Math 7590: Braid Portfolio Schedule: Proposal & First Review: Thursday, 4 March. By end of mid-term week, turn in one written review & your selected presentation topic with a brief summary and initial list of references 20 Minute Presentation: Week following Spring Break: April 14, 16 & 21 Portfolio: Final Exam: 12 Noon, Sat., May 9 Electronic Version of Project, Reviews, Exercise (All files needed for total reconstruction: LaTeX, Figures, Source Code, PDF) in one .gz(ip) (Mac/Linux) or ZIP (Windows) archive and attached via email. Grades: Grading will be weighted as follows: Project: (see below) 50% Research Paper Reviews: 20% Other "portfolio" items: 30% Worked Exercises or Examples Revised & Expanded Lecture Notes for a single topic Algorithmic Development: Normal Forms: Thurston, Artin, Handle Reduction, Garside Growth Functions for Braid/Garside Groups (see Dehornoy/Weist) Review/Summary of Research Article: In the style of a Math Review: http://www.ams.org/mathscinet write three one-page summaries of published articles on braids. Resources: Birman Bibliograph: see Course Portfolio Web Page Braids: A Survey by Birman, Joan S. & Brendle, Tara E. http://arxiv.org/abs/math/0409205 Open Problem Lists: See appropriate sections of the lists in the Arkiv Benson Farb book: Problems on Mapping Class Groups Robion Kirby math.GT/0406190 Problems on invariants of knots and 3-manifolds. Other On-Line Courses on Braids: http://www.math.uiuc.edu/~kwhittle/math415.html http://www.math.ucla.edu/~radko/191.1.05w/ See references in "Possible Project Topics" link http://math.bard.edu/greg/math191.html Project: Develop a topic related to braids. An oral presentation of your project will be made after Spring Break. A written report will be submitted with the portfolio by the Final Exam date. (Precise dates above.) Jones Polynomial and Yang Baxter Equation for Quantum sl_2: q-alg/9506002 Title: Links, Quantum Groups, and TQFT's Authors: Stephen Sawin Abstract: The Jones polynomial and the Kauffman bracket are constructed, and their relation with knot and link theory is described. The quantum groups and tangle functor formalisms for understanding these invariants and their descendents are given. The quantum group $U_q(sl_2)$, which gives rise to the Jones polynomial, is constructed explicitly. Representations of Braids Non-injectivity of the Burau Representation: Moody/Long/Bigelow Reidemeister-Schreier Proof of Pure Braid Presentation Thompson Group: Greenberg/Sergiescu Comm. Math. Helv. vol. 66 Artin Groups/Garside Groups: Papers of Patrick Dehornoy & Ruth Charney Algebras Related to Braid Theory: Cord Algebra: math/0407071 Framed knot contact homology. Lenhard Ng Birman-Murakami-Wenzl Algebra Braids detected by Finite Type Invariants: Hutchings/Bar Natan/ Quandles & Braids: M Eisermann: Yang-Baxter deformations of quandles and racks Algorithm Development Braid Ordering Algorithm: Dehornoy/Weist Free Differential Calculus: Group Cohomology of Semi-Direct Products Cohen/Suciu Applications to Physics: Braids, Yang-Baxter & Quantum Groups Statistical Mechanics: Subfactors and Knots (CBMS Regional Conference Series in Mathematics) by Vaughan F. R. Jones Lou Kauffman: Book: Knots & Physics Section 1.8, 1.10 Other Applications: DNA: Dewitt L. Sumners Protein Folding: [ps] arXiv:0902.1025 Fatgraph Models of Proteins. Braid Group Cryptography. arXiv:0711.3941 David Garber. Braids & Juggling: Satyan L. Devadoss math.GT/0602476 Visualization of Seifert Surfaces or Turaev Surfaces