Several Complex Variables

There are few differences between single and multivariable calculus with real variables, and most single-variable results extend nicely to higher dimensions. This may lead one to believe that studying several complex variables is simply extending the beautiful one-dimensional results, but nothing could be further from the truth. In fact, there are many results in several complex variables which are wildly different from those in one-variable complex analysis. A few examples are the following:

The zeros of a nonconstant analytic function of one complex variable are always isolated points, and cannot limit in the domain of the function. The zeros of an analytic function of several complex variables can never be contained in a compact subset of its domain.

For every domain in the complex plane, there is an analytic function which cannot be extended analytically to a larger domain. In Cn, only certain domains (called pseudoconvex domains) have this property.

This project will study the basics of several complex variables, exploring the two examples above (zeros of analytic functions and pseudoconvexity) and more topics as time allows.

Written by:  Tomoki Isogai
Advisor: Jon Armel

Queueing Theory

Have you ever stood in a line in a grocery store and fumed when the other lines seemed to move faster? Have you been tempted to switch lines? How long would you have to wait to be served if you didn’t switch lines?

In this comps project, we will learn basic principles of modeling queues. You will see applications in customer service, traffic, and computer networks. You will also have the opportunity to model any system on the Carleton campus where lines are an issue!

Written by:  Dan Lojovich, Ben Haynor, and Sadaf Sultan
Advisor: Laura Chihara

Got otters

Ecology is mainly concerned with the distribution and abundance of plants and animals. The objective of this statistical project is to explore the use of aerial track surveys to model, estimate and monitor the otter population in southeast Minnesota. The data and the questions posed in this project were provided to us by the Minnesota Department of Natural Resources (DNR). Students will meet and consult with statisticians at the DNR and prepare a final report for them.

A key feature in many wildlife surveys is that the probability of detecting an animal that is present in the region can be quite low. Thus, when an animal is not detected at a site it could be because no animal was there or an animal was there but the observer did not see it. Occupancy models attempt to distinguish between these two scenarios and will be studied in this project.

The data to be analyzed was obtained by researchers who flew in a helicopter above different rivers in Minnesota. Whenever they say what appeared to be otter tracks, they pressed a button to record a GPS waypoint associated with the location of the track. While flying they continued to record waypoints every five seconds. Flights were repeated several times.

The data collection presents a host of interesting inference questions. The goal of this project is to provide the DNR some insight into the reliability of using this type of survey to monitor otter populations. Students will also work to develop models that fit the way data are collected. Once developed models could be fit to the actual data. There is also likely to be a simulation component where models are fit using simulated data.

In the past five years numerous papers and books have been written on occupancy models. This project will begin with students becoming familiar with the existing literature and the statistical methods presented there. Possible statistical methods used in this project include maximum likelihood estimation, Monte Carlo simulation, logistic modeling, and Bayesian methods, to name a few.

Written by:  Chrisna Aing, Sarah Halls, and Kiva Oken
Advisor: Bob Dobrow

Title: Building Puzzles from Finite Simple Groups

In a certain sense, the classical 15-puzzle is a physical manifestation of A15, which is one of the fabled finite simple groups. Similarly, the 3x3x3 Rubik’s cube embodies a finite group which is not simple. Recently, mathematician Igor Kriz and undergraduate Paul Siegel have constructed puzzles governed by two other finite simple groups, namely the Mathieu groups M12 and M24. Our goal in this project will be to construct puzzles which embody yet more finite groups, especially simple groups. To accomplish this, we’ll need to learn a bit about the classification of the finite simple groups, permutation representations of groups, and representations of groups over finite fields. Puzzles one could actually build will be our ideal, but we’ll probably aim to have a collection of puzzles we can put on the web.

To play with the puzzle this group produced for the Higman-Sims group, follow the link from Eric Egge’s web page.

