Joe Johns

Max Planck Institute	Phone:(49)341-9959-880
Inselstraße 22, Leipzig 04103, Germany	Email:johns[AT]mis.mpg.de

Teaching: Analysis IV, April-July 2012, University of Leipzig

• Homework 1 (Analysis IV), due Wed May 2
• Homework 2 (Analysis IV), due Wed May 9

Below, find some notes from Analysis II. And for Analysis III, find syllabus and homework, and some notes we will roughly follow, called "textbook", see Chapters 11,12,13,15.

• Summary of key concepts from Analysis II
• More detailed summary of line and surface integrals (Analysis II).
• Syllabus for Analysis III
• Final grade will be 100% based on final exam, and you must have 50% of homework points to write the exam.
• Final exam date: Thurs Feb 16, in room R221, 10:00-13:00.
• Review sheet for Analysis III
• Practice problems for Analysis III
• textbook
• Someone requested "a nice simple explanation of manifolds", with examples. Here are two fairly short extracts from two books, which might be helpful. See also the wikipedia entry, or planet math.
• Manifolds in R^n. Has some examples, pictures, and basic facts. The exercises at the end are good if you want to sharpen your knowledge some day.
• Abstract manifolds as a set (not embedded in R^n). Has a few examples too.

  Below there is a reference for differential forms. Here are some others: Rudin's Principles of mathematical anaylsis Chapter 10; and our textbook above (chapter 15 section 2.2.2); and my notes above "summary of key concepts from analyis II" (week 14).

• Differential Forms. This is a textbook you might find useful to review differential forms by David Bachman.
There is a lot of explanation of the geometric meaning of differential forms in here, which is good for you in the long run, but which we did not cover in this class. There is also some good explantion of how to compute things, which is more like what we did in our class: see sections 3.4-3.6 for integration and section 4.4 for differentiation. See chapter 5 for Stoke's theorem, and also a nice discussion of how the classical theorems (Stoke's, divergence thm) connect to the general Stoke's formula. For a nice short discussion of manifolds see Ch. 7. (For later: check out Ch. 6 for physics like Electro-Magnetism.)
• This is section 1.1 (a review of ODE's) from "Basic Partial Differential Equations" by Bleeker and Csordas.
• This is sections 1.3, 2.1 (on seperation of variables and 1st order linear PDEs) from "Basic Partial Differential Equations" by Bleeker and Csordas.
• This is section 2.2 (on variable coefficents and the parametric form of the solution) from "Basic Partial Differential Equations" by Bleeker and Csordas.
• Homework 1, due Monday Oct 17
• Homework 2, due Monday Oct 24
• Homework 3, due Tuesday November 1
• Homework 4, due Monday November 7
• Homework 5, due Monday November 14
• Homework 6, due Monday November 21
• Proof of Theorem B (omitted from lecture): Read and understand this for extra practice with the concepts and/or if you want to have an idea why Theorem B is true.
• Homework 7, due Monday November 28
• Homework 8, due Monday December 5
• Homework 9, due Monday December 12
• Homework 10, due Monday December 19
• Homework 11, due Monday January 9
• Homework 12, due Monday January 16
• Homework 13, due Monday January 23
• Homework 14, due Monday January 30

Research Interests
• Symplectic topology, Lefschetz fibrations, Lagrangian submanifolds, Floer cohomology
• A summary of recent research.
• My C.V.

Publications and preprints

• Lefschetz fibrations on Cotangent bundles of 2-manifolds. To appear in Gokova Geom. Topol. GGT (2012)
• Complexifications of Morse functions and the directed Donaldson-Fukaya category. J. Symplectic Geom. 8 (2010), no. 4, pp. 403-500.
• The Picard-Lefschetz theory of complexified Morse functions. (68 pages) Submitted to Algebraic and Geometric Topology (AGT), January 2010
• Three symplectic Morse-Bott operations: handle attachments, plumbing, and Lagrangian surgery (17 pages, preprint) This is used in the AGT paper. Will probably submit to a different journal.
• The Flow category and the Fukaya category (14 pages, preprint) This has been subsumed in the JSG paper.
• An open mapping theorem for o-minimal structures, Joseph Johns, The Journal of Symbolic Logic, vol. 66, no.4 (Dec. 2001) pp. 1817-1820.