Notes on the adjunction, compactification, and mapping space topologies from John Terilla's topology course. There are other subfields of topology. One subfield is algebraic topology, which uses algebraic tools to rigorously express intuitions such as “holes.” For example, how is a hollow sphere different from a hollow 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 “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. It is great to study topology at Princeton. Princeton has some of the best topologists in the world; Professors David Gabai, Peter Ozsvath and Zoltan Szabo are all well-known mathematicians in their fields. The junior faculty also includes very promising young topologists. Prof. Gabai has been an important figure in low-dimensional topology, and is especially known for his contributions in the study of hyperbolic 3-manifolds. Profs. Ozsváth and Szabó together invented Heegaard Floer homology, 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. It may also be beneficial to learn other related topics well, including basic abstract algebra, Lie theory, algebraic geometry, and, in particular, differential geometry. Courses Access-restricted-item true Addeddate 2022-01-25 17:07:37 Autocrop_version 0.0.5_books-20210916-0.1 Bookplateleaf 0008 Boxid IA40327619 Camera Sony Alpha-A6300 (Control) Collection_set printdisabled External-identifierMunkres, James R. (2000). Topology (Seconded.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260. 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. Unless one is (and you are not!) planning to write a PhD thesis in General Topology, Munkres is (more than) enough.

Firstly I apologize if this is a bit of a soft question, it's hard for me to ask this quite concretely so I do apologize if this post doesn't seem like I'm asking something immediately. This seminar is an introduction to knot theory, and there is often one each year. Like other junior seminars, students are expected to learn and 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 homology invariants, such as the Khovanov homology and the Heegaard Floer homology.Munkres completed his undergraduate education at Nebraska Wesleyan University [2] and received his Ph.D. from the University of Michigan in 1956; his advisor was Edwin E. Moise. Earlier in his career he taught at the University of Michigan and at Princeton University. [2] Ocr tesseract 5.0.0-1-g862e Ocr_detected_lang en Ocr_detected_lang_conf 1.0000 Ocr_detected_script Latin Ocr_detected_script_conf 0.9936 Ocr_module_version 0.0.14 Ocr_parameters -l eng Old_pallet IA-WL-0000203 Openlibrary_edition While I certainly have a lot more Differential Topology and Algebraic Topology to learn (and I look forward to it), I also feel like I should learn a bit more of General Topology.