which a group of randomly selected voters, rather than the entire electorate, cast ballots to decide the election outcome. They specifically investigated what sample sizes would be needed to produce representative election results, considering both theoretical and simulation-based results, as well as analysis of data from actual elections. While the minimum necessary sample size depends on a number of factors, including closeness of the election as measured by a variety of metrics, the vast majority of elections studied could be effectively decided by a group of 1000 or fewer voters—small enough to fit in an auditorium and engage in educational or deliberative processes prior to voting.**Degrees of Nonseparability:**(Jacob Roth, Bill Zan, and Professor Jonathan Hodge): This team studied nonseparability in voter preferences for multiple-question referendum elections. Nonseparability describes the phenomenon in which a voter's preferences on one or more proposals may depend on the outcomes of other proposals. This project developed new mechanisms for distinguishing between varying levels of nonseparability. In contrast to prior results, maximally nonseparable preferences—those that are as ill-behaved as possible with regard to preference interdependence—seem to be relatively rare. This work also extended prior research relating degrees of nonseparability to the desirability of referendum election outcomes.**Circle Packing:**This team (Daniel Ralston, Sarah Van, and Professor William Dickinson) investigated circle packing with two classes of radii on all flat tori. We discovered all the locally and globally optimally dense arrangements of three circles with radii ratio Sqrt(2)-1 on any torus where there is one circle with a larger radius and two with a smaller radius. The most dense packing in this family has 7 tangencies and a density of about 88%. Additionally, we discovered that there are two families of tori on where there are two different packings that realize the globally maximally dense arrangements: one has no free circles and the other contains only free circles.**Extremal Numbrix Puzzles:**This team (Faith Hensely, Ashley Peper, and Professor Lauren Keough) investigated the minimum number of clues one needs to give so that a Numbrix puzzle on a rectangular grid has a unique solution. They proved that one clue will not yield a unique solution for a*m*x*n*Numbrix puzzle when*m*and*n*are each greater than 1. When changing the game to allow for diagonal moves or imagining the grid is on a torus, they also showed that 1 clue will never result in a unique solution.**Spherical and Hyperbolic Geometry:**This team (Dylan O’Connor, Ramon Suris-Rodriguez, and Professor William Dickinson) investigated and proved Lexell’s Theorem in the hyperboloid model of the hyperbolic plane. Lexell’s problem asks if you are given a triangle ABC with fixed base AB, what is the locus of points, Gamma, so that for all X in Gamma, Area(ABC) = Area(ABX). The solution on the sphere is due to Anders Lexell, a student of Euler. The solution in the hyperbolic plane has been determined in the Poincare disk model by Papadopoulos and Su. Their work is not analogous to Lexell’s solution on the sphere. Our work in the hyperbolic model is exactly analogous to the sphere and actually proves the result simultaneously on the sphere, Euclidean plane, and hyperbolic plane. Further, our work leads naturally to a conjecture about the solution to the Lexell problem in S³ and H³.**Hausdorff metric geometry:**This team (Roman Vasquez, Rachel Wofford, and Professor Steven Schlicker) added to the knowledge of the geometry of the metric space of all non-empty compact subsets of n-dimensional real space under the Hausdorff metric. Specifically, they discovered infinitely many previously unknown integer sequences that represent the number of sets at each location on a Hausdorff segment between two finite sets that form a configuration. The approach they took was to connect finite configurations to edge covers of bipartite graphs, where the number of edge covers of a bipartite graph is the same as the number of sets at each location between the end sets in the related configuration.**Symplectic Geometry:**This team (Isaiah Silaki, Allen Yang and Professor Filiz Dogru) studied main theorems in this new geometry such as Darboux’s Theorem and Gromov’s Non-Squeezing theorem, explored similarities and differences between Symplectic geometry and the Riemannian geometry. Additionally, this team explored its relation to Hamiltonian mechanics and physics.

