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Math
9120: Mapping Class Groups and Braid Groups
Fall 2011 - Spring 2012
Meets: | Tue/Thu
11:00 AM - 12:20 PM in
Wachman Hall, room
527 |
Instructor: | David
Futer |
Office: | 430
Wachman Hall |
Office
Hours: | by
appointment |
E-mail: | dfuter
at
temple.edu |
Phone: | (215)
204-7854 |
Course content:
This course will focus on the symmetry groups of manifolds and cell
complexes. We will look at both the geometric and algebraic properties of the
mapping class group, which can roughly be thought of as the group of
symmetries of a surface. One of the ways in which we will study the group is
by designing a certain cell complex on which it acts by isometries. This
principle has wide generalizations: one can derive a great deal of information
about a group by constructing a geometric object for the group to act on.
In the spring semester, we will focus our attention on the braid groups, which
are a special class of mapping class groups. In particular, we will study the
representation theory of these groups, with connections to invariants of knots
and links.
Textbooks:
Prerequisites: Math 8061-62.
Class format: This will be a reading seminar, with rotating presentations by
all the students.
Grading: Grades will be based on the quality of
presentations, as well as participation during others' presentations.
Presentation schedule (Fall)
All page numbers refer to Farb-Margalit version 5.0, available online.
Day |
Topic |
Reading |
Presenter |
8/30 | Overview | Chapter 0 | Dave
| 9/1 | Hyperbolic geometry | P. 17-23 | Christian
| 9/6 | Geodesics | P. 23-26 | Christian, Beca
| 9/8 | Simple closed curves | P. 27-31 | Beca
| 9/13 | The bigon criterion | P. 32-38 | Beca, Jessie
| 9/15 | Change of coordinates | P. 38-45 | Jessie
| 9/20 | MCG of the disk, punctured disk, annulus | P. 46-53 | Austin
| 9/22 | MCG of the torus, punctured torus, 4-holed sphere |
P. 54-59 | Austin, Brian
| 9/27 | The Alexander Method | P. 59-66 | Brian
| 9/29 | Dehn twist basics | P. 67-72 | Christian
| 10/4 | Dehn twists and intersecton numbers | P. 72-77 | Christian, Beca
| 10/6 | The center of the mapping class group | P. 77-81 | Beca
| 10/11 | Relations between Dehn twists | P. 81-88 | Beca, Jessie
| 10/13 | Cutting, capping, including | P. 88-92 | Jessie
| 10/18 | The complex of curves | P. 94-100 | Austin
| 10/20 | Birman exact sequence | P. 99-106 | Austin
| 10/25 | Finite generation | P. 107-112 | Brian
| 10/27 | Explicit generators | P. 112-120 | Brian, Christian
| 11/1 | Lantern relation, abelianization | P. 121-128 |
Christian
| 11/3 | The arc complex is contractible | P. 139-141 | Beca
| 11/8 | Finite presentability | P. 141-145 | Beca
| 11/10 | Symplectic basics; algebraic intersection number |
P. 167-173 | Brian
| 11/15 | The Euclidean algorithm | P. 173-177 | Brian, Jessie
| 11/17 | The symplectic representation; congruence subgroups |
P. 177-181, 184-187 | Jessie
| 11/22 | Residual finiteness | P. 187-192 | Jessie, Austin
| 11/29 | The Torelli group | P. 193-199; Putman paper | Austin
| 12/1 | Braid groups | P. 251-258 | Christian
| 12/6 | Algebraic structure of braid groups | P. 258-264 | Christian
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Presentation schedule (Spring)
Pages are from Kassel-Turaev unless noted otherwise. FM = Farb-Margalit, CB = Casson-Bleiler.
Day |
Topic |
Reading |
Presenter |
1/17 | Braids, knots, and links | P. 47-50, 58-60 | Dave
| 1/19 | An algorithm for braiding | P. 61-66 | Dave
| 1/24 | Closed braids in a solid torus | P. 51-55 | Beca
| 1/26 | Closed braids; Burau representation | P. 56-57,
93-97 | Beca, Brian
| 1/31 | Twisted homological representations | P. 98-102,
105-106 | Brian
| 2/2 | Nonfaithfulness of Burau; reduced Burau | P. 102-105, 107-111 | Brian, Jessie
| 2/7 | The Alexander-Conway polynomial | P. 111-118 | Jessie
| 2/9 | Lawrence-Krammer-Bigelow representation | P. 118-124 |
Jessie, Christian
| 2/14 | Noodles and spanning arcs | P. 125-129 | Christian
| 2/16 | Faithfulness of LKB | P. 137-149 | Dave
| 2/21 | Properties of the symmetric group | P. 151-163 | Dave
| 2/23 | Iwahori-Hecke Algebras | P. 163-169 | Jessie
| 2/28 | Ocneanu traces, HOMFLY-PT Polynomial | P. 170-175 | Jessie, Brian
| 3/1 | Semi-simple algebras | P. 176-192 | Brian
| 3/13 | Partitions and tableaux | P. 195-203 | Beca
| 3/15 | The Young lattice | P. 203-209 | Beca, Christian
| 3/20 | Simplicity of seminormal representations | P. 210-217 | Christian
| 3/22 | Simplicity of reduced Burau; intro to Nielsen-Thurston
theory | P. 219-221; FM p. 387-390 | Jessie, Brian
| 3/27 | Measured foliations and pseudo-Anosov maps |
FM P. 314-321, 388-390 | Brian, Dave
| 3/29 | The Nielsen-Thurston trichotomy | FM P. 390-398, 420-422 | Dave
| 4/3 | Geodesic laminations | CB P. 60-68, 79-80 | Dave
| 4/5 | Stable & unstable laminations | CB P. 82-86 | Dave
| 4/10 | Braids and contact geometry | | Beca
| 4/12 | Braid groups in cryptography | | Christian
| 4/17 | Braids, operads, and the Grothendieck-Teichmuller
group | | Brian
| 4/19 | Stable & unstable foliations | CB, P. 89-94 | Dave
| 4/24 | Transverse measures | CB, P. 95-102 | Dave
| 4/26 | The word and conjgacy problems | | Jessie
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Last modified: Fri Aug 21 13:41:22 PDT 2009
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