Geometry Projects

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: 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: Convex Optimization, by Stephen Boyd and Lieven Vandenberghe

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

Convexity is the secret sauce behind the success of modern optimization techniques, which have in turn generated myriad applications from circuit design to machine learning. Despite its widespread diffusion across the disciplines of engineering, finance, and computer science, convex optimization possesses at its center a startlingly elegant theory. Convexity is an algebraic property of Euclidean spaces which manages to interact cleanly with their topological structure. Duality of convex programs leads not only to deeper insights into the geometry of the problems but also to consequential optimization algorithms. In this project, the mentee will develop an acquaintance with convexity and learn to recognize problems which are convex in nature. They will study Lagrange duality and its many interpretations. They will also explore different algorithms for unconstrained and constrained optimization problems. Depending on the mentee’s interests and maturity, this project can either adopt a more theoretical flavor with an excursion into subdifferential analysis, or a more applied flavor with “hands on” computational problems.

Book: Elementary Differential Geometry (2nd Edition), by Andrew Pressley

Prerequisites: Multivariable calculus (e.g. MAT 203), and a proof based course (e.g. MAT 200).

The first goal is to carefully study curves in the plane and in space: definitions, revisiting the concept of the length of a curve, and reparametrization. The next goal will be to define curvature and understand it through the study of curvature in the plane and then in space. Finally, the ultimate goal will be to explore interesting theorems such as the isoperimetric inequality, the four-vertex theorem, or simply study very thoroughly an example (such as the helix) in a detailed manner. The latter will depend on the tastes and preferences of the student.

Book: Differential Geometry – Curves - Surfaces - Manifolds (2nd Edition) by Wolfgang Kuhnel

Prerequisites: Calculus III-IV (e.g. MAT 203, MAT 303), linear algebra (e.g. MAT 211), and real analysis (e.g. MAT 319/320). Complex analysis (MAT 342) is a bonus.

A study of curves in R^n: both the local and global theory, in order to see how the study of curves in 2D and 3D can generalize to many dimensions. This will allow us to study the local theory of surfaces at a more precise level. The goal will be to grasp well the fundamentals about the local theory of surfaces, and then study very carefully one specific type of surfaces of interest. These include but are not limited to: ruled surfaces, minimal surfaces, and hypersurfaces in R^{n+1}.

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: The Geometry of Fractal Sets, by Kenneth Falconer.

Prerequisites: A course in analysis (e.g. MAT 319/320). Enrollment in a class in measure theory (e.g. MAT 324) would be helpful, but is not necessary, as the minimal measure theory needed can be developed during the project.

We are used to the idea that points are zero dimensional, lines have one dimension, and planes have two dimensions. However, defining dimension of general sets in a careful way is not so easily done. Take for example, the Middle Thirds Cantor set, obtained by taking the unit interval and removing the middle third, and continuing this process indefinitely to the remaining intervals. What is the dimension of this set?

Such “fractal” sets occur naturally in analysis, probability, and dynamical systems, and mathematicians have developed several notions of dimension to study these sets. In this project, we will define various notions of dimension for sets in the plane, discuss some general properties of these dimensions, and use them to study interesting examples of fractal sets like the one described above. Possible directions for this project include constructing continuous yet nowhere differentiable functions, studying when fractal sets are a subset of a path in the plane, studying projections of fractal sets onto lines, or whatever other possible topic the student is interested in exploring. There are many possible directions.

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.