**Lie Algebras:**This team (Jacksyn Bakeberg, Kathryn Blaine, and Prof. Firas Hindeleh) investigated the classification problem of the seven-dimensional solvable Lie algebras. In particular, they focused on the case where the nilradical is five-dimensional abelian. They were able to classify all the Lie algebras in this class. A paper that the team wrote with their results has been submitted.**Voting Theory:**This team (Ben Becker, Colby Brown, Kaleigh Roach, Lee Trent, and Prof. Jonathan Hodge) used graph theory to model how voter preferences on the various questions in a referendum election can influence each other. They used directed graphs and hypergraphs to represent influence in both individual voter preferences and entire electorates, proving a variety of structural results. They also showed via simulation that influence digraphs can be used to intelligently sequence multiple-question elections, yielding better results than those obtained by simultaneous voting or random sequencing.**Circle Packing:**This team (Susanna Manning, Bridget Parker, and Prof. William Dickinson) investigated circle packing with two classes of radii on a square flat torus and unit triangular flat torus. They discovered all the locally and globally optimally dense arrangements of three circles with radii ratio √2 - 1 on both of these tori. There is a unique optimally dense packing in each case except on the unit triangular flat torus. Interestingly in this case, with two small and one large circle, there is a locally non-globally maximally dense arrangement with room for a free circle.**Spherical and Hyperbolic Geometry:**This team (Geneva Collins, Lucas Perryman-Deskins, and Prof. William Dickinson) generalized one Japanese temple problem into spherical and hyperbolic geometry. In addition, they figured out the geometric construction of the associated arrangement of five circles and one line segment in Euclidean, spherical, and hyperbolic geometries. This problem was originally inscribed on a tablet (sangaku) from 1888 and hung in a shrine in the Fukusima prefecture. Besides working this problem, they also investigated another sangaku problem from the Aichi prefecture (1837) that involved an arrangement of 5 circles and 3 ellipses, and they supplied a proof of it in Euclidean geometry.**Graph Coloring:**This team (Hannah Critchfield, Maia Wichman, and Prof. Emily Marshall) investigated the chromatic number of families of graphs whose edges can be partitioned into certain layers. In particular, they found the chromatic number of all graphs whose edges can be partitioned into two cyclic layers and all graphs whose edges can be partitioned into a cyclic layer and an outerplanar layer. In addition, they found the chromatic number of several specific types of graphs whose edges can be partitioned into two outerplanar layers.

**Voting Theory:**This team (Tasha Fu, David Shane, and Prof. Jonathan Hodge) proved results about the kinds of conclusions that can be drawn from incomplete information about voter preferences in referendum elections, with particular emphasis on characterizing separability and influentiality relationships. They also investigated ways to intelligently sequence multiple-question elections in order to yield better election outcomes.**Equal Circle Packing:**This team (Sean Haight, Quinn Minnich, and Prof. William Dickinson) explored equal circle packings on flat Klein bottles. Building on the previous work of the 2012, 2014, and 2015 REUs, we found and proved the optimality of the packings of four equal circles on any flat Klein bottle.**Anti-Games:**This team (Sophie Mancini, Jacob Van Hook, and Prof. David Clark) spent the summer studying Steiner Triple Systems. They first determined how to generalize the game of Anti-SET to triple systems. They then developed and proved an explicit winning strategy for the second player on the infinite family of Projective Steiner Triple Systems. This strategy relies on the geometric properties of these systems.**Generalized Catalan Numbers:**This team (Emily Dautenhahn, Hannah Pieper, and Prof. Shelly Smith) created sets counted by the Raney numbers, which are a generalization of the Catalan numbers based on two parameters,*k*and*r*. Starting with known sets of combinatorial objects counted by the Catalan numbers, they were able to generalize 23 sets for the parameter*k*(*k*-Catalan objects) and were also able to generalize 17 of those sets for the parameter*r*(Raney objects). They created bijections between these sets to demonstrate that the sets have the same cardinality.

**Voting Theory:**This team (Nicole Buczkowski, Stephanie Thrash, and Prof. Jonathan Hodge) investigated lottery voting, a voting system in which winners are determined by random selection of ballots rather than counting votes. They proved that, for multi-seat elections under lottery voting, a political party's optimal strategy is to run as many candidates as possible, and to run candidates that are equally popular.**Graph Embeddings:**This team (Rachel Barnett, Rose Johnson-Leiva and Prof. William Dickinson) spent the summer investigating the embeddings of graphs in the cubic three torus (the quotient of real three space by a lattice generated by three unit pairwise orthogonal vectors). After deciding on an appropriate definition of a 3-cell embedding of a graph in a three torus (motivated by considerations from sphere packing), they proved that a double cube graph has a unique (up to homotopy) 3-cell embedding on a three torus. The double cube graph is the usual cube edge graph where each edge is replaced by a pair of multi-edges.**Data Analysis:**This team (Daniel Harper, Brianna King, and Prof. Dan Frobish) compared seven different dimension reduction methods in the context of survival analysis (time until event of interest, e.g. time until develop cancer), allowing for the possibility that some individuals may be cured (not at risk for the event). The goal was to predict survival probability and cure probability based on thousands of predictor variables (e.g. gene expression levels). They simulated thousands of data sets, and showed that no one dimension reduction method outperformed all others, based on the four criteria of bias (accuracy) and variability (precision) of estimates of both survival and cure probability.-
**Outer Billiards and Fundamental Domains:**This team (Stephanie Loewen, Tristan Wells, and Prof. Filiz Dogru) explored the possible connections between the outer billiard map, Fuchsian groups, and its fundamental regions. They discovered that, in the hyperbolic plane, the regular tiling polygons coincides with the fundamental domains of corresponding outer billiard map. Further investigation showed that the result is also true for quasiregular tilings in the hyperbolic plane.

