Math Comps 2007-08

Elliptic Functions and Modular Forms

This is an opportunity for one or two (preferably two) people to delve deeply into material combining algebra and/or number theory with geometry and/or complex analysis and to get substantial (weekly) experience in presenting that material at the blackboard. Abstract algebra I is definitely a prerequisite; other courses that may be useful include number theory, functions of a complex variable, the current algebraic geometry seminar, and abstract algebra II. Specific topic(s) will be chosen, according to the participants' background and interests, from possibilities that range from elliptic functions and modular forms through elliptic curves to commutative algebra and algebraic geometry.

Written by: Jonah Ostroff

Advisor: Mark Krusemeyer

Cover Times of Random Walks on Finite Graphs

The cover time of a random walk on a finite graph is the number of steps it takes to hit all of the vertices in the graph. We investigated the problem of finding whatever information we could (expectation, variance, or exact distribution) about the cover times for random walks on certain types of graphs, in particular, the n-cycle, the star, the "sparkler", and the Petersen graph, deriving new results for the last three graphs. We utilized a variety of techniques to study the cover time, including a general method of exhaustion, gambler's ruin absorption times, recurrence relations, and simulation.

Written by: Michael Duyzend, Rebecca Ferrell, and Miranda Fix

Advisor: Bob Dobrow

L’Hospital Translation Project

This is an ongoing project at Carleton to produce the first English translation of the Marquis de l’Hospital’s Analyse des Infiniment Petits. This work, published in 1696, was the first calculus textbook and has never been translated into English. French 204 or equivalent is a prerequisite for this project.

Written by:  Kristi Welle

Advisor: Sam Patterson

Fifth-Grade Challenge Math Curriculum

This comps group created and wrote a set of fifty lesson plans for challenging mathematical investigations for fifth graders in the Northfield School District. These lesson plans will be used by parent volunteers in the fifth grade classrooms to provide once weekly one-hour pull-outs for children who are not being challenged by the standard fifth grade mathematics curriculum. The content covers an impressive range of mathematical ideas from geometry, number theory, combinatorics, algebra, computer science, probability, topology, and estimation, made understandable to fifth graders.

Written by: Gabe Hart, Alissa Pajar, Melissa Schwartau, Lily Thiboutot

Advisor: Deanna Haunsperger

Wavelets

Wavelets are functions that cut up data into different frequency components, and then study each component with a resolution matched to its scale. They have advantages over traditional Fourier methods in analyzing physical situations where the signal contains discontinuities and sharp spikes.

This project will begin with directed reading on the subject. From there we will explore the many applications of wavelets. Examples include analysis (such as detection of crashes or edges), data compression, and reconstruction after compression (for example, creating a complete fingerprint from a partial print left on woven fabric).

Written by: Amanda Brown, Trevor Burnham, Adam Steege, Nathan Williams

Advisor: Gail Nelson