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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MAT 355: Introduction to Differential Geometry
This course is taught by Professor Yang, and its topics are known to vary from year to year, especially those covered toward the end of the semester. Prof. Yang covered, with some level of detail, the first four (out of the five) chapters of do Carmo’s Differential Geometry. In particular, there was a heavy emphasis on the the Gauß map (involving discussion of the first and second fundamental forms) from chapter three, and the intrinsic geometry of surfaces R3 gone over in chapter four. Aside from do Carmo’s book, there was reliance on other sources to cover material, like discussion about minimal surfaces and the materials of the last couple of weeks. The last two weeks had a heavy emphasis on looking at the Laplacian on those surfaces, and the uniformization of surfaces.
MAT 416: Introduction to Algebraic Geometry (Kollar)
This is a course on varieties, which are sets of solutions to polynomial equations. Commutative algebra is a prerequisite, either in the form of MAT 447 or by reading Atiyah and MacDonald’s classic text and doing lots of exercises to get comfortable with the tools used in algebraic geometry. The course follows Shafarevich’s text and focuses on aspects of varieties, their local and global geometry, embeddings into projective space, and the specific case of curves which is extremely well-understood. The final third of the course consisted of student presentations about various special topics like elliptic curves, surfaces, resolutions of singularities, algebraic groups and others. This course is fast-paced and challenging, but worth the effort. Homeworks tended to vary in length, frequency and difficulty.
MAT 416: Introduction to Algebraic Geometry (Katz)
This is a course on sheaves, schemes and the cohomology of coherent sheaves on projective varieties. It follows the well-known text by Hartshorne. Commutative algebra is an absolute prerequisite and an introduction to varieties is highly recommended since schemes and sheaves are very abstract objects and having a good stock of examples in hand is vital to understand the material well. One unusual feature of this course which adds to its difficulty is that the bulk of the material on schemes and sheaves is relegated to the readings and homework while the instructor lectures on the cohomology of projective varieties. Thus in some sense, it is two courses rolled into one and one would be wise to treat it as such. This is arguably the most challenging course offered by the mathematics department due to the constantly steep learning curve and the exceptionally heavy workload. A final expository project makes for a fun finish to what certainly will be a grueling semester.
MAT 455: Advanced Topics in Geometry – Lie Theory
The goal of this course is to study the structure theory of Lie groups and Lie algebras. These objects are ubiquitous in mathematics and are studied using a variety of algebraic, analytic and geometric techniques. This course covers the geometry, structure theory, classification and touches upon their representation theories. Some background in differential geometry is essential, mostly material from the first few weeks of MAT 355. Alternatively, reading through the first few chapters of Spivak’s book on Differential Geometry should suffice. A variety of textbooks are useful — in particular, Adams, Humphreys and Bump.
Max Rabinovich ’13 (mrabinov[at]princeton[dot]edu)