


This symposium was extended to cover the full period of the grant,
namely 1 August 1999 to 31 July 2002, with main activities taking
place in the academic years 1999/2000 and 2001/2002. It was
especially useful to be able to do this, because shortly after the
symposium commenced one of the coorganisers (Victor Pidstrigatch)
moved to Germany and later Stavros Garoufalidis moved to Warwick and
was able to use symposium funds to organise a very wellattended and
wellreceived workshop in Quantum Topology (7) below.
To support the symposium the Mathematics Department gave a temporary
lectureship to Max Forester, who is now an EPSRC research fellow.
Research related to the symposium centred around a series of eight
concentrated periods of activity (listed below) with background
activity in the form of weekly seminars and longer term visitors.
 EPSRC/LMS Instructional Conference for Graduate Students :
30 August3 September 1999
 Introductory Workshop on Topology :
617 September 1999
 Twoday meeting on LowDimensional Topology :
67 January 2000
 Workshop on Topology Geometry and Physics :
815 April 2000
 Workshop on Geometry and Topology :
922 July 2000
 Workshop on Topology, Operads and Quantisation :
1014
December 2001
 Workshop on Quantum Topology :
1822 March 2002
 Workshop on Geometric Topology :
112 July 2002
The appendices to this report contain full details of programmes and
participants for (2) to (8). Appendices are numbered to match the
list above.
Note Activity (1) was supported by the LMS and a separate grant
from the EPSRC. It is included in this report because it was part of
the planned activity of the symposium and ran straight into the
Introductory Workshop.
The symposium was extremely successful in its main aim of bringing
first class topologists together to work on the central problems in
the area. The long (and incomplete) list of papers advanced by the
interaction enabled by the symposium (see below) and the very positive
feedback (also below) both testify to this fact. The money spent by
the EPSRC on this symposium is clearly money very well spent and we
are extremely grateful.
Output from the symposium
Many pieces of work were advanced significantly during the symposium
period and new projects begun both between visitors and between local
people and visitors. Below is a list of preprints published at
Warwick and on arxiv.org as well as articles submitted to journals,
so far as they are known to the organisers. It contains papers which
our visitors listed on feedback forms as arising from or as having
been significantly advanced by their stay at Warwick. The list is
partial because it is not possible to ensure that participants fill in
these feedback forms with appropriate care!
 H Akiyoshi, M Sakuma, Comparing two convex hull
constructions of cusped hyperbolic manifolds (I), to appear in the
Proceedings of the Workshop ``Kleinian groups and hyperbolic
3manifolds'', held in Warwick, 2002, London Math. Soc. Lecture Note
Series, Warwick preprint 10/2002
 H Akiyoshi, M Sakuma, M Wada, Y Yamshita,
Comparing two convex hull constructions of
cusped hyperbolic manifolds (II),
ibid, Warwick preprint 11/2002
 H Akiyoshi, H Miyachi, M Sakuma, A refinement of
McShane's identity for quasifuchsian punctured torus groups, Warwick
preprint 21/2002
 Jorgen Ellegaard Andersen, Asymptotic faithfulness of
the quantum SU(n) representations of the mapping class groups,
Invent. Math. to appear, arXiv:math.QA/0204084
 Ivelina Julianova Bobtcheva, Maria Grazia Messia,
HKRtype invariants of 4dimensional thickenings and 2dimensional
CWcomplexes, arXiv:math.QA/0206307
 S Boyer, B Wiest, D Rolfsen, Orderable 3manifold
groups, to appear
 M Cahen, S Gutt, J Horovitz, J Rawnsley, Moduli
space of symplectic connections of Ricci type on $T^{2n}$; a formal
approach, J Geom Phys, to appear, arXiv:math.SG/0201167
 John Crisp, Luis Paris, Artin groups of type B and D,
to appear
 John Crisp, Luis Paris, Representations of the braid
group by automorphisms of groups, invariants of links, and Garside
groups, to appear
 Mieczyslaw K Dabkowski, Jozef H Przytycki, Burnside
obstructions to the MontesinosNakanishi 3move conjecture, Geom.
Topol. 6(2002) 355360, arXiv:math.GT/0205040
 Adam Epstein, Topological matings of quadratic polynomials,
in preparation
 Paul Feehan, PU(2) monopoles and 4manifold invariants,
to appear
 R Fenn, M Jordan, L Kaufmann, The Birack: an Invariant
of Virtual Knots and Links, to appear
 R Fenn, A Bartholomew, New Polynomials for (Virtual)
Knots and Links, in preparation
 Roger Fenn, Colin Rourke, Brian Sanderson, The
classification of classical links, in preparation
 Max Forester, On the isomorphism problem for generalized
BaumslagSolitar groups, to appear
 David Gabai, Smale conjecture for hyperbolic 3manifolds:
${\rm Isom}(M^3)\simeq{\rm Diff}(M^3)$, J. Differential Geom. 58
(2001) 113149
 C McA Gordon, Dehn fillings of large hyperbolic 3manifolds,
J. Differential Geom. 58 (2001) 263308
 C McA Gordon, Nonintegral toroidal Dehn surgeries,
to appear
 C McA Gordon, DD Long, A Reid Surface subgroups of
Coxeter groups, to appear
 Bill Harvey, Mustafa Korkmaz, Homomorphisms between
mapping class groups, in preparation
 JDS Jones, On the definition of operads, to appear in
Contemp Math. (2003)
 Ebru Keyman, Ordering on singular braids, in preparation
 Simon King, The size of triangulations supporting a given link,
to appear, Geom. Topol. 5 (2001), 369398, arXiv:math.GT/0007032
 Simon King, How to make triangulations of $S^3$ polytopal,
arXiv:math.GT/0009216
 John Klein, Moduli of suspension spectra, to appear
 John Klein, On embeddings up to homotopy in the sphere,
to appear
 Eun Soo Lee, The support of Khovanov invariants for
alternating knots, arXiv:math.GT/0201105
 Brian Mangum, Theodore Stanford, Brunnian links are
determined by their complements, Algebr. Geom. Topol. 1 (2001)
143152 arXiv:math.GT/9912006
 Sergei Matveev, Special spines and an invariant of
4manifolds, to appear
 Paul Melvin, Blake Mellor, A geometric interpretation
of Milnor's triple linking numbers, submitted to
Geom. Topol. arXiv:math.GT/0110001
 Paul Melvin, Rob Kirby, Local surgery formulas for
quantum invariants, to appear
 Justin Roberts, Simon Willerton, The RozanskyWitten
weight systems, to appear
 Justin Roberts, Justin Sawon, Simon Willerton, The
RozanskyWitten TQFT, to appear
 Colin Rourke, Knotted triangulations of the threeball,
in preparation
 Colin Rourke, Brian Sanderson, The compression theorem
II: directed embeddings, Geom. Topol. 5 (2001) 431440,
arXiv:math.GT/003026
 Colin Rourke, Brian Sanderson, The compression theorem
III: applications, for a preliminary version see section 5 onwards
of arxiv:math.GT/9712235:v2
 Colin Rourke, Brian Sanderson, A proof of Eliashberg's
folding theorem, in preparation
 Colin Rourke, Brian Sanderson, Adrian Varley, The
$C^0$singularity theorem, in preparation
 David Spring, Directed embeddings and the simplification of
singularities, Commun. Contemp. Math. 4 (2002) 107144
 Andras Stipsicz, Commutators, Lefschetz fibrations and the
signature of surface bundles, Warwick preprint 16/2000
 Andras Stipsicz, Surface bundles: an interesting example,
Warwick pre\print 15/2000
 Andras Stipsicz, A note on the transverse isotopy problem,
Warwick pre\print 18/2000
 A Voronov, Notes on universal algebra,
Warwick preprint 26/2001, arXiv:math.QA/0111009
 Andreas Zastrow, Higher homology groups of planar sets do not behave anomalously, to appear
 Andreas Zastrow, Higher homology groups of subsets of surfaces do not behave anomalously, to appear
Feedback from participants
Participants also used the feedback forms to make general comments
about the symposium activities and also (sometimes) to describe the
research undertaken here. There were no negative comments whatsoever.
Here is a selection of comments chosen for information or interest
value.

