Number Theory Projects

Book: Algebra, by Michael Artin

Prerequisites: Proofs and calculus. Multivariable calculus (e.g. MAT 203).

The main goal of this project is to cover the basics of groups, matrices, eigenvalues and eigenspaces, linear transformations, symmetry groups. We will emphasize applications such as modular arithmetic, the rotational symmetries of a cube/platonic solid, or Burnside’s lemma, which involves permutation groups. The student could also explore symmetry groups of plane tilings, such as deriving the crystallographic restriction, something that is not covered in a standard abstract algebra class. Another possibility is to bring in some applications to topology, such as describing restrictions on the fundamental group of a surface.

Book: Undergraduate Algebraic Geometry, by Miles Reid

Prerequisites: Proofs. A course in ring theory and abstract algebra (e.g. MAT 313, MAT 314) is highly recommended.

Algebraic Geometry at its core is the study of polynomial equations. These polynomial equations define shapes called algebraic varieties. This project will begin with the case of plane curves: polynomial equations defined on a suitable compactification of the complex plane, known as projective space. We will ’play’ with various toy problems, like how many cubics (curves defined by degree 3 polynomials) can pass through a given set of points, with the main aim to study Bezout’s theorem: a theorem that tells us exactly how many intersection points there are between two curves. Using this, we will be able to solve some fun numerical problems, such as Pascal’s mystic hexagon.

If time permits, we will study higher dimensional varieties, and hopefully end with a famous theorem: every smooth cubic surface contains 27 lines.

Book: A Modern Introduction to Dynamical Systems, by Richard Brown

Prerequisites: A course in real analysis (e.g. MAT 319/320)

Continuous maps of the circle to itself are relatively simple to define and visualize. However, many systems of mathematical or physical interest are modeled extremely well by iterated application of such a map on a point on the circle. For example, asking how an initial point moves under the rotation map by some angle θ will lead us to an investigation of the continued fraction expansion, which is of great interest in number theory. Expanding on this will lead us to define the rotation number of a map and then to the stability of the solar system! On the other hand, the doubling map, which sends a point on the circle to a point with twice its angle from a reference point, is a fantastic example of a chaotic map. Investigation into this map leads to symbolic dynamics, connecting circle maps to linear algebra and information theory. Various connections with other parts of dynamical systems may be discussed, according to interest.

Book: The Poincare Half Plane: A Gateway to Modern Geometry, by Saul Stahl

Prerequisites: Calculus and proofs. Courses in linear algebra, topology, and complex analysis (e.g. MAT 310, 322, 342, 360, 362, 364) will allow the mentee to delve further into the material, but are not necessary.

In Euclidean geometry, one way to state the parallel postulate is to assume that given a line l and a point x not on that line, there exists exactly one line l’ containing x which does not intersect l.

For many years in the 18th and 19th centuries, mathematicians tried to deduce the parallel postulate from the other axioms of geometry, but a valid proof alluded them. Indeed, there are non-euclidean geometries (a fancy way of describing how to define the main objects in geometry like lines) which satisfy all the axioms of Euclidean geometry except the parallel postulate.

One such model is the hyperbolic ball or hyperbolic half plane. In hyperbolic space, the fastest way to get between two points may not be in a Euclidean straight line, but rather in certain circular arcs. The sum of the angles of a hyperbolic triangle is not necessarily equal to 180 degrees. In this project, we will study the basics of hyperbolic geometry in the planar setting, comparing theorems in Euclidean geometry and their hyperbolic counterparts. After getting through some basics, the direction will be up to the interests of the student.

Book: Irrational Numbers, by Iven Niven

Prerequisites: Calculus and Proofs

You've likely heard that many common numbers are irrational, meaning that they can't be expressed by dividing two whole numbers. Some examples are √2, π, and e. But why are they irrational? In this project the mentee will prove that these numbers in fact are not rational and will learn about how to mathematically approach irrational numbers. In doing so, they will begin developing techniques from the more advanced mathematical subjects of analysis and algebra.

Book: A Brief Introduction to Spectral Graph Theory, by Bogdan Nica or Spectral Graph Theory, by Fan R. K. Chung

Prerequisites: A course in linear algebra (e.g. MAT 211)

Graphs are mathematical objects used to represent societies and networks. The aim of spectral graph theory is to understand certain properties of graphs, by examining certain related matrices and their eigenvalues. For example, a classic theorem proved using spectral graph theory is the following: In a society where every two people have exactly one mutual friend, there is one person who is friends with everybody. This project can take on a more algebraic or a more analytic flavor depending on the interests of the student.