Algebra 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: Algebraic Topology, by Allen Hatcher

Prerequisites: Experience with metric spaces and groups.

What's the difference between a sphere and a doughnut? What about a circle and a punctured disk? The notion of homotopy, intuitively that of moving a path from one place to another, gives us a way of mathematically answering these questions. We will develop topological spaces and manifolds with the goal of studying the fundamental group of manifold. We will go through the van Kampen theorem and discuss covering spaces as well. If time permits a short introduction to homology is possible.

Book: The Banach-Tarski Paradox, by Grzegorz Tomkowicz and Stan Wagon

Prerequisites: A course in real analysis (e.g. MAT 319/320). Topology (e.g. MAT 364) and group theory (e.g. MAT 313) are not required, but are encouraged.

Can you imagine splitting a ball into five pieces, re-arranging them somehow, and be able to get two distinct balls identical to the original one? This can be done mathematically. This fact is called the Banach-Tarsky Paradox/Theorem.

To prove this, we need to use Measure Theory, understand some topological facts of the sphere, and make a brief detour into Group Theory. The goal of this project is to understand the proof of Banach-Tarsky’s Theorem.

Book: Category Theory in Context, by Emily Riehl

Prerequisites: A background in abstract algebra (e.g. MAT 313). Interested students will greatly benefit from a strong background in linear algebra (e.g. MAT 315) or topology (e.g. MAT 364).

As one pursues mathematics, a feeling of déjà vu can occur. The newest class covers a new “kind” of mathematical object and studies it intensively. Then, one moves on to study functions between objects of this type. Sets and functions, groups and group homomorphisms, vector spaces and linear maps, topological spaces and continuous maps: all are repetitions of this general trend. Moreover, similar kinds of constructions and proofs seem to reappear in these different contexts, always with a mostly similar appearance. For example, sets have a Cartesian product, vector spaces have the product space, and topological spaces also have a product space. Finally, many mathematical notions have some notion of “naturality” or “canonicality” to them. For example, a finite-dimensional vector space and its dual space are known to be isomorphic, but this isomorphism is not “natural”. This project will aim to introduce the body of mathematics which can explain all of these curious facts: category theory. We will introduce the concept of categories, discuss the process of diagram-chasing, and arrive at the idea of a limit. Other topics in category theory and relationships with other parts of mathematics may be discussed, according to interest.

Book: Clifford Algebras: an Introduction, D. J. H. Garling

Prerequisites: A course in linear algebra (e.g. MAT 310). A course in abstract algebra (MAT 313, MAT 314 encouraged)

The Clifford algebra associated to a given vector space with a quadratic form (like Euclidean space) is a very useful object to use to study the geometry of that quadratic form. Another way to say that this is a useful concept is to see that it comes up in multiple different contexts, like algebra, geometry, topology and physics. Our goal in this project is to get acquainted with this concept and its usefulness. We will aim at classifying the real and complex Clifford algebras, introduce the spin groups and study their relevance to the representation theory of orthogonal groups. This last fact comes up in physics, as discovered by Dirac, which we will also discuss.

Book: Office Hours with a Geometric Group Theorist, by Matt Clay and Dan Margalit

Prerequisites: Linear algebra (e.g. MAT 211). Some group theory would help but not required

Geometric group theory is a relatively new field concerning the geometric aspects of groups. It is a subject that began in the 80s that is an area of active research today. This project will start with a crash course on groups and eventually familiarize the mentee with the basic ideas of the field, while strongly emphasizing a geometric point of view. Time permitting, the project can cover more complicated topics, such as the mapping class group and hyperbolic geometry.

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: Naive Lie Theory, by John Stillwell

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

Lie groups and Lie algebras are mathematical objects that have both a geometric structure, and an algebraic one. The classical example of a Lie group is the set of real numbers - we can add and subtract real numbers from each other, but there is also a geometric interpretation of the real numbers as an infinite line. Broadly speaking, Lie theory is concerned with finding relations between these aspects.

We will study specific examples of Lie groups and Lie algebras, focusing on those which arise as symmetries of vector spaces, and examine how to express these objects as collections of matrices.

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.

Book: The Symmetries of Things, by J. Conway, H. Burgiel, and C. Goodman-Strauss

Prerequisites: Only an enthusiasm for mathematics is required for this project. This project can very much be tailored to the mathematical background of the mentee.

Symmetries and symmetric patterns surround us in all aspects of our lives. We would like to be able to describe symmetrical patterns mathematically, and the first task will be to enumerate them. We will first discuss a new method, Conway’s enumeration, using four fundamental classifying features, seemingly non-mathematical in nature. We will then prove that this indeed classifies all possible planar patterns, using Euler’s theorem and some topology. The more advanced students will begin with spherical and frieze patterns, and will discover how group theory governs the possible symmetries that can occur. A more advanced student can go even further, venturing into hyperbolic space and flat universes.