**Equal Circle Packing**: This team (Samantha Moore, Robert Dickens, and Prof. William Dickinson) explored equal circle packings on flat Klein bottles. Building on the previous work of the 2012 and 2014 REU, we found and proved the optimality of the packings of three equal circles on any flat Klein bottle.**Voting Theory:**This team (Selene Chew, T.J. Warner, and Prof. Jonathan Hodge) used graph theoretic models to generate multidimensional voter preferences with a variety of interdependence structures, or*characters*. They introduced the*character spectrum*of a graph, which is the set of all possible characters that can be generated by the Hamiltonian paths in the graph. They then explored the relationships between the properties of a graph (such as number of edges) and the size and nature of the graph’s character spectrum.**Combinatorial Sums and Identities****:**This team (Ryan Bianconi, Marcus Elia, and Prof. Akalu Tefera) explored, conjectured and formulated formulas for new and old combinatorial summation expressions. We learned and mastered fundamental and useful skills for combinatorial identity proving: combinatorial (counting) methods and computerized proof techniques (Gopher’s and the Wilf-Zeilberger Methods), and applied various computer summation algorithms for discovering and proving challenging and interesting combinatorial-identities.**Frame Sudoku****:**This team (Emily Alfs, Susanna Lange, and Prof. Shelly Smith) explored a Frame Sudoku, a variation of Sudoku, using combinatorial counting techniques, partitions of integers, and equivalence relations. We fully analyzed and counted 4x4 Frame puzzles with unique solutions, and determined the minimal number of clues needed to reconstruct the entire frame of a 6x6 Frame puzzle. We wrote computer programs to count the number of solutions for a given Frame, and to generate Frame Sudoku puzzles with unique solutions.

**Equal Circle Packing**: This team (Julia Dandurand, Brenna Baker, and Prof. William Dickinson) explored equal circle packings on a square flat Klein bottle. We successfully adapted the techniques that previous groups have used on flat tori to flat Klein bottles. Building on the work of the 2012 REU group, we found and proved the optimality of the packing of three equal circle on a square flat Klein bottle.**Voting Theory:**This team (Brealyn Beals, Ian Calaway, and Prof. Jonathan Hodge) continued and extended prior investigations of graph theoretic models of interdependent preferences in referendum elections. In particular, they defined a class of graphs called bead graphs and studied the interdependence structures of preferences generated by Hamiltonian paths in bead graphs. The project resulted in proofs of important structural properties, along with a complete characterization of bead graph preferences for elections with up to 4 questions. A manuscript is currently in preparation.**Gridline Graphs in Three and Higher Dimensions:**This team (Joshua Mireles, Adam Volk, and Prof. Feryal Alayont) focused on extending understanding of gridline graphs, which form a subclass of perfect graphs, from 2 dimensions to higher dimensions. We studied a new way of defining gridline graphs in three and higher dimensions, and characterized these graphs in terms of decompositions into certain smaller graphs and line graphs. We also obtained a partial list of forbidden subgraphs in the three-dimensional gridline graphs.**Hausdorff Metric Geometry:**This team (Pallavi Aggarwal, Ryan Swartzentruber, and Prof. Steven Schlicker) added to the knowledge of the geometry of the metric space of all non-empty compact subsets of n-dimensional real space under the Hausdorff metric. Specifically, they studied orthogonality in this space in an attempt to determine a well-defined notion of angle between rays in this space. Using orthogonality this group defined Pythagorean triples of compact sets and determined the conditions under which sets could form these triples. In the process they discovered a surprising result that there are instances when such Pythagorean triples are not possible. In addition, they found some unusual results related to orthogonality and rays that present problems creating a well-defined notion of the measure of an angle. Among these strange results this group found is that it is possible to drop infinitely many perpendiculars from a compact set that is not on a ray to that ray. The resulting paper, "Pythagorean orthogonality of compact sets," was published in*Involve*.

**Equal Circle Packing:**This team (Madeline Brandt, Hanson Smith and Prof. William Dickinson) found all optimally dense packings of four of equal circles on any flat torus. There turn out to be several two parameter regions in the moduli space of flat tori where there are two or more optimal arrangements (one globally dense and the others locally dense). This is the first example where there have been multiple optimal packings on a single torus with packing graphs that are not homeomorphic as subsets of the torus. The behavior of the optimally dense packings agrees the work of A. Heppes from 1999. A manuscript is in preparation.**Voting Theory:**This team (Beth Bjorkman, Sean Gravelle, and Prof. Jonathan Hodge) investigated graph theoretic representations of multidimensional binary preferences associated with referendum elections. We specifically studied preferences that can be represented by Hamiltonian paths in cubic graphs with the Gray Code labeling. We characterized the algebraic structure of the sets of preferences that can be generated in this manner, and we also proved results about the interdependence structures that result from such preferences. A manuscript is currently in preparation.**Mathematics and 3D Printing:**At the beginning of the summer, the mathematics department at GVSU acquired a MakerBot Replicator 2 3D-printer, and research of this team (Melissa Sherman-Bennett, Sylvanna Krawczyk, and Prof. Edward Aboufadel) revolved around this technology. The goal was to develop novel techniques using mathematics to design objects appropriate for the printer. The team developed techniques to “print” algebraically-defined surfaces, manifolds defined from real data (such as elevation data from geography), and friezes based on data collected by the Kinect camera. Using ideas from linear algebra, the team then developed a method to identify depth data for an object based on two photographs, and “print” these objects, such as a human hand. At the end of the summer, the team wrote a primer, “3D Printing for Math Professors and Their Students,” that was made available for free on the Internet. A second manuscript is also available.**Extended Outer Billiards in the Hyperbolic Plane:**This team (Sanjay Kumar, Austin Tuttle and Prof. Filiz Dogru) focused on analyzing the extended outer polygonal billiard map in the hyperbolic plane. We classified polygonal tables with respect to their rotation numbers, whether rational or irrational, and we wrote programs to investigate our conjectures for periodic orbits of this special circle map generated from polygonal tables. A manuscript is in preparation.