"The enthusiasm of the lecturers focussed my attention on the
talks. The expertise of the lecturers was visible in every single talk."
Graduate student attending Instructional Conference

"The lecture were well prepared with the lecturers willing to
answer questions and go over material again when necessary."
Graduate student attending Instructional Conference

"The meeting was a welcome opportunity to learn at first hand
of many recent results in Topology." Andrew Ranicki, September 1999

"Great place, great conference." Stefan Bauer, April 2000

"Really great! The MRC provides excellent facilities. The
workshop in Geometry and Topology was extremely successful.
Altogether a very worthwhile visit." Cameron Gordon, June 2000

"A wellorganised workshop on Topology and Geometry. I had a
wonderful time. The support staff at the Institute were always
helpful and efficient. I enjoyed working here during the workshop."
David Spring, July 2000

"Great as usual." David Gabai, July 2000

"I enjoyed my stay at the MRC and it was very fruitful both
because of the good working conditions and the nice housing. The MRC
houses are really great, both the location and the interior of the
houses are unique and extremely nice." Andras Stipsicz, August 2000

"The workshop in Geometry and Topology was great fun and very
inspiring." Brian Magnum, July 2000

"Very nice conference and atmosphere." Oliver Dasbach, July 2000

"A very interesting conference on a high level." Simon King, July 2000

"Congratulations to the woderful staff, Mrs. McAllister, Colin Rourke,
Victor Pidstrigatch and John Jones for organising and running a great
conference!" Paul Feehan, July 2000

