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.

Advisor: Sam Patterson

Terms: Fall and Spring

Fifth-Grade Challenge Math Curriculum

Students in this comps group will run one or more Challenge Math Groups (pull-outs for talented children) for Northfield 5th graders. (These pull-outs will be located in a local elementary school.) During fall term, students will read math education literature on Discovery-Based Learning, psychological development, along with books and articles about running Math Circles and the like. During winter term, students will do independent reading in several areas of mathematics (like Graph Theory, Combinatorics, Number Theory), and devise lesson plans based on these areas. During the spring term, students will write up a book for use by future parent-volunteers to run Challenge Math groups.

Advisor: Deanna Haunsperger

Terms: Fall, Winter, and Spring (2 credits each)

Directed reading: We'll pick the topic(s) together!

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.

Advisor: Mark Krusemeyer

Terms: Fall and Winter

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).

Advisor: Gail Nelson

Terms: Fall and Winter