**Arrow Path Sudoku:**This team (Ellen Borgeld, Elizabeth Meena, and Prof. Shelly Smith) explored a variation of a Sudoku puzzle that uses an arrow in each cell, pointing to the cell containing the subsequent number, in additional to only a small number of numerical clues. We use inclusion-exclusion and computer programs that we wrote to count 2x2, 2x3, and 3x3 number blocks that admit valid arrow paths, Arrow Path blocks that are solvable, and the number of solutions for each block. We developed an equivalence relation on the set of blocks of each size, then partitioned the sets into equivalence classes to facilitate combining blocks to form 4x4, 6x6, and 9x9 Arrow Path Sudoku boards. We described and counted all possible 4x4 boards that are solvable, and determined the maximum number of numerical clues required to create Arrow Path puzzles of each size with a unique solution. A manuscript is in preparation.**Combinatorial Sums and Identities:**This team (Sean Meehan, Michael Weselcouch and Prof. Akalu Tefera) explored, conjectured and formulated various challenging and interesting combinatorial sums and identities. To do this the team spent a great deal of time studying powerful combinatorial (counting) methods and computer assisted proof techniques of Wilf-Zeilberger and other symbolic computation techniques. Using various computer summation algorithms the team was able to discover and prove challenging and interesting old and new combinatorial identities.**Equal Circle Packing on a Flat Square Klein Bottle:**This team (Matthew Brems, Alexander Wagner and Prof. William Dickinson) explored one and two equal circle packings on a square flat Klein bottle. To do this using the same methods as in packings on flat tori, we had to explicitly calculate the identity component of isometry group of any flat Klein bottle and we rewrote a program to compute the possible packing graph structures on a Klein bottle. Along the way, we proved a theorem that every flat Klein bottle is isometric to a flat Klein bottle where the generating vectors are orthogonal (i.e. a rectangular flat Klein bottle). This enabled us to discover and prove the optimality of the one and two equal circle packings on a square flat Klein bottle and to conjecture the optimal packing arrangement for three equal circles on a square flat Klein bottle. A manuscript is planned in the future.**Single-Peaked Preferences in Multiple-Question Elections:**This team (Lindsey Brown, Hoang Ha, and Prof. Jonathan Hodge) applied the concept of single-peaked preferences to the multidimensional binary alternative spaces associated with a variety of multiple-criteria decision-making problems, including referendum elections. They generalized prior work on cost-conscious preferences in referendum elections, showing that single-peaked binary preferences are nonseparable except in the most trivial cases, and that electorates defined by single-peaked preferences always contain weak Condorcet winning and losing outcomes. They also developed a general method for enumerating single-peaked binary preference orders, finding exact counts for 2, 3, and 4-dimensional alternative spaces. A manuscript, "Single-peaked preferences over multidimensional binary alternatives," was published in*Discrete Applied Mathematics*.

**Applications of Wavelets:**This team (Nathan Marculis, SaraJane Parsons, and Prof. Edward Aboufadel, with help from Clark Bowman) worked on the following problem: using accelerometer, GPS, and other data collected by smartphones while driving, how can this data be used to identify the location and severity of potholes? The team made use of wavelet filters, Kruskal’s algorithm, and other mathematical tools to develop an algorithm to solve the problem. In February 2012, the City of Boston announced that they would be using the Wavelet-Kruskal solution, along with algorithms from other researchers, in their Street Bump app.**Equal Circle Packing:**This team (AnnaVictoria Ellsworth, Jennifer Kenkel and Prof. William Dickinson) found all optimally dense packings of three of equal circles on any flat tori. For all but a two parameter region in a moduli space of tori there is exactly one optimally dense arrangement. Inside this region there are two optimally dense packings (one globally dense and the other locally dense). The behavior of the optimally dense packings agrees with the previous summer's work and the work of Heppes.**Outer Billiards in the Hyperbolic Plane:**This team (Neil Deboer, Daniel Hast, and Prof. Filiz Dogru) analyzed the orbit structures and the geometric properties of the outer (dual) billiard map in the hyperbolic plane. We geometrically constructed the 3-periodic orbit for small triangles. This construction led us to define a new term to describe the strong criterion, "triangle-small polygon" to classify polygons in the hyperbolic plane. As a result, we have discovered the special class of polygons which have at least one 3-periodic orbit inside the hyperbolic plane. A manuscript based on this work has been submitted for publication, and another is in preparation.**Voting Theory:**This team (Clark Bowman, Ada Yu, and Prof. Jonathan Hodge) developed an iterative voting method for referendum elections. Our method allows voters to revise their votes as often as they would like during a fixed voting period, with the current results of the election displayed in real time. Through extensive computer simulation, we showed that our method yields significant improvements from standard simultaneous voting and in many cases solves the separability problem, a phenomenon that is known to yield undesirable and even paradoxical outcomes in referendum elections. A paper based on this work, "The potential of iterative voting to solve the separability problem in referendum elections," was published in*Theory and Decision*.

