Undergraduate Topology - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T08:14:20Z http://mathoverflow.net/feeds/question/105246 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105246/undergraduate-topology Undergraduate Topology Wondering 2012-08-22T16:12:55Z 2012-08-23T00:55:25Z <p>I am developing an introductory topology course for undergraduates, and I am wondering what topics to cover. At my institution, real analysis is not a prerequisite for the course, so it is more than likely that the intended audience has not been exposed to this material. Does anyone have any suggestions?</p> http://mathoverflow.net/questions/105246/undergraduate-topology/105254#105254 Answer by Aeryk for Undergraduate Topology Aeryk 2012-08-22T17:39:05Z 2012-08-22T17:39:05Z <p>I've found that doing low-dimensional manifold topology is very appealing to undergraduates. I used the "Topology Now!" text by Messer and Straffin and, while the text isn't perfect, the approach was wonderfully successful.</p> http://mathoverflow.net/questions/105246/undergraduate-topology/105258#105258 Answer by Theo Johnson-Freyd for Undergraduate Topology Theo Johnson-Freyd 2012-08-22T18:06:59Z 2012-08-22T18:06:59Z <p>I second Aeryk's suggestion to focus on low-dimensional topology. More generally, I think that algebraic topology can be more exciting than point-set topology.</p> <p>That said, when I was an undergraduate, I do remember being quite excited about the Bourbaki program and point-set definitions and so on. I remember one "first course in topology" that alternated days: low-dimensional topology on even days and point-set on odd. Except the point-set portion began with set theory, cardinal and ordinal numbers, and the axiom of choice; then moved on to metric spaces; and only then introduced point-set topology. I basically think that to motivate the point-set definitions, you had better start with metric spaces.</p> <p>If on the other hand you focus more on broadly-defined algebraic topology, then in addition to the low-dimensional topology of manifolds (surfaces, knots, etc.), another good topic is Brower fixed-point theorem as an application of fundamental group functor on pointed spaces. Perhaps, if you are very ambitious, you can prove that 2dTQFT = commmutative Frobenius algebra, and talk more generally about cobordism equivalence. Oh, and especially given the recent <a href="http://terrytao.wordpress.com/2012/08/22/bill-thurston/" rel="nofollow">sad news</a>, be sure to include a little Morse theory and <a href="http://www.geom.uiuc.edu/docs/outreach/oi/" rel="nofollow">Outside In</a>.</p> http://mathoverflow.net/questions/105246/undergraduate-topology/105260#105260 Answer by Kris Williams for Undergraduate Topology Kris Williams 2012-08-22T18:24:28Z 2012-08-22T18:24:28Z <p>As a resource for low-dimensional topology, I would suggest <a href="http://www.amazon.com/Shape-Space-Chapman-Applied-Mathematics/dp/0824707095/ref=sr_1_1?s=books&amp;ie=UTF8&amp;qid=1345659598&amp;sr=1-1&amp;keywords=shape+of+space%20%22The%20Shape%20of%20Space%22" rel="nofollow">The shape of space</a> by Jeffrey Weeks. It covers how to build compact two-manifolds and some three manifolds. Exercises include playing tic-tac-toe on surfaces and forming sums of surfaces. It then moves on to some three dimensional topology with a heavy focus on attempting to visualize the spaces.</p> http://mathoverflow.net/questions/105246/undergraduate-topology/105266#105266 Answer by Steve Huntsman for Undergraduate Topology Steve Huntsman 2012-08-22T19:55:28Z 2012-08-22T20:35:01Z <p>I think you could do a lot worse than to focus on modern applications by using the texts of <a href="http://books.google.com/books?id=MDXa6gFRZuIC" rel="nofollow">Edelsbrunner and Harer</a> and/or <a href="http://books.google.com/books?id=oKEGGMgnWKcC" rel="nofollow">Zomorodian</a> as touchstones and an avenue towards current work in topological data analysis. These books are self-contained treatments that focus on Morse theory and homology over \$\mathbb{Z}/2\mathbb{Z}\$, and there are a lot of materials and software available for a course to be built from.</p> http://mathoverflow.net/questions/105246/undergraduate-topology/105267#105267 Answer by Ronnie Brown for Undergraduate Topology Ronnie Brown 2012-08-22T20:18:36Z 2012-08-22T20:18:36Z <p>In the 1970s I developed an undergraduate course on knots, (source book was by Crowell and Fox) to replace general topology and homology, as it was very easy for students to understand the point of the course, there were interesting relations with group theory, and lots of specific calculations and other things to do. The course was eventually taken over by others, and resulted in a book, </p> <p>Knots and Surfaces, by N.D. Gilbert and T. Porter</p> <p>which had good reviews. </p> <p>For me, it led to giving popular talks on "How mathematics gets into Knots", and eventually to the exhibition you can see on the web site for the <a href="http:///www/popmath.org.uk" rel="nofollow">Centre for the Popularisation of Mathematics</a>. In these talks I could also talk <em>about</em> mathematics, including, for example, the importance of analogy in mathematics. This led to one boy at a talk for children, some aged 12, asking: "Are there infinitely many prime knots? " Wow! So giving this undergraduate course has led to all sorts of fun and rewarding things! My copper pentoil knot used with string to demonstrate the ideas of the fundamental group has also travelled to many countries, see for example the pdf of a <a href="http://pages.bangor.ac.uk/~mas010/WJS-beamer-handout.pdf" rel="nofollow">William J. Spencer Lecture</a> in Kansas, April, 2012. </p> http://mathoverflow.net/questions/105246/undergraduate-topology/105278#105278 Answer by Gerry Myerson for Undergraduate Topology Gerry Myerson 2012-08-22T23:54:05Z 2012-08-22T23:54:05Z <p>We give a Geometry and Topology course at Macquarie for students with no real analysis, using <a href="http://maths.mq.edu.au/~chris/notes/geometry_topology.html" rel="nofollow">notes</a> written by a colleague. <a href="http://rutherglen.science.mq.edu.au/math300s112/" rel="nofollow">Here</a> is the homepage for the course, so you can get an idea of what we do. </p> http://mathoverflow.net/questions/105246/undergraduate-topology/105281#105281 Answer by Gerhard Paseman for Undergraduate Topology Gerhard Paseman 2012-08-23T00:55:25Z 2012-08-23T00:55:25Z <p>If the intent is to provide breadth, then many of the suggestions others have made are quite appealing, especially if it is made clear what branches of topology are being introduced and what a student should do outside of class to develop depth in any or all of the branches.</p> <p>If the intent is to provide depth, there are likely several texts out there, one for each branch, with suggestions. I remember covering Munkres first course in Topology starting with chapter 2; even though we skipped over the set theory and foundations, I was intrigued enough by them to study set theory and foundations while in graduate school. Although the class did not go all the way through the book that first semester, we got exposed to quite a bit, and I developed more ofa taste for formalism from that class more than from any other that I took as an undergraduate.</p> <p>If the intent is to provide both depth and breadth, I suggest part of it be run as a student seminar. A later toplogy course I took had me present Sard's theorem; if nothing else came from that course I at least know how to prepare to explain Sard'd theorem for my next opportunity.</p> <p>Gerhard "And This Was Decades Ago" Paseman, 2012.08.22</p>