Written by:  Erica Chesley, Zach Starer-Stor, and Emma Zhou
Advisor: Eric Egge

A Journey in Teaching Calculus

For this comps project, a student or students would get hands-on experience as a Teaching Assistant in Calculus with Problem Solving, offered 3a in Fall 2009. The student would run Tuesday/Thursday problems sessions 10:45 – 11:50 all fall, coordinate with Eric Egge who’s teaching the course, write up lesson plans and review sheets and help the students learn Calculus. Most of the work for this comps project would be fall term, with a very small amount during winter term compiling the lesson plans and such from fall. The comps student would also have a campus job of holding office hours during fall term for the students in 101.

Written by:  Daniel Bernal
Advisor: Deanna Haunsperger & Eric Egge

Directed Reading: You can help pick the topic!

This is an opportunity for people, either individually or (preferably) in a group of two, to delve deeply into material that goes beyond the standard undergraduate curriculum. Topics can be chosen from analytic number theory (where functions of a complex variable are used to investigate and solve problems in number theory), algebraic geometry (where abstract algebra is used to study the geometry of solution sets to polynomial equations), combinatorics, and possibly other areas. Specific examples of topics include 1) elliptic functions and modular forms; 2) Dirichlet’s theorem on primes in an arithmetic progression; 3) the Rogers-Ramanujan identities. Participants will get substantial (weekly) experience in presenting material at the blackboard.

Written by:  Francis Adams and Ernest Liu
Advisor: Mark Krusemeyer

Shadows of the Cantor Set

The traditional Cantor Set (also known as Cantor’s middle third set) is formed by taking the closed interval [0,1] and deleting the open middle third, then deleting the open middle thirds of the resulting two intervals, then the open middle third of the resulting four intervals, and so on and so forth. Surprisingly, the resulting set turns out to be uncountable. Yet, those who have seen Lebesgue measure also know that the Cantor set has measure 0.

This process can be carried out in higher dimensions. In the plane, a “plus sign” is deleted from a unit square. This process is repeated on the four remaining squares, etc., to yield a two-dimensional Cantor set. This set, when projected down onto a horizontal line is the usual one-dimensional Cantor set. However, if one can imagine shining a flashlight through this set at a 45 degree angle, the shadow left on a horizontal line will be an interval.

What happens when one shines a flashlight through this two-dimensional Cantor set at other angles? Does this generalize to higher dimensions? These are some of the questions we will tackle.

Written by:  Matt Cordes, Becky Patrias, Beatrice White, and Zheng Zhu
Advisor: Gail Nelson

Mathematical Models in Biology – Circadian Rhythms

The team will survey current mathematical models of the regulation of circadian rhythms (e.g. the timing of wake-sleep cycles in mammals) and pursue one model in depth.

Written by:  Grace Elwell, Henry Heitzer, and Erik Williams
Advisor: Samuel Patterson

Statistical Forecasting and Time Series Analysis

Students will work to improve the forecast accuracy for the demand of certain 3M products using historical data provided by the company. Each month 3M must forecast the demand for the tens of thousands of diverse products that they produce across many divisions of the company. The forecasting models that are constructed during this project must not only be accurate, but they also must be manageable using existing 3M resources. Students will have to consider how their forecasting models can be implemented and maintained by 3M. A formal presentation of these models will be given at 3M and a research paper will be produced.

Written by:  Robert Carlton, Gorkem Celebioglu, Daniel O’Connell, and Eric Tiede
Advisor: Katie St. Clair

Knot: the Same?

Any five year old knows about knots- they are what happens when we’re not too careful tying our shoelaces. But as any scout can attest, knots come in many, many types, and one of the goals of mathematical knot theorists is to distinguish different types of knot from one another. Unfortunately, we are far from a satisfactory answer, with advances in fits and spurts and often driven by seemingly unrelated areas of algebra, analysis, and physics. In this comps project, we will study invariants of knots, as well as of related objects like links, braids, and tangles. We will investigate not only the effectiveness of certain invariants but also how they are related to each other. Time permitting, we will also consider applications of knot theory to other sciences and to other branches of mathematics.

Written by:  Alex Fisher, Rosemary Phelps, and Danny Wells
Advisor: Helen Wong