**Higher Dimensional Rook Polynomials**: This team (Rachel Moger-Reischer, Ruth Swift, and Prof. Feryal Alayont) focused on generalizations of 2-dimensional rook polynomials to three and higher dimensions. The theory of 2-dimensional rook polynomials is concerned with counting the number of ways of placing non-attacking rooks (no two in a row or a column) on a 2-dimensional board. The theory can be generalized to three and higher dimensions by letting rooks attack along hyperplanes. In 2 dimensions, the rook numbers of certain families of boards correspond to known number sequences, including Stirling numbers, number of derangements, number of Latin rectangles and binomial coefficients, and provide other combinatorial interpretations of these sequences. Our focus this summer was exploring similar correspondences for three and higher dimensional rook numbers. Building upon research conducted in 2009 funded by a GVSU S3 Grant, we found a family of boards in higher dimensions generalizing the 2-dimensional boards with Stirling numbers as their rook numbers. The rook numbers of these higher dimensional boards and those of their complements resulted in generalized central factorial numbers and the generalized Genocchi numbers. We also found a family of boards in higher dimensions that generalize the staircase boards in 2 dimensions. The rook numbers of these boards are binomial coefficients as are those of the 2-dimensional staircase boards. These examples provide new combinatorial interpretations of these sequences. An manuscript based on this work has been submitted for publication.**Orthogonality in the Space of Compact Sets**: This team (Mychael Sanchez, Jon VerWys, and Prof. Steve Schlicker) focused on the topic of Pythagorean orthogonality in the space H of all nonempty compact subsets of n dimensional real space. Our ultimate goal was to develop a trigonometry on H. The space H is a metric space using the Hausdorff metric h, and previous REU groups have learned much about line segments in H. If A, B, and C are elements of H, we defined the segments AB and AC to be orthogonal if their lengths satisfy the Pythagorean identity, that is the square of h(B,C) is the sum of the squares of h(A,B) and h(A,C). When this happens we say that A, B, and C form the vertices of a right triangle in H with segment BC as hypotenuse and segments AB and AC as legs. This group made progress on a characterization of exactly when a segment BC can be the hypotenuse of a right triangle in H ( this is not always possible) and when a segment AB can be a leg. This group also discovered many different ways that orthogonality in H is different than orthogonality in n dimensional real space. Progress was made on defining the concept of spread in H which may lead us to an interesting and useful notion of trigonometry in H.2-dimensional staircase boards. These examples provide new combinatorial interpretations of these sequences.**Equal Circle Packing on Flat Tori**: This team (Brandon Tries, Jon Watson, and Prof. Will Dickinson) focused on equal circle packings of small numbers of circles in a one-parameter family of flat tori. The family of tori that we worked on were those that are the quotient of the plane by a lattice generated by two unit vectors with an angle of between 60 and 90 degrees. (A packing of circles on a torus is an arrangement of circles that do not overlap and are contained in the torus. The density of a packing is the ratio of the area covered by the circles divided by the area of the torus.) We focused on finding all the optimally dense (both locally and globally) arrangements of 1 to 4 circles on this one-parameter family of tori. Roughly speaking, a packing is locally optimally dense if it has equal or greater density than all near by arrangements. A packing is globally maximally dense if it is the most dense locally optimally dense packing. For 1 and 2 circles packed on any torus in the one-parameter family of tori, we proved there is a unique optimally dense arrangement for each torus. For 3 circles packed on a torus, the number of and type of arrangement depend on the angle of the torus. For any angle strictly between 60 and 90 degrees, we proved that there are exactly two locally maximally dense arrangements. When the angle is 90 degrees the two arrangements are identical and at 60 degrees, the one of the arrangements is no longer locally maximally dense. The behavior of the optimally dense packings at the extreme angles of 60 and 90 degrees agrees with previous REU [2007 - DMS 0451254] and other research.**Rank Disequilibrium in Multiple-Criteria Evaluation Schemes**: This group (Faye Stevens, Jamie Woelk, and Prof. Jonathan Hodge) developed a mathematical model of the concept of rank disequilibrium, which occurs when individuals are evaluated over multiple criteria and have different perceptions of the relative value of each criterion. Rank disequilibrium has been shown to be a significant source of organizational conflict, and this project was the first attempt to formally model and investigate rank disequilibrium from within a mathematical framework. The main objects of study were rank aggregation functions, which assign overall rankings to combinations of rankings on individual criteria. The group defined several desirable properties for rank aggregation functions to satisfy and proved necessary and sufficient conditions for the existence of rank aggregation functions with these properties. The group proved that while it is nearly impossible to avoid all forms of inequity, certain forms of inequity can be avoided by limiting the number of possible rankings on each criterion, restricting the set of possible ranking profiles, or by exploiting known information about the evaluees’ preferences. A paper based on this work, "Rank Disequilibrium in Multiple-Criteria Evaluation Schemes," was published in*Involve.*

