Program on Braids

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PRIMA Summer School

( 4 - 29 Jun 2007)

Overview · Outline · Student Seminar · Schedule · Application

Overview

The Summer School aims to attract graduate students to the study of braids and their varied applications. It starts with very basic introductory lectures, and progresses to present the mathematical theory of braid groups, configuration spaces and simplicial sets. Following this theoretical background, visiting experts will highlight applications of braids to the diverse topics of cryptography, magnetohydrodynamics, and robotics.

Outline

The first week begins with a 4-hour introduction to the topology and group theory needed to define braids and braid groups. This is suitable for students not familiar with say W.S. Massey "Algebraic Topology: An Introduction" Springer GTM 56. Then Dale Rolfsen presents 4 lectures on the basics of braids and braid groups: his Brazilian workshop lecture notes http://www.math.ubc.ca/~rolfsen/papers/brasil/BrasilBraid.pdf will be useful. Jie Wu introduces the fundamentals of simplicial sets and simplicial groups (4 hours), a key ingredient in recent research at NUS linking braids to homotopy groups of spheres. Fred Cohen rounds off the theoretical background with 2 hours on configuration spaces, an important related topic for physical applications and also an important way of approaching braid groups. Then attention switches to applications. Mitch Berger (4 hours) discusses how braids are applied in astrophysics to the study of solar physics and magnetic fields. David Garber's 2 talks look at probabilistic algorithms to solve word problems in braid groups, as a contribution to cryptography. Finally, Robert Ghrist spends 2 hours on the role of braid theory in robotics.

Week 1: 4 - 8 Jun 2007
(a) Preliminaries in topology and algebra, by E-Jay Ng: 4 hours (PDF)
(b) Braids - definitions and braid groups, by Dale Rolfsen: 4 hours (PDF)

Week 2: 11 - 15 Jun 2007
(a) Simplicial objects and homotopy groups, by Jie Wu: 4 hours (PDF)
(b) Configuration spaces, by Fred Cohen: 2 hours (PDF 1, PDF 2)

Week 3: 18 - 22 Jun 2007
(a) Magnetohydrodynamics, by Mitchell Berger: 4 hours
(b) Configuration spaces and robotics, by Robert Ghrist: 2 hours (PDF)
(c) Braid groups and cryptography, by David Garber: 2 hours (PDF)

Student Seminar

During the three weeks of tutorials, graduate students (and others) are encouraged to present seminars on related material.

Schedule

4 - 8 Jun 2007 · 11 - 15 Jun 2007 · 18 - 22 Jun 2007

Monday, 4 June 2007
01:30pm - 02:00pm	Registration of participants
02:00pm - 03:00pm	Preliminaries in topology and algebra (Tutorial Lecture 1) (PDF) Ng E-Jay, National University of Singapore
03:00pm - 03:30pm	--- Coffee Break ---
03:30pm - 04:30pm	Preliminaries in topology and algebra (Tutorial Lecture 2) (PDF) Ng E-Jay, National University of Singapore
Tuesday, 5 June 2007
10:30am - 10:45am	--- Coffee Break ---
10:45am - 11:45am	Preliminaries in topology and algebra (Tutorial Lecture 3) (PDF) Ng E-Jay, National University of Singapore
11:45am - 02:00pm	--- Lunch Break ---
02:00pm - 03:00pm	Preliminaries in topology and algebra (Tutorial Lecture 4) (PDF) Ng E-Jay, National University of Singapore
Wednesday, 6 June 2007
10:30am - 10:45am	--- Coffee Break ---
10:45am - 11:45am	Braids - definitions and braid groups (Tutorial Lecture 1) (PDF) Dale Rolfsen, University of British Columbia, Canada
11:45am - 02:00pm	--- Lunch Break ---
02:00pm - 03:00pm	Braids - definitions and braid groups (Tutorial Lecture 2) (PDF) Dale Rolfsen, University of British Columbia, Canada
03:00pm - 03:30pm	--- Tea at Department of Mathematics ---
Thursday, 7 June 2007
10:30am - 10:45am	--- Coffee Break ---
10:45am - 11:45am	Braids - definitions and braid groups (Tutorial Lecture 3) (PDF) Dale Rolfsen, University of British Columbia, Canada
11:45am - 02:00pm	--- Lunch Break ---
02:00pm - 03:00pm	Braids - definitions and braid groups (Tutorial Lecture 4) (PDF) Dale Rolfsen, University of British Columbia, Canada
03:00pm - 03:30pm	--- Coffee Break ---
03:30pm - 04:30pm	An algorithm to compute Seifert Matrices from Braids Julia Collins, University of Edinburgh, UK
Friday, 8 June 2007
10:30am - 10:45am	--- Coffee Break ---
10:45am - 11:45am	The integral homology ring of the based loop space on a flag manifold Jelena Grbic, University of Manchester, UK
11:45am - 02:00pm	--- Lunch Break ---
02:00pm - 03:00pm	Orderability of braids Zhixian Zhu, Zhejiang University, China
03:00pm - 03:30pm	--- Coffee Break ---
03:30pm - 04:30pm	Free differential calculus and representations Peng Zheng, Zhejiang University, China

