# Geometry

Geometry is a branch of mathematics that studies the properties of space. This includes the usual three-dimensional space of ordinary experience—suitably formalized, of course—but it includes many more exotic spaces. You might have heard of the Mobius strip or the Klein bottle, for example. These are both examples of spaces with interesting geometric properties. They are by no means the only ones. Going beyond these types of spaces, which resemble ordinary space on a small scale, geometry also studies a range of other types of spaces: varying from spaces that share the small scale structure of the complex plane to spaces defined purely in algebraic terms. This variety of spaces can be roughly divided into those studied by differential geometry and those studied by algebraic geometry.

Differential geometry is a part of geometry that studies spaces, called “differential manifolds,” where concepts like the derivative make sense. Differential manifolds locally resemble ordinary space, but their overall properties can be very different. Think of the surface of a donut: on a small scale, it looks like a slightly bent piece of a plane, but globally, it is nothing like a plane. Besides being bounded, it also has the unusual property that a string can be rolled up on it in a way that does not allow it to be unraveled. Differential geometry is a wide field that borrows techniques from analysis, topology, and algebra. It also has important connections to physics: Einstein’s general theory of relativity is entirely built upon it, to name only one example.

Algebraic geometry is a complement to differential geometry. It’s hard to convey in just a few words what the subject is all about. One way to think about it is as follows. A line, or a circle, or an ellipse, are all certainly examples of geometric structures. Now these can be thought of intrinsically, the way differential geometry might consider them, or they can be thought of as subsets of a larger space: the plane. Moreover, they are subsets with the very special property of being describable using Cartesian coordinates as the set of solutions to a collection of polynomial equations. Such sets are called “algebraic varieties,” and they can be studied not only in the setting of real-valued coordinates, but with coordinates that are complex numbers or, really, take values in any field. This is the classical face of algebraic geometry, and it is very likely to be your first introduction to the area. If you go further in it, you will be brought over to the abstract, modern point of view, which gives a way to define the geometries of algebraic varieties without reference to any outside space, or any polynomial equations. The vehicle for doing so is the notorious and unjustly vilified “scheme.” Algebraic geometry has connections just as far ranging as those of its differential cousin. It’s particularly important as a field in its own right and in algebraic number theory, but it has found uses in theoretical physics and even biology, as well.

In order to graduate with a degree in mathematics, it is required to take one course in discrete mathematics, geometry or topology.

Courses

MAT 350: Differential Manifolds
This course covers analysis in several variables, differential manifolds, differential forms, Stokes’ theorem, and some selected topics. It covers the later material of MAT218 and goes a little farther. It supposedly has significant overlap with MAT218 so one is recommended not to take both courses. However, we don’t know anyone who can confirm or deny this. Pre-requisites are MAT215.
Offered in the Fall.

MAT 355: Introduction to Differential Geometry
This course traditionally covers the classical theory of curves and surfaces in R3. Topics vary but common topics include the first and second fundamental form, Gauss map, Gauss-Bonnet theorem, minimal surfaces, affine connections, geodesics, exponential map. The usual textbook for this class is differential geometry of curves and surfaces. This course is a good preparation for a later course on Riemannian geometry, but it is also helpful for any further study in differential geometry. Pre-requisites are introductory analysis.
Offered in the Spring.

MAT 526: Topics in Geometric Analysis and General Relativity
The last couple of years, Professors Dafermos and Rodnianski have taught Topics graduate classes; the name seems rather self-evident. Very few undergraduates seem to take these courses, however.
Offered both semesters.

MAT 531: Introduction to Riemann Surfaces
Professor Gunning has taught this class for the last many years, until his retirement. The class has subsequently been taught by postdocs, and generally surveys complex manifolds, algebraic curves, de-Rham cohomology, projective spaces, and differential forms on complex manifolds. The theory is built up to prove the uniformization theorem, which states that any simply connected Riemann surface is equivalent to the open unit disk, the complex plane, or the extended complex plane. Complex Analysis and Topology are prerequisites, while Algebraic Topology, undergraduate differential geometry and Algebraic Geometry would be helpful to know but certainly not required. Offered in the Spring.

MAT 550: Differential Geometry
This course starts with Riemannian geometry. The textbook Riemannian Geometry by Do Carmo would be a good reference for this part of the course. The later part of the course covers special topics depending on the professor. Geometric analysis and CR geometry are likely topics. Prerequisites are familiarity with basics of smooth manifolds although there hasn’t really been a course that teaches this. It would be pretty helpful to take MAT355 first.
Offered in the Fall.

MAT 558: Topics in Conformal and Cauchy-Riemann Geometry
Taught alternatingly by Professors Chang and Yang for the last many years. Very few undergraduates seem to take these courses, however.
Offered both semesters.

Algebraic Geometry
In recent years, many undergraduates have learned Algebraic Geometry by constructing their own reading courses with professors. For advice on doing so, contact any of the following people:

Contacts

tbd

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