**Wavelets and Diabetes**: This group (Rob Castellano, Derek Olsen, and Prof. Ed Aboufadel) created a new measurement developed to quantify the variability or predictability of blood glucose in type 1 diabetics. Using continuous glucose monitors (CGMs), this measurement -- called a PLA index -- is a new tool to classify diabetics based on their blood glucose behavior and may become a new method in the management of diabetes. The PLA index was discovered while taking a wavelet-based approach to study the CGM data. This wavelet-based approach emphasizes the shape of a blood glucose graph. Their article, "Quantification of the variability of continuous glucose monitoring data algorithms," appeared in*Algorithms*.**Greater Than Sudoku**: This group (Ana Burgers, Katherine Varga, and Prof. Shelly Smith) investigated aspects of a variation of a Sudoku puzzle that uses inequalities between adjacent cells rather than numerical clues. They showed that the cells of an m by n inequality block are a partially ordered set, and that an inequality block is solvable if and only if it is acyclic. They also defined an equivalence relation on the set of solvable 2 by 2 inequality blocks and proved that there are 224 Greater Than Shidoku (4 by 4) boards with unique solutions. Their article, “Analysis of a Sudoku variation using partially ordered sets and equivalence relations,” appeared in Involve, 7:2(2014), pp 187-204.**Hypergeometric Sums and Identities**: This group (Samantha Dahlberg, Tim Ferdinands, and Prof. Akalu Tefera) studied computerized proof techniques, specifically Gosper's and Zeilberger's algorithms and the Wilf-Zeilberger proof method, and counting (combinatorial) proof techniques. Using computerized and counting proof techniques the team investigated several hypergeometric sums, found closed forms of various challenging and interesting hypergeometric sums and proved identities involving binomial coefficients. Using counting techniques, the team was able to find a beautiful proof for one of the challenging problems proposed in the June 2009 issue of*Mathematics Magazine*(the solution is submitted for publication). The team also gave an elementary and elegant approach, using the WZ method, for sums of Choi, Zoring and Rathie. Their article, "A Wilf-Zeilberger Approach to Sums of Choi, Zornig and Rathie," appeared in*Quaestiones Mathematicae.***Cost-conscious voters**: This group (Kyle Golenbiewski, Lisa Moats, and Prof. Jonathan Hodge) developed an axiomatic model of cost-conscious voters in referendum elections. They used this model to prove a variety of useful results, including: (i) that the probability of cost-consciousness approaches zero as the number of questions grows without bound; (ii) that, under certain conditions, elections in which voters are cost-conscious will always contain at least a weak Condorcet winning outcome (and a weak Condorcet losing outcome); and (iii) that the interdependence structures within the preferences of cost-conscious voters can be varied and unpredictable. Their article, "Cost-conscious voters in referendum elections," appeared in*Involve*.

**Wavelets**: This group (Sarah Boyenger, Clara Madsen, and Prof. Ed Aboufadel) developed a new method to create portraits that imitate the style of artist Chuck Close. Wavelets were used for detecting and classifying edges. While Chuck Close uses a diamond tiling of the plane for his portraits, this new method can use regular tilings by triangles and other objects. Their article, "Digital creation of Chuck Close block-style portraits using wavelet filters", appeared in 2010 in the Journal of Mathematics and the Arts. In addition, the research group created a digital "Chuck Close-like" print of Ramanujan that was featured at the art exhibition at the 2011 Joint Mathematics Meetings.**Hausdorff Metric Geometry**: This group (David Montague and Prof. Steve Schlicker) obtained some fascinating results on finite betweenness in the Hausdorff metric geometry. More specifically, they found necessary and sufficient conditions under which there can be a finite set between two sets A and B. One unexpected consequence of this characterization is that for some sets A and B, there can be finite sets at some locations between A and B but not at others. This group also found some interesting preliminary results on convexity in this geometry.**The Geometry of Polynomials**: This group (Haggai Nuchi, Aaron Shatzer, and Prof. Steve Schlicker) investigated several problems related to the Sendov conjecture: a two-circle type theorem for polynomials with all real zeroes, finding polynomials with minimal deviation from their roots to their critical points, and maximizing the deviation from the roots of a polynomial with real roots (and its derivative) to their centroids. The most interesting results were obtained in the latter problem where the group has made significant progress in proving its conjecture of the polynomials with maximal deviation.**Gerrymandering**: This group (Emily Marshall, Geoff Patterson, and Prof. Jonathan Hodge) explored the notion of convexity as it relates to the problem of detecting gerrymandering and producing optimal congressional redistrictings. They defined the convexity coefficient of a district or region, and used Monte Carlo simulation to approximate the convexity coefficient of each of the 435 congressional districts in the United States. They explored several theoretical questions pertaining to approximation of convexity coefficients and the effect of subdividing regions using straight-line cuts. They also explored ways to modify the convexity coefficient to account for population density and irregularities in state boundaries. Their article,"Gerrymandering and Convexity" appeared in the September 2010 issue of the*College Mathematics Journal*, and the convexity coefficient introduced therein has its own entry on Wolfram MathWorld.

