{"id":89,"date":"2017-01-18T14:59:21","date_gmt":"2017-01-18T18:59:21","guid":{"rendered":"http:\/\/blogs.nd.edu\/chris-schommer-pries\/?page_id=89"},"modified":"2017-01-18T14:59:21","modified_gmt":"2017-01-18T18:59:21","slug":"topics-in-topology-summer-2013","status":"publish","type":"page","link":"https:\/\/sites.nd.edu\/chris-schommer-pries\/courses\/topics-in-topology-summer-2013\/","title":{"rendered":"Topics in Topology, Summer 2013"},"content":{"rendered":"<p>Summer 2013, Central Lecture of the <a href=\"http:\/\/www.math.uni-bonn.de\/people\/grk1150\/index.html\" target=\"_blank\">Graduiertenkolleg 1150<\/a> (Homotopy and Cohomology). Thur 11-13:00 (SemR 0.011) (<a href=\"https:\/\/basis.uni-bonn.de\/qisserver\/rds?state=verpublish&amp;status=init&amp;vmfile=no&amp;publishid=72123&amp;moduleCall=webInfo&amp;publishConfFile=webInfo&amp;publishSubDir=veranstaltung\" target=\"_blank\">University link<\/a>). First lecture: April 11th.<\/p> <p style=\"text-align:center\"><strong>Lecture Notes and Exercises<\/strong><\/p> <p style=\"text-align:center\"><a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BordismOldAndNew2013.pdf\" target=\"_blank\">Full lecture notes<\/a> (in one pdf, last update July 17, 2013)<\/p> <ul> <li><a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BordismOldAndNew2013-Course-Syllabus.pdf\"><span style=\"color:#444444;line-height:1.7\">Course Syllabus<\/span><\/a><\/li> <li><a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BO%26N-Lect1.pdf\">Lecture 1: Overview<\/a> (April 11, 2013) <ul> <li> <a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BO%26N-Exercises1.pdf\">Exercises 1 (pdf)<\/a><\/li> <li>Does anyone have a link to a good video of the &#8220;belt-trick&#8221; or something equivalent?<\/li> <\/ul> <\/li> <li><span style=\"color:#444444;line-height:1.7\"><a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BO%26N-Lect2.pdf\">Lecture 2: Pontryagin&#8217;s Construction<\/a> (April 18,2013)<\/span> <ul> <li>Exercises 2 (<a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BO%26N-Exercises2.pdf\">pdf<\/a>) (29.4.2013 &#8211; begining of exercise 2.1 corrected)<\/li> <li><a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BO%26N-Lect2-Addendum.pdf\">Addendum (pdf)<\/a> on the relationship between stable tangential framings and stable normal framings. (there are still some errors in this. 29.4.2013)<\/li> <\/ul> <\/li> <li><span style=\"color:#444444;line-height:1.7\"><a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BO%26N-Lect3.pdf\">Lecture 3: Introduction to Jet Transversality<\/a>. (April 25, 2013) (update: 29.4.2013)<\/span> <ul> <li><a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BO%26N-Exercises3.pdf\">Exercises 3 (pdf)<\/a> (exercise 3.2 corrected)<\/li> <\/ul> <\/li> <li><a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BO%26N-Lect4.pdf\">Lecture 4: The Pontryagin-Thom Construction<\/a> (May 2, 2013) <ul> <li><a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BO%26N-Exercises4.pdf\">Exercises 4 (pdf)<\/a> (There are some really fun ones this time!)<\/li> <\/ul> <\/li> <li>(No Lecture 9.5.2013)<\/li> <li><a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BO%26N-Lect5.pdf\">Lecture 5:  A General Proof of Density Theorems<\/a> (May 16, 2013) (like Thom&#8217;s Jet Transversality Theorem) (update 16.5.2013: Incorrect Lemma 5.1.8 stricken, proof of Globalization Thm 5.2.3 altered slightly to avoid using the faulty lemma). <ul> <li><a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BO%26N-Exercises5.pdf\">Exercises 5 (pdf)<\/a><\/li> <\/ul> <\/li> <li>(No Lecture 23.5.2013 or 30.5.2013)<\/li> <li>Lecture 6: End of Proof of Jet Transversality (June 6, 2013) (see previous week for lecture notes) No exercises this week.<\/li> <li><a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BO%26N-Lect7.pdf\" target=\"_blank\">Lecture 7: The h-principle, part I.<\/a> (June 13,2013) <ul> <li><a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BO%26N-Exercises7.pdf\" target=\"_blank\">Exercises 7 (pdf)<\/a><\/li> <\/ul> <\/li> <li>Lecture 8: The h-principle, part II: (June 20, 2013)<\/li> <li>Lecture 9: The h-principle, part III (June 27, 2013) <ul> <li><a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BO%26N-Fibrations.pdf\">Notes on fibrations<\/a><\/li> <li><a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BO%26N-Flexible_sheaves.pdf\">Notes on flexible sheaves<\/a><\/li> <\/ul> <\/li> <li><span style=\"color:#444444;line-height:1.7\">Lecture 10: The h-principle , part IV (July 4, 2013)<\/span> <ul> <li><a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BO%26N-Lemma-E-Borel.pdf\">A surprising lemma of \u00c9mile Borel<\/a><\/li> <\/ul> <\/li> <li>Lecture 11: The h-principle, part V (July 11, 2013) <ul> <li><a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BO%26N-Gromov%27s-Theorem.pdf\">Smooth functions satisfy the h-principle<\/a><\/li> <\/ul> <\/li> <li>Lecture 12: The h-principle, part VI (July 18, 2013) <ul> <li><a href=\"https:\/\/dl.dropboxusercontent.com\/u\/8165870\/BO%26N2013\/BO%26N-Gromov%27s-Theorem-II.pdf\" target=\"_blank\">Gromov&#8217;s Theorem for open partial differential relations<\/a><\/li> <\/ul> <\/li> <\/ul> <h5 style=\"text-align:center\">Abstract<\/h5> <p><span style=\"color:#444444;line-height:1.7\">A basic problem in topology is to gain an understanding of the structure of both manifolds and maps between them. A powerful tool for this is the theory of bordism, as initiated in the mid-20th century by Ren\u00e9 Thom and Lev Pontryagin. This theory provides a powerful feedback loop between manifolds on the one hand and maps between them, on the other.