Topology, in broad terms, is the study of those qualities of an object that are invariant under certain deformations. Such deformations include stretching but not tearing or gluing; in laymen’s terms, one is allowed to play with a piece of paper clay without poking holes in it or joining two separate parts together. (A popular joke is that for topologists, a doughnut and a coffee mug are the same thing, because one can be continuously deformed into the other.)
A topology on an object is a structure that determines which subsets of the object are open sets; such a structure is what gives the object properties such as compactness, connectedness, or even convergence of sequences. For example, when we say that [0, 1] is compact, what we really mean is that with the usual topology on the real line R, the subset [0, 1] is compact. We could easily give R a different topology (e.g., the lower limit topology), such that the subset [0, 1] is no longer compact. Point-set topology is the subfield of topology that is concerned with constructing topologies on objects and developing useful notions such as separability and countability; it is closely related to set theory.
There are other subfields of topology. One subfield is algebraic topology, which uses algebraic tools (i.e., groups, rings, etc.) to rigorously express intuitions such as “holes”. For example, how is a hollow sphere different from a hollow doughnut (known also as a torus)? One may say that the torus has a “hole” in it while the sphere does not. This intuition is captured by the notion of the fundamental group, which, (very) loosely speaking, is an algebraic object that counts the number of “holes” of a topological space. There are other useful algebraic tools, including various homology and cohomology theories. These can all be viewed as a mapping from the category of topological spaces to algebraic objects, and are very good examples of functors in the language of category theory; it is for this reason that many algebraic topologists are also interested in category theory.
Another subfield is geometric topology, which is the study of manifolds, spaces that are locally Euclidean. For example, hollow spheres and tori are 2-dimensional manifolds (or, simply, 2-manifolds). Because of this Euclidean feature, very often (although unfortunately not always), a differentiable structure can be put on manifolds, and geometry (which is the study of local properties) can be used as a tool to study their topology (which is the study of global properties). A very famous example in this field is the Poincaré conjecture, which was proven using (advanced) geometric notions such as Ricci flows. Of course, algebraic tools are still useful for these spaces.
The study of 1- and 2-manifolds is arguably complete – as an exercise, you can probably easily list all 1-manifolds without much prior knowledge—and inexplicably, much about manifolds of dimension greater than 4 is known. However, for a long time, many aspects of 3- and 4-manifolds had evaded study; thus developed the subfield of low-dimensional topology, the study of manifolds of dimension 4 or below. This is an active area of research, and in recent years has been found to be closely related to quantum field theory in physics.
Topology Courses [Show]Topology Courses [Hide]
There is only one course in topology at the undergraduate level at Princeton.
MAT 365: Topology
This is the first course in topology that Princeton offers, and has been taught by Prof. Zoltan Szabó in the last few years. The course, following the classic textbook by Munkres, is a careful study of point-set topology. It begins with examining different topologies one can put on familiar spaces, and constructions such as product, metric and quotient topologies. The second part of the course is concerned with developing notions of “desirable” properties, including separability axioms like T0 , T1 and the Hausdorff condition, as well as countability axioms like the first- and second-countable conditions. The final part of the course is an introduction to the fundamental group π1; after some initial calculations (including that of π1(S1), the fundamental group of the circle), more general tools such as covering spaces and the Seifert-van Kampen theorem are used to calculate π1 of more complicated spaces. This course is a prerequisite for all other topology courses at Princeton. Naturally, I recommend all students interested in topology take this; moreover, since this course deals with many basic notions that one will certainly meet in the future, I would think that almost every mathematics student should take this course. As a bonus, this course satisfies the geometry requirement of the department.
Occasionally, there is a junior seminar about a particular topic in topology; usually it is about knot theory.
Junior Seminar in Knot Theory
This seminar is an introduction to knot theory. Like other junior seminars, students are expected to present a topic on their own. Topics covered vary, but typically include tri-colorability of knots and links, numerical knot invariants such as the crossing number, unknotting number and bridge number, and polynomial invariants such as the Jones polynomial and the Alexander–Conway polynomial. More advanced students may learn about categorification of these polynomials (i.e. homology invariants), such as the Khovanov homology and the Heegaard Floer homology.
There are five main graduate courses in topology. (There may be other occasional courses.)
MAT 560: Algebraic Topology
This course is an introduction to algebraic topology. Depending on the instructor, the materials covered may vary. Prof. Zoltan Szabó usually follows the (again) classic textbook by Hatcher, while Prof. William Browder, who recently retired, uses the more calculus-flavored textbook by Madsen and Tornehave. When following Hatcher, the course covers notions such as CW complexes, the fundamental group, simplicial and singular homology, as well as tools to compute these homologies. The dual notion of singular cohomology may also be covered. When following Madsen and Tornehave, the course is an in-depth examination of the theory of de Rham cohomology – which, for smooth manifolds, coincides with singular cohomology. De Rham cohomology originates from calculus, and makes heavy use of the rigorous notion of differential forms, which has been less rigorously introduced in MAT 218 when discussing integration and generalized Stokes’ theorem. Homology and cohomology are dual notions, and are related by Poincaré duality, which, time permitting, is covered in both approaches.
This course is designed for more serious students of topology. Algebraic tools are so well-developed within the realm of topology, that this subject is essentially a prerequisite for any more advanced study. After taking MAT 365, students should have some idea about their abilities to visualize topological spaces and their interests in the subject; I would suggest those who consider pursuing a possible research career in the field to immediately follow up with this course.
MAT 568 and MAT 569: Gauge Theory, Low-Dimensional Topology and Symplectic Geometry
This course is more likely to be taught by Prof. Peter Ozsváth or Prof. Zoltan Szabó. The course is probably about Heegaard Floer homology, developed by the two professors, and/or other homology theories in low-dimensional topology, as well as their interactions with symplectic geometry, gauge theory and other related topics. MAT 365 is definitely a prerequisite, and almost definitely MAT 560 too; it is also probably good to know Morse theory before taking the course.
MAT 571 and MAT 572: Low-Dimensional Topology
This course is more likely to be taught by Prof. David Gabai. The course probably takes a more “classic” approach towards low-dimensional topology, covering notions such as foliations, hyperbolic structures and their applications. I have never taken this course, so it may be wise to check with the professor before enrolling in it.
Outlook and Advice [Show]Outlook and Advice [Hide]
It is great to study topology at Princeton. Princeton has some of the best topologists in the world; Prof. William Browder (who has just retired), Prof. David Gabai, Prof. Zoltan Szabó and Prof. Peter Ozsváth are all well-known mathematicians in their fields. The junior faculty also includes very promising young topologists.
Prof. William Browder was a pioneer and a major contributor to the classification of high-dimensional manifolds, using surgery theory. Prof. David Gabai has been an important figure in low-dimensional topology, and is especially known for his contributions in the study of hyperbolic 3-manifolds. Prof. Zoltan Szabó and Prof. Peter Ozsváth together invented Heegaard Floer homology, which is a homology theory for 3-manifolds.
After finishing the sequence MAT 365 and MAT 560, topology students can consider taking a junior seminar in knot theory (or some other topic), or, if that is not available, writing a junior paper under the guidance of one of the professors. (Both junior and senior faculty members are probably willing to provide supervision.) It is also a good idea to learn Morse theory, which is an extremely beautiful theory that decomposes a manifold into a CW structure by studying smooth functions on that manifold. The graduate courses are challenging, but not impossible, so interested students are recommended to speak to the respective professors early.
For a student of topology, it may also be beneficial to learn other related topics well. These include basic abstract algebra, Lie theory, algebraic geometry, and, especially, differential geometry.