**Group A**(Katrina Honigs, Vincent Martinez, and Prof. Steve Schlicker) defined and investigated three different types of connectedness in the geometry of the Hausdorff metric and obtained some preliminary results about sets that satisfy each type of connectedness. Each finite configuration in this geometry has an associated bipartite graph. Several results obtained about edge covers of graphs by this group gave insight into the possible number of sets at each location between the end sets in any configuration. A paper based on this research, "Missing edge coverings of bipartite graphs and the geometry of the Hausdorff metric," was published in the*Journal of Geometry*.**Group B**(Mark Krines, Jennifer Lahr, and Prof. Jonathan Hodge) explored questions pertaining to the notion of separability in voter preferences over multiple questions. Specifically, they generalized methods for counting preseparable extensions, which are used to build larger preference orders from smaller ones, and developed a characterization of the types of preferences that can be constructed via preseparable extensions. They also characterized the algebraic structure of the sets of permutations that preserve the separability of a given separable preference order, and discovered a connection between separable preference orders and Boolean term orders, which have applications to abstract algebra and comparative probability theory. Their article, "Preseparable extensions of multidimensional preferences," appeared in the journal*Order*.**Group C**(Anna Castelaz, Daniel Guillot, and Prof. Will Dickinson) focused on finding all the locally and globally dense packings of 1 to 5 circles on the standard triangular torus. Roughly, a packing of*n*equal circles is locally maximally dense if there exists a positive epsilon, so that if each circle center is moved by less than epsilon, the density must decrease (i.e. the smallest pairwise distance between the centers must decrease). A packing of*n*equal circles is globally maximally dense if it is the most dense locally maximally dense packing. Two published articles resulted from this research: "Optimal Packings of up to Five Equal Circles on a Square Flat Torus" (with Sandi Xhumari), which appeared in*Beiträge zur Algebra und Geometrie*; and "Optimal Packings of up to Six Equal Circles on a Triangular Flat Torus" (with Sandi Xhumari), which appeared in the*Journal of Geometry*.**Group D**(Daniel Gorski, Hanna Komlos, and Prof. Filiz Dogru) investigated the dynamics of the dual billiard map in the Euclidean plane and the hyperbolic plane. The orbit behavior of the map changes with respect to the shape and size of the table. Their work concentrated on regular polygonal tables which tessellate the Euclidean or hyperbolic plane.

**Group A**(Timothy Armstrong, Liz Smietana, and Prof. Ed Aboufadel) investigated position coding and invented two new position codes -- one based on binary wavelets, while the other uses a base-12 system combined with binary matrices. A manuscript has been posted to the ArXiV.**Group B**(Brian Lerch, Jonah Leshin, and Prof. Clark Wells) addressed the problem of finding an optimal seating strategy to maximize acquaintances made at successive events.**Group C**(Daniel Schultheis, Lisa Morales, and Prof. Steve Schlicker) created a computer program to automate the computation of the number of elements at each location on a Hausdorff segment between two finite sets. They also created new integer sequences which arise from special types of Hausdorff configurations. Their paper, "Polygonal chain sequences in the space of compact sets," appeared in 2008 in the*Journal of Integer Sequences***.****Group D**(Christy Hediger, Daniel Taylor, and Prof. Will Dickinson) solved San Gaku geometry problems from Japan, ones that involved parallel lines. They generalized these problems and solutions to spherical and hyperbolic geometry.

**Group A**(Justin From, Samuel Kolins, and Prof. Matt Boelkins) investigated several problems in the geometry of polynomials focused on the relationship between the set of zeroes and the set of critical numbers. The article, "Polynomial Root Squeezing" appeared in*Mathematics Magazine*in February 2008.**Group B**(Scott Bachman, Amber DeMore, and Prof. Paul Fishback) investigated various problems involving "least squares derivatives" associated with various random variables and corresponding families of orthogonal polynomials.**Group C**(Beverly Lytle, Caroline Yang, and Prof. Ed Aboufadel) applied wavelets and statistics to match handwriting samples and determine forgeries. The article, "Detecting Forged Handwriting with Wavelets and Statistics", written by these researchers, appeared in the Spring 2006 issue of the*Rose-Hulman Undergraduate Mathematics Journal*.**Group D**(Chantel Blackburn, Alex Zupan, and Prof. Steve Schlicker) investigated the geometry the Hausdorff metric imposes on the collection of non-empty compact subsets of*n*-dimensional real space. A major finding for this group was that, although there are configurations in this space that allow*k*elements at each location between two sets for infinitely many different values of*k*(including examples for*k*between 1 and 18), there is NO possible configuration that allows exactly 19 elements at each location. A paper based on their work and work from the 2004 REU, “A Missing Prime Configuration in the Hausdorff Metric Geometry,” appeared in 2009 in the*Journal of Geometry*.

