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 Moebius 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.

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Max Rabinovich ’13 (mrabinov[at]princeton[dot]edu)