2021-2022 Catalog

# MATH 400 Senior Colloquium

An exploration of advanced topics in mathematics, chosen by the instructor. Assessment varies by seminar, but could include written assignments, oral presentations, or exams. Seniors typically take one seminar each semester. Completion of two seminars partially fulfills the comprehensive requirement for graduation with a degree in mathematics. May be repeated once for credit.

Introduction to Knot Theory

We learn how mathematics is used to understand knots and links, covering topics such as prime and composite knots, knot colorability and other invariants, and links in graphs.

Applied Mathematics

This weekly seminar course is an introduction to some of the classic techniques of applied mathematics. Topics will include scaling, dimensional analysis, regular and singular perturbations, and asymptotic matching. The goals of the course are to expose students to important techniques widely used in applied mathematics; to present a different mode of classroom instruction and student involvement where students see themselves as equal participants in the learning process; to provide students with opportunities to practice and hone oral presentation skills; and to become more familiar with using modern computational tools to solve mathematical problems.

Euler to Uhlenbeck (History of Mathematics)

This weekly seminar course is a survey of selected topics in the history of mathematics. The Spring semester seminar will cover Euler to Mirzhakani and Uhlenbeck. Through exposure and access to primary historical sources and other materials, students will gain deeper insights into mathematical concepts they have seen before, be introduced to new mathematical ideas, and learn about the history and development of mathematics as it used today. The objective of the course is for students to synthesize and connect various mathematical ideas as a capstone experience for the major. Students will present a project in written and oral form at the end of the course.

Euclid to Newton (History of Mathematics)

This weekly seminar course is a survey of selected topics in the history of mathematics. The Fall semester seminar will (roughly) cover Euclid to Leibnitz and Newton. Through exposure and access to primary historical sources and other materials, students will gain deeper insights into mathematical concepts they have seen before, be introduced to new mathematical ideas, and learn about the history and development of mathematics as it used today. The objective of the course is for students to synthesize and connect various mathematical ideas as a capstone experience for the major. Students will present a project in written and oral form at the end of the course.

Introductory Algebraic Number Theory

One of the most basic facts in all of mathematics is the Fundamental Theorem of Arithmetic:  given a positive integer $\dpi{300}\inline n$, it has a unique (up to ordering) factorization into primes. The integers are not the only interesting set of numbers to work with though.  For instance, suppose you are interested in solutions to the equation $\dpi{300}\inline x^2 + 1 = 0$. This naturally leads us to consider the number $\dpi{300}\inline i = \sqrt{-1}$ as well. We can then define a new collection of numbers called the Gaussian integers $\dpi{300}\inline \mathbb{Z}[i]$ which consist of elements of the form $\dpi{300}\inline a + b i$ for $\dpi{300}\inline a,b \in \mathbb{Z}$.  Given a prime $\dpi{300}\inline p \in \mathbb{Z}$, we also have $\dpi{300}\inline p \in \mathbb{Z}[i]$. Is $\dpi{300}\inline p$ still prime in $\dpi{300}\inline \mathbb{Z}[i]$? It turns out that sometimes, but not always! For instance, in $\dpi{300}\inline \mathbb{Z}[i]$ we have $\dpi{300}\inline 2 = (1 + i)(1 - i )$ and $\dpi{300}\inline 5 = (2 + i) (2 - i)$, but $\dpi{300}\inline 3$ is still prime! This fact is related to writing $\dpi{300}\inline p$ as a sum of two squares as well as how the polynomial $\dpi{300}\inline x^2 + 1$ factors when considered modulo $\dpi{300}\inline p$.  The study of how primes factor when considered in larger rings is referred to as algebraic number theory. In this course we will study algebraic number theory for the simplest cases of $\dpi{300}\inline \mathbb{Z}[i]$ and possibly other rings of the form $\dpi{300}\inline \mathbb{Z}[\sqrt{d}]$ to see the underpinnings of the overall vast theory.

2 units