**Group A**(Amanda Morris, Abby VanHouten, and Prof. Jodi Sorensen) created "Bubble Bifurcations" in dynamical systems.**Group B**(Matt Katschke, Jodi Simons, and Prof. Will Dickinson) solved San Gaku geometry problems from Japan, and generalized these problems and solutions to spherical and hyperbolic geometry.**Group C**(Kevin Brink, Drew Colthorp, Prof. Ed Aboufadel) applied wavelets and other tools to the problem of finding airplanes in aerial photographs.**Group D**(Kris Lund, Patrick Sigmon, and Prof. Steve Schlicker) investigated the geometry the Hausdorff metric imposes on the collection of non-empty compact subsets of n-dimensional real space. In particular, they found many interesting occurrences of Fibonacci and Lucas numbers in this geometry. A paper based on their work, “Fibonacci sequences in the space of compact sets,” was published in*Involve*. Another paper, “A Missing Prime Configuration in the Hausdorff Metric Geometry,” appeared in 2009 in the*Journal of Geometry*.

**Group A**(Christopher Bay, Amber Lembcke, and Prof. Steve Schlicker) investigated the geometry of the Hausdorff space, in particular the lines of that space. The article "When Lines Go Bad in Hyperspace," written by these researchers, appeared in*Demonstratio Mathematica*in 2005.**Group B**(Julie Olsen, Jesse Windle, and Prof. Ed Aboufadel) applied wavelets and other tools to the problem of breaking CAPTCHAs. The article "Breaking the Holiday Inn Priority Club CAPTCHA," written by these researchers, appeared in the March 2005*College Mathematics Journal*.**Group C**(Jen Miller, Ben Vugteveen, and Prof. Matt Boelkins) investigated polynomial root dragging. The article, "From Chebyshev to Bernstein: A Tour of Polynomials Large and Small", written by these researchers, appeared in the May 2006*College Mathematics Journal*.**Group D**(Lindsey Bromenshenkel, Justin Hogg, and Prof. Clark Wells) conducted work in Lie Theory and its relations to rotations in 3-dimensional space.

**Group A**(Jon Mayberry, Audrey Powers, and Prof. Steve Schlicker) investigated the geometry of the collection of non-empty compact subsets of n-dimensional real space. The article, "A Singular Introduction to the Hausdorff Metric Geometry", written by these researchers and D. Braun from the 2000 REU, appeared in 2005 in the*Pi Mu Epsilon Journal*.**Group B**(Keith Dailey, Lisa Driskell, and Prof. Ed Aboufadel) worked on problems involving wavelets and steganography. The article "Wavelet-Based Steganography," written by Lisa Driskell, appeared in the*Cryptologia*in 2004.**Group C**(Jon Armel, Rana Mikkelson, and Prof. Jodi Sorensen) investigated the complete bifurcation diagram in dynamical systems. This work was noted in the article, "Sprinkler Bifurcations and Stability," by Jody Sorensen and Elyn Rykken, which appeared in the November 2010 issue of the*College Mathematics Journal*.**Group D**(Noah DeLong, Emily Fagerstrom, and Prof. Clark Wells) conducted work in Lie Theory and its relations to rotations in 3-dimensional space.

**Group A**(Dominic Braun and Prof. Steve Schlicker) investigated the geometry of the Hausdorff space.**Group B**(Amanda Cox, Amy Oostdyk, and Prof. Ed Aboufadel) developed bivariate Daubechies scaling functions.**Group C**(Dawn Ashley, Hillary Van Spronsen, and Prof. Jodi Sorensen) conducted work in "The Real Bifurcation Diagram." Their article, "Symmetry in Bifurcation Diagrams," appeared in the*Pi Mu Epsilon Journal*.**Group D**(Matt Horton and Prof. Paul Fishback) researched Mandlebrot sets for ternary number systems. Their article, "Quadratic Dynamics in Matrix Rings: Tales of Ternary Number Systems," appeared in*Fractals*.**Group E**(Tim Pierce, Courtney Taylor, and Prof. Clark Wells) investigated rotations that arise from chemistry.

*Any publications listed are based upon work supported by: the National Science Foundation under Grants Nos. DMS-1262342, DMS-1003993, DMS-0451254, DMS-0137264, and DMS-9820221; and the National Security Agency under Grant No. H98230-16-1-0030**. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF) or the National Security Agency (NSA).*