"The workshop helped the finishing of the paper by Andersen [4].
This is a major result, which proves that the representation of the
mapping class groups (of a fixed but arbitrary genus) that comes from the
TQFT $SU(N)_q$ (for all $N$ and $q$, a complex root of unity) is faithful.
Josef (Przytycki) used the workshop and the computing facilities at
Warwick to simplify his original proof (of [10])."
Stavros Garoufalidis, March 2002

"The Workshop on Quantum Topology was very interesting: I have
greatly appreciated the talks, the opportunity to discuss with colleagues
and the hospitality of the Institute." Christian Blanchet, March 2002

"This meeting was a first class affair. I gave a talk on my
current research and this together with feedback helped me to
reformulate my ideas and push them further. In addition three of my
current students attended and gained a great deal from the talks and
ambience. Also one of my ex students from Turkey attended and gave a
talk." Roger Fenn, July 2002

"Part of the results of my second paper [9] were found after
talking with some knot theorists in Warwick." Luis Paris, July 2002

"I attended the workshop as a newcomer to the area of
geometric topology. I found the talks to be very interesting and well
presented. While I have no new results as a direct result of my
participation, I feel that the conference allowed me to get a feel for
what is happening in the area and to meet the many of the people
working the area. I certainly feel that the workshop has inspired me
to research in geometric topology." James Cruickshank, July 2002

"The conference was, as always at Warwick topology meetings,
very smoothly organised and the atmosphere very conducive to both
working alone and, more importantly, to interacting with the other
visitors and locals." Justin Roberts, July 2002

"One thing I learned (from Colin Rourke and Martin Dunwoody's
lectures) is the basic idea behind the recognition problem of $S^3$
and the inherent difficulty of an elementary approach to the PC. This
is not likely, though, to lead to any publication. I also had some
new ideas inspired by James Hughes' lecture on link homotopy and
string links. In discussions with him later, we realized that the
groups involved in the Milnor, as well as LinHabegger approach to
classifying links up to homotopy, are biorderable groups!" Dale
Rolfsen, July 2002

"During the workshop this summer, Rob Kirby and I continued
our search for a topological/combinatorial approach to OzsvathSzabo
Floer homology of 3manifolds. In particular we are close to a
combinatorial formula for the Maslov index of certain holomorphic
disks that arise in the theory, and we spent our time investigating
whether this formula (or some variant) is consistent with known
results on the Euler characteristic of the OSFloer homology."
Paul Melvin, July 2002

"The workshop was highly enjoyable compared to others I have
been to. The quality of the talks was uniformly very high. The wide
range of topics meant that I learned many new things. I met several
colleagues for the first time. I was glad to be able to talk about my
work to an audience that had not heard about it before. And high marks
for the live entertainment!" Max Forester, July 2002

"The second paper [46], as it is acknowledged by a footnote
in the appropriate preprint, has been inspired by discussions that I had
with David Spring during the first Warwick workshop.
Always having enjoyed my coming to Warwick, its unique
discussion atmosphere, and its permanently accessible
computing and printingnetwork." Andreas Zastrow, July 2002

"One of the problems I was studying at the time of the
workshop (which has now resolved itself into two shortlytosurface
preprints) is the question of when a spectrum has the homotopy type of
a suspension spectrum. The results of this investigation are now being
applied to study embeddings up to homotopy of finite complexes $K$ in the
$n$sphere (here we are interested in representing the functional
stable dual $F(K,S^{n1})$ as a suspension spectrum)." John Klein,
October 2002

"In the conferences on `Geometry, Topology and Physics' that took place in Warwick in the year 2000 I began to understand techniques in topology that were usefull for my Phd thesis (`Generalised Einstein Equations on Kähler Manifolds', Imperial College, 2002).
 The July 2002 conference was extremely interesting. In particular the
talks given by Fenn, Kirby, Rolfsen, Rourke and Sanderson (covering topics
that probably no one has ever thought as much as they had) were memorable.
The discussions that took place after and during the talks were enlightening
for me.
 I am indebted to the organizers for setting up these conferences where young mathematicians had the opportunity to learn and discuss with some of the living legends in the subject." Daniel Pons, October 2002
Appendices :
 Appendix 1: Flyer for 1999/2000 activities, flyer and programme
for EPSRC/LMS Instructional Conference, August 1999
 Appendix 2: Programme and participants' list for Introductory
Workshop on Topology, September 1999
 Appendix 3: Programme and participants' list for twoday meeting
on LowDimensional Topology, January 2000
 Appendix 4: Programme and participants' list for workshop on
Topology Geometry and Physics, April 2000
 Appendix 5: Programme and participants' list for workshop on
Geometry and Topology, July 2000
 Appendix 6: Flyer, programme and participants' list for workshop on
Topology, Operads and Quantisation, December 2001
 Appendix 7: Flyer, programme and participants' list for workshop on
Quantum Topology, March 2002
 Appendix 8: Programme and participants' list for workshop on
Geometric Topology, July 2002
Colin Rourke
Coinvestigator 14 October 2002


Symposium Index
Other reports
