**Multivariate Explorations of Zooplankton and Environment in Minnesota**

This project will explore the relationship between zooplankton populations and environmental conditions in Minnesota lakes. Plankton is the basis of freshwater and saltwater ecosystems—all aquatic life is dependent upon the energy and oxygen it provides. Over the past several years, the Department of Natural Resources has built a rich dataset sampling zooplankton monthly in 24 diverse lakes throughout Minnesota. The DNR is interested in using these data to derive indicators of changing environmental quality. Some questions they would like to answer include: (i) Which characteristics of zooplankton communities change in consistent and predictable patterns in response to human disturbances? (ii) Are patterns in Minnesota zooplankton communities similar to those observed in other regions? (iii) What patterns can be discerned among zooplankton communities across different environmental variables?

Students will study and use multivariate statistical methods, such as principal components, factor and discriminant analysis, and canonical correlation analysis, for exploring and analyzing these data. They will also meet and communicate with DNR biologists and statisticians, who will be analyzing these data using different statistical approaches.

Many of the multivariate techniques that will be studied are used when one observes a high-dimensional data set, collecting data on a large number of variables, and wishes to develop a smaller number of variables that account for most of the variation in the observed data. The techniques are heavily based on linear algebraic methods, particularly projections and inner product spaces, eigen-analysis and the singular value decomposition.

This project will take place over Winter and Spring Terms. It will culminate in both a presentation to the Carleton community at the end-of-year comps gala, as well as a presentation of findings to the DNR. Prerequisite is Math 275 Introduction to Statistical Inference. Knowledge of R and comfort with computing software is a plus. It is recommended that students take Math 332 Advanced Linear Algebra during Fall Term, where some of these methods and applications will be introduced.

Written by: Prasit Dhakal, Aman Gupta, and Ben Langfeldt

Advisor: Bob Dobrow

**Glimpses of Soliton Theory**

A nice story about the history and the underlying physical properties of the Korteweg-deVries equation can be found at an Internet page of the Herriot-Watt University in Edinburgh (Scotland). The following text is taken from that page:

Over one hundred and fifty years ago, while conducting experiments to determine the most efficient design for canal boats, a young Scottish engineer named John Scott Russell (1808-1882) made a remarkable scientific discovery. Here his original text:

*I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair* *of horses, when the boat suddenly stopped – not so the mass of water in the channel which it had* *put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then* *suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large* *solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course* *along the channel apparently without change of form or diminution of speed. I followed it on* *horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation*.

It was not until the mid 1960’s when applied scientists began to use modern digital computers to study nonlinear wave propagation that the soundness of Russell’s early ideas began to be appreciated. Russell viewed the solitary wave as a self-sufficient dynamic entity, a “thing” displaying many properties of a particle. The phenomenon described by Russell can be expressed by a non-linear Partial Differential Equation of the third order. The phrase “solitary wave” has evolved into the term “soliton”.

At first glance, it might sound like we will be solving Partial Differential Equations, generally a very hard thing to do. Although we will be studying these equations, the closest we will come to solving one is to verify that known solutions are indeed solutions (it is actually quite remarkable that many solutions are known), but no experience PDEs beyond Multivariable Calculus is required. The plan is to start by working through the text **Glimpses of Soliton Theory: The Algebra and Geometry of nonlinear PDEs**, by Alex Kasman. The focus here is more on differential operators and (eventually) pseudo-differential operators. Along the way we will use *Mathematica* to generate pretty pictures of these waves, but we will also be visiting elliptic curves, wedge products, and Grassmann cones. The book ends with a list of ideas for independent projects. This comps experience with a project from this list or an appropriate alternative.

This is not an analysis type project. I think of this as a combination of multivariable calculus and linear algebra taken to a higher level.

Written by: Michael Coughlin, Brian James, Cory Fauver, and Jon Aranda

Advisor: Gail Nelson

**Voting Theory**

In voting theory, there are two main areas that many of the voting theory discussions are divided into: 1) Preferential Voting: Given three or more candidates, *n* voters, and a voting system (such as plurality or Borda’s method), what sort of unfortunate things can happen when a candidate drops out of the race or a voter decides not to vote? What paradoxes can arise? (We know that there are many.) 2) Weighted Voting: Given three or more candidates, *n* voters, and each voter’s vote has a weight associated with it, how much power does each particular voter have? That is, is the power to decide the outcome of the election proportion to the weights assigned to the voters? (We know the answer is “no.”) Calculating how much power each voter has was only done for elections using plurality in the past (despite knowing from preferential voting theory that other methods create fewer paradoxes) because the calculations become messy. In the 90s, a co-author and I devised a scheme to calculate power when the Borda method is used.

I would like some students to begin reading about weighted voting and preferential voting, ending up reading my paper about power, and then extend those results to see what things can happen when we use other weighting systems, or even perhaps Single Transferable Vote (a.k.a. Instant Runoff Voting; which is gaining in popularity, especially in Minnesota) with weights. This might end up being a reading course in what already exists, but I’m hoping that we’ll find interesting questions to attack along the way. No prerequisites, though it would be necessary that at least one person in my group would have basic computing skills.

Written by: Kian Flynn, Erin Jones, and Erika Mackin

Advisor: Deanna Haunsperger

**Creating and Exploring Symmetry**

Description can be found here.

**Modular Subgroups and Symmetry in the Hyperbolic Half-Plane** – written by Frank Firke and Abram Jopp

**What, the Hecke Group?: Fractions and Symmetry in Hyperbolic Space** – written by Rebecca Cordes, Jonathan Hahn, and Collin Hazlett

Advisor: Frank Farris/Stephen Kennedy

**Voronoi Diagrams and the Medial Axis**

Description can be found here.

Written by: Ben Anderson, Erika Warrick, Katie Storey

Advisor: Jack Goldfeather

**Bayesian Modeling with Ecological Applications**

Students in this comps group will use Bayesian inference techniques to complete one of the two following projects:

1. Analyze black bear harvest (hunting) data using a Bayesian approach to monitor the abundance (or relative abundance) of black bears in Minnesota over a number of years.

2. Construct a Bayesian model to determine how many moose reside in Minnesota. There will be data from two sources (a detection survey and a stratified aerial survey) that will need to be incorporated into the model and the goal is to find a model that yields more precise estimates than the current Horvitz-Thompson type estimator of abundance.

Students in this project will learn about Bayesian modeling and inference techniques. They will study how to fit a model in R or WinBUGS and to assess how well their model fits a particular data set.

Written by: Michael Alexander, Scarlett Tse, and Tanner Martin

Advisor: Katie St. Clair

**Individual Comps Presentations**

**Unsolvability, Unprovability, and Hilbert’s 10th Problem**

Russ Buehler

**Partial One-Dimensional (POD) Regression Models**Jun Young Park

**Computational Topology: Homology and Persistence**Sen Zhao

**Counting Below the Curve: Combinatorial Proofs of Integral Formula**

Isaac Hodes

**Counting Perfect Matchings in a Planar Graph**Danny Chen