Time:  2:15  3:30pm, Tuesdays and Thursdays, Fall 2009. 

Place:  Gates 498 
Instructor:  Dmitriy Morozov (dmitriy@mrzv.org) 
 Homework 1 is now available.
 Homework 2 is now available.
 Homework 3 is now available.
 Course project reports are due Sunday, December 6.
Date  Topic  Notes 

Tu, Sep 22  Introduction. Connected Components. 

Th, Sep 24  Curves and Knots. 
On Wikipedia: Jordan Curve Theorem, Knot Theory, Links, Complexity. 
Tu, Sep 29  Surfaces.  [EH09] Section II.1. See [M00] Sections 74,76,77 for rigorous treatment of polygonal schema, cutting pasting, and surface classification. On Wikipedia: Surface, Crosscap, Projective plane, Klein bottle. 
Th, Oct 1  Fundamental Group. Homotopy.  HE's notes on Fundamental Group are not part of [EH09]. Afra Zomorodian's notes provide concise and clear review of the relevant topics in Group Theory. He also hosts English translation of Markov's paper on insolubility of homeomorphy. 
Tu, Oct 6  Simplicial Complexes. 
On Wikipedia: Simplex, Simplicial complex, Abstract simplicial complex, Barycentric subdivision. 
Th, Oct 8  Nerves.  [EH09] Sections III.2, III.3, III.4. [H01] Section 4.G, BronKerbosch algorithm. On Wikipedia: Nerve, VietorisRips complex, Voronoi diagram, Delaunay triangulation. 
Tu, Oct 13  Homology. Matrix Reduction. 
On Wikipedia: Homology, Reduced homology, Smith normal form. 
Th, Oct 15  Relative Homology.  [EH09] Section IV.3. On Wikipedia: Relative homology, Exact sequence. 
Tu, Oct 20  Exact Sequences.  [EH09] Section IV.4. On Wikipedia: Snake Lemma, MayerVietoris sequence. 
Th, Oct 22  Cohomology.  [EH09] Section V.1. On Wikipedia: Cohomology. 
Tu, Oct 27  Poincare Duality.  [EH09] Section V.2. On Wikipedia: Poincare duality. 
Th, Oct 29  Alexander Duality.  [EH09] Section V.4. On Wikipedia: Alexander duality. 
Tu, Nov 3  Smooth Generic Functions. 
On Wikipedia: Morse Theory. 
Th, Nov 5  PL Functions. 

Tu, Nov 10  Reeb graphs. 

Th, Nov 12  Persistent Homology. 

Tu, Nov 17  Stability. 

Th, Nov 19  Zigzag Persistence.  [CdS08]. 
Tu, Dec 1  Extended Persistence 

Th, Dec 3  Levelset Zigzag. 
[EH09]  (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18) Herbert Edelsbrunner and John Harer. Computational Topology: an Introduction. AMS Press, 2009. Note This book will not be available until January. However, it is a superset of course notes which can serve as a good supplement until the book is out. 
[CdS08]  Gunnar Carlsson and Vin de Silva. Zigzag Persistence. Manuscript, 2008. 
[M02]  Yukio Matsumoto. An Introduction to Morse Theory. AMS Press, 2002. 
[H01]  Allen Hatcher. Algebraic Topology. 2001. 
[CLRS01]  Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein. Introduction to Algorithms. MIT Press, 2nd edition, 2001. 
[M00]  James R. Munkres. Topology. Prentice Hall, 2000. 
[M84]  James R. Munkres. Elements of Algebraic Topology. Perseus, 1984. 
Leonidas Guibas and Dmitriy Morozov gratefully acknowledge the support to the Geometry Group provided by the Computer Forum during the 200910 academic year.