<\/span><\/p> <p><span style=\"color:#444444;line-height:1.7\">Topological field theories are a modern extension of bordism invariants. They  <\/span><span style=\"color:#444444;line-height:1.7\">provide a further bridge linking topology and algebra together. The <\/span><span style=\"color:#444444;line-height:1.7\">classification of topological field theories, which combines methods and ideas from differential topology, homotopy theory, and  higher category theory, has lead to a greater understanding of this back-and-forth interplay.  <\/span><\/p> <p>In this lecture course we will take stroll along these bridges. We will focus on some of the techniques and ideas which play a part in the classification of TFTs, beginning with the classical theory of bordism and progressing into more modern developments.<\/p> <p><strong>Prerequisites<\/strong>: Algebraic topology (homology, cohomology, and homotopy theory). Smooth manifolds, tangent bundles. Classifying spaces. Exposure to characteristic classes, spectra, or spectral sequences would be helpful.<\/p> <p><strong>Homework:<\/strong> The best way to learn is by doing, and so I will provide (optional) homework problems every week.<\/p> <p><strong>Text<\/strong>: There is no single text. I will give some references and I am hoping to provide lecture notes. I am drawing from lectures given by:<\/p> <ul> <li><span style=\"line-height:14px\">Dan Freed, <a href=\"http:\/\/www.ma.utexas.edu\/users\/dafr\/M392C\/index.html\" target=\"_blank\">Bordism Old and New<\/a> (UT Austin, Fall &#8217;12). Yes, I stole his title. <\/span><\/li> <li>John Francis, <a href=\"http:\/\/www.math.northwestern.edu\/~jnkf\/classes\/hprin\/\" target=\"_blank\">The h-Principle in Topology<\/a> (Northwestern, winter &#8217;10\/&#8217;11)<\/li> <li>Michael Weiss, <a href=\"http:\/\/wwwmath.uni-muenster.de\/u\/mweis_02\/preprints\/hprin23jan13.pdf\" target=\"_blank\">Immersion Theory for Homotopy Theorists<\/a> (U. of Aberdeen, winter &#8217;04\/&#8217;05)<\/li> <\/ul> <h4 style=\"text-align:center\"><span style=\"font-size:.83em;line-height:1.7\">A Preliminary List of Topics\/Lectures&#8230;  subject to change<\/span><\/h4> <ol> <li><span style=\"line-height:14px\">Overview<\/span><\/li> <li>Morse Theory and Handles, the second stable stem.<\/li> <li> Thom Transversality, &#8220;Transversality unlocks the secrets of manifolds&#8221;<\/li> <li>The Pontryagin-Thom construction<\/li> <li><span style=\"color:#444444;line-height:1.7\">Weak fibrations and h-principles (Segal categories?)<\/span><\/li> <li>The Hirsch-Smale theorem (a.k.a. Immersion theory)<\/li> <li>Configuration Spaces, scanning,  and more.<\/li> <li>The third stable stem, via geometry.<\/li> <li>The Cobordism Hypothesis, part 1: ideas and statements<\/li> <li>The Galatius-Madsen-Tillmann-Weiss theorem<\/li> <li>The Cobordism Hypothesis, part 2: overview of the proof<\/li> <li>leftovers and applications.<\/li> <\/ol> ","protected":false},"excerpt":{"rendered":"<p>Summer 2013, Central Lecture of the Graduiertenkolleg 1150 (Homotopy and Cohomology). Thur 11-13:00 (SemR 0.011) (University link). First lecture: April 11th. Lecture Notes and Exercises Full lecture notes (in one pdf, last update July 17, 2013) Course Syllabus Lecture 1: Overview (April 11, 2013) Exercises 1 (pdf) Does anyone have a link to a good [&hellip;]<\/p>\n","protected":false},"author":2296,"featured_media":0,"parent":56,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-89","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sites.nd.edu\/chris-schommer-pries\/wp-json\/wp\/v2\/pages\/89","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.nd.edu\/chris-schommer-pries\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sites.nd.edu\/chris-schommer-pries\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/sites.nd.edu\/chris-schommer-pries\/wp-json\/wp\/v2\/users\/2296"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.nd.edu\/chris-schommer-pries\/wp-json\/wp\/v2\/comments?post=89"}],"version-history":[{"count":1,"href":"https:\/\/sites.nd.edu\/chris-schommer-pries\/wp-json\/wp\/v2\/pages\/89\/revisions"}],"predecessor-version":[{"id":90,"href":"https:\/\/sites.nd.edu\/chris-schommer-pries\/wp-json\/wp\/v2\/pages\/89\/revisions\/90"}],"up":[{"embeddable":true,"href":"https:\/\/sites.nd.edu\/chris-schommer-pries\/wp-json\/wp\/v2\/pages\/56"}],"wp:attachment":[{"href":"https:\/\/sites.nd.edu\/chris-schommer-pries\/wp-json\/wp\/v2\/media?parent=89"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}