Topology Projects

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: 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: Topology from a Differentiable Viewpoint, by John Milnor

Prerequisites: Point-set topology (e.g. MAT 364)

Project goal: The field of differential topology concerns the topological aspects of smooth manifolds and smooth mappings. It is a subject that began in the 50s which has evolved into various other subjects over the last several decades (gauge theory, 4-manifold theory, Seiberg-Witten theory, symplectic geometry, etc.). This project will situate the mentee with the fundamental basics of the field (tools, ideas, methods), while strongly emphasizing a geometric point of view. Time permitting, the project can cover more complicated topics, such as exotic spheres and cobordism theory.

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: 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: 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: Calculus on Manifolds, by Michael Spivak

Prerequisites: A course in multivariable calculus (e.g. MAT 203)

In a standard course in calculus in multiple variables one will encounter three theorems: Green's theorem, relating the flow of a vector field in a planar region to the circulation on the boundary; Stokes' theorem, relating the curl of a vector field on a surface to the circulation on the boundary; and the Divergence theorem, relating the divergence of a vector field in a 3-D region to the flux on the boundary. These are all special cases of what is known as Stokes' theorem on manifolds. This project will develop the concept of a manifold and that of a differential form. We will learn how to integrate these forms on manifolds and eventually prove the most general version of Stokes' theorem.

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.

Book: Topology, Geometry, and Gauge Fields: Foundations, by G. Naber

Prerequisites: A course in linear algebra (e.g. MAT 310). A course in real analysis (e.g. MAT 319/320).

This project can adapt itself to the student’s interests and background. We will start with some topology: introduction to topological spaces, homotopy and homology, and depending on the student this can be enough for the whole reading project. On the other hand, we can also move on to differentiable manifolds and Lie groups. The chosen book is especially approachable, because these are introduced in a more explicit, hands-on, way. The book then moves on to the application of all of these to the study of Gauge Fields, which lie at the heart of modern physics, especially particle physics, and which became a stage for deep collaborations between geometers and theoretical physicists.