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.

Applications of Graph Theory

We will learn a selection of interesting results in graph theory, together with their wide-ranging applications. Prior familiarity with graph theory is helpful but not required; necessary concepts and definitions from graph theory will be covered.

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.

Cryptology

Cryptology is the study of sending secret messages over insecure communication channels. Cryptographic capabilities are important to politics and foreign affairs, and underlie the functioning of a great deal of the modern economy. In this course we will study the mathematical theories and techniques that make cryptography function, and also those that break it. We will learn how these mathematical tools influence the use of cryptographic systems, and practice explaining these implications to people without developed mathematical backgrounds.

History of Mathematics (Euler to Uhlenbeck)

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.

History of Mathematics (Euclid to Newton)

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.

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.

Introductory Algebraic Number Theory

One of the most basic facts in all of mathematics is the Fundamental Theorem of Arithmetic: given a positive integer $n$ , 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 $x^2 + 1 = 0$ . This naturally leads us to consider the number $i = \sqrt{-1}$ as well. We can then define a new collection of numbers called the Gaussian integers $\mathbb{Z}[i]$ which consist of elements of the form $a + b i$ for $a,b \in \mathbb{Z}$ . Given a prime $p \in \mathbb{Z}$ , we also have $p \in \mathbb{Z}[i]$ . Is $p$ still prime in $\mathbb{Z}[i]$ ? It turns out that sometimes, but not always! For instance, in $\mathbb{Z}[i]$ we have $2 = (1 + i)(1 - i )$ and $5 = (2 + i) (2 - i)$ , but $3$ is still prime! This fact is related to writing $p$ as a sum of two squares as well as how the polynomial $x^2 + 1$ factors when considered modulo $p$ . 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 $\mathbb{Z}[i]$ and possibly other rings of the form $\mathbb{Z}[\sqrt{d}]$ to see the underpinnings of the overall vast theory.

Mathematical Applications of Linear Algebra

We will use the book Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra, by Jiřì Matoušek. We will concentrate mostly on the mathematical applications (as opposed to the algorithmic applications). In the first two class meetings we will learn about (generalized) vector spaces, as preparation for the remainder of the course.

Mathematical and Scientific Computing

This course is an introduction to the field of Mathematical and Scientific Computing. As such, we will investigate complex mathematical problems, models and simulations, combined with related computer algorithms and advanced tools, used to generate solutions for these problems. Such problems come from a variety of areas in mathematics, as well as from Biology, Economics, and more. Topics include very large and very precise numbers, measures of algorithmic efficiency, a variety of programming languages and platforms, memory – speed tradeoffs, and more.

Credits

2 units