4 - 8 Jun 2007 · 11 - 15 Jun 2007 · 18 - 22 Jun 2007

Monday, 11 June 2007
10:30am - 10:45am	--- Coffee Break ---
10:45am - 11:45am	Configuration spaces (Tutorial Lecture 1) (PDF 1) Fred R. Cohen, University of Rochester, USA
11:45am - 02:00pm	--- Lunch Break ---
02:00pm - 03:00pm	Simplicial objects and homotopy groups (Tutorial Lecture 1) (PDF) Jie Wu, National University of Singapore
03:00pm - 03:30pm	--- Coffee Break ---
03:30pm - 04:30pm	String links and orderability Ekaterina Yurasovskaya, The University of British Columbia, Canada
Tuesday, 12 June 2007
10:30am - 10:45am	--- Coffee Break ---
10:45am - 11:45am	Configuration spaces (Tutorial Lecture 2) (PDF 2) Fred R. Cohen, University of Rochester, USA
11:45am - 02:00pm	--- Lunch Break ---
02:00pm - 03:00pm	Simplicial objects and homotopy groups (Tutorial Lecture 2) (PDF) Jie Wu, National University of Singapore
03:00pm - 03:30pm	--- Coffee Break ---
03:30pm - 04:30pm	Some nontrivial elements in the stable homotopy groups of sphere Yuyu Wang, Nankai University, China
Wednesday, 13 June 2007
10:30am - 10:45am	--- Coffee Break ---
10:45am - 11:45am	Batalin -- Vilkovisky algebras, braid groups and free loop spaces (PDF) Miguel Xicotencatl Merino, Centrode Investigacion Estudios Avanzados, Mexico
11:45am - 02:00pm	--- Lunch Break ---
02:00pm - 03:00pm	Simplicial objects and homotopy groups (Tutorial Lecture 3) (PDF) Jie Wu, National University of Singapore
03:00pm - 03:30pm	--- Tea at Department of Mathematics ---
Thursday, 14 June 2007
10:30am - 10:45am	--- Coffee Break ---
10:45am - 11:45am	The cohomology and homology of a lie algebra Qibing Zheng, Nankai University, China
11:45am - 02:00pm	--- Lunch Break ---
02:00pm - 03:00pm	Simplicial objects and homotopy groups (Tutorial Lecture 4) (PDF) Jie Wu, National University of Singapore
03:00pm - 03:30pm	--- Coffee Break ---
03:30pm - 04:30pm	A new construction of Anick's fibration Stephen Theriault, University of Aberdeen, UK
Friday, 15 June 2007
10:30am - 10:45am	--- Coffee Break ---
10:45am - 11:45am	Homotopy and cohomology of spaces of homomorphisms Alejandro Adem, University of British Columbia, Canada
11:45am - 02:00pm	--- Lunch Break ---
02:00pm - 03:00pm	On the inverse braid monoid Volodia Vershinin, Universite Montpellier II, France
03:00pm - 03:30pm	--- Coffee Break ---
03:30pm - 04:30pm	The braid groups of the projective plane Laura Rocio González Ramírez, Centro de Investigación y Estudios Avanzados del Instituto, Mexico

4 - 8 Jun 2007 · 11 - 15 Jun 2007 · 18 - 22 Jun 2007

Monday, 18 June 2007
10:30am - 10:45am	--- Coffee Break ---
10:45am - 11:45am	The Leray spectral sequence and the cohomology of configuration spaces Cristhian Emmanuel Garay López, Centro de Investigación y Estudios Avanzados del Instituto, Mexico
11:45am - 02:00pm	--- Lunch Break ---
02:00pm - 03:00pm	On hopf algebras associated to groups and quandles Maria Guadalupe Castillo Perez, Bonn Universität, Germany
03:00pm - 03:30pm	--- Coffee Break ---
03:30pm - 04:30pm	Combinatorial description of homotpy groups of wedge of spheres Hao Zhao, Nankai University, China
Tuesday, 19 June 2007
10:30am - 10:45am	--- Coffee Break ---
10:45am - 11:45am	Magnetohydrodynamics (Tutorial Lecture 1) Mitch A Berger, University College London, UK
11:45am - 02:00pm	--- Lunch Break ---
02:00pm - 03:00pm	Configuration spaces and robotics (Tutorial Lecture 1) (PDF) Robert Ghrist, University of Illinois, Urbana-Champaign, USA
03:00pm - 03:30pm	--- Coffee Break ---
03:30pm - 04:30pm	Orbit configuration spaces and surface braid groups Miguel Xicotencatl Merino, Centrode Investigacion Estudios Avanzados, Mexico
Wednesday, 20 June 2007
10:30am - 10:45am	--- Coffee Break ---
10:45am - 11:45am	Magnetohydrodynamics (Tutorial Lecture 2) Mitch A Berger, University College London, UK
11:45am - 02:00pm	--- Lunch Break ---
02:00pm - 03:00pm	Configuration spaces and robotics (Tutorial Lecture 2) (PDF) Robert Ghrist, University of Illinois, Urbana-Champaign, USA
03:00pm - 03:30pm	--- Tea at Department of Mathematics ---
Thursday, 21 June 2007
10:30am - 10:45am	--- Coffee Break ---
10:45am - 11:45am	Braid groups cryptography (Tutorial Lecture 1) (PDF) David Garber, Holon Institute of Technology, Israel
11:45am - 02:00pm	--- Lunch Break ---
02:00pm - 03:00pm	Magnetohydrodynamics (Tutorial Lecture 3) Mitch A Berger, University College London, UK
03:00pm - 03:30pm	--- Coffee Break ---
03:30pm - 04:30pm	Spaces of group homomorphisms and group cohomology Torres Giese Enrique, The University of British Columbia, Canada
Friday, 22 June 2007
10:30am - 10:45am	--- Coffee Break ---
10:45am - 11:45am	Braid groups cryptography (Tutorial Lecture 2) (PDF) David Garber, Holon Institute of Technology, Israel
11:45am - 02:00pm	--- Lunch Break ---
02:00pm - 03:00pm	Magnetohydrodynamics (Tutorial Lecture 4) Mitch A Berger, University College London, UK
03:00pm - 03:30pm	--- Coffee Break ---
03:30pm - 04:30pm	Patterns generated during the transition to turbulence Hua-Shu Dou, Temasek Laboratories, National University of Singapore

4 - 8 Jun 2007 · 11 - 15 Jun 2007 · 18 - 22 Jun 2007

Application

Limited funds are available for support of participation of graduate students, for either 3 weeks or the whole 4 week period of the Summer School. (Potential NUS graduate students are especially welcome.) When applying, please submit abstract of seminar, if any, that you can present (need not be original research).

Application form : MSWord | PDF | PS

Application deadline: 31 March 2007.

Overview · Outline · Student Seminar · Schedule · Application