Titles
and abstracts
Thurday
December 13:
 Cristina
Anghel (Oxford
University): 13:0513:50+14:0014:30
Title: Coloured Jones
polynomials and topological intersection pairings.
Abstract: The world
of quantum invariants started with the discovery of the Jones
polynomial. Then, ReshitikhinTuraev introduced a purely algebraic
construction that having as input a quantum group produces link
invariants. The coloured Jones polynomials {J_N(L,q)}_N are sequences
of link invariants constructed in this way using the quantum group
U_q(sl(2)), whose first term is the original Jones polynomial. R.
Lawrence introduced a sequence of topological braid group
representations based on the homology of coverings of configuration
spaces. Using that, Bigelow and Lawrence gave a homological model for
the Jones polynomial, using its Skein relation nature. We will present
a topological model for all coloured Jones polynomials. We will show
that J_N(L,q) can be described as graded intersection pairings between
two homology classes in a covering of the configuration space in the
punctured disc. This shows that the Lawrence representations are rich
objects that contain enough information to encode all coloured Jones
polynomials and possibly more. In the last part of the talk we will
present some directions towards a geometrical categorification for
J_N(L,q) that can be defined out of this topological model.
 Paolo
Aceto (Oxford University): 14:4515:45
Title: Rational
cobordisms and integral homology.
Abstract: We prove
that every rational homology cobordism class in the subgroup generated
by lens spaces contains a unique connected sum of lens spaces whose
first homology embeds in any other element in the same class. As a
consequence we show that several natural maps to the rational homology
cobordism group have infinite rank cokernels, and obtain a
divisibility condition between the determinants of certain 2bridge
knots and other knots in the same concordance class. This is joint
work with Daniele Celoria and JungHwan Park.
 Mark
Powell (Durham University): 16:1517:15
Title: CassonGordon
invariants and Blanchfield pairings.
Abstract: I will give
an introduction to CassonGordon invariants and then explain their
relationshop to twisted Blanchfield pairings.
Friday
December 14:
 Kamila
Winnicka
(University of Warsaw): 09:0010:00
Title: Forms over
$\mathbb{C}[t,t^{1}]$ and $\mathbb{R}[t,t^{1}]$.
Abstract:
In this
introductory talk we recall a classification of hermitian forms over
$\mathbb{F}[t,t^{1}]$ for $\mathbb{F}=\mathbb{R},\mathbb{C}$. The
real case is wellknown, while there are some subtleties concerning
the complex case. The talk is based on a paper by Borodzik, Conway and
Politarczyk.

Arunima Ray (Max Planck Institut): 10:3011:30
Title: The
4dimensional sphere embedding theorem.
Abstract: The disc
embedding theorem for simply connected 4manifolds was proved by
Freedman in 1982 and forms the basis for his proofs of the topological
hcobordism theorem, the topological 4dimensional Poincaré
conjecture, 4dimensional topological surgery, and the classification
of simply connected 4manifolds. The disc embedding theorem for more
general manifolds is proved in the book of Freedman and Quinn.
However, the geometrically transverse spheres claimed in the outcome
of the theorem are not constructed. We close this gap by constructing
the desired transverse spheres. We also outline where transverse
spheres appear in surgery and the classification of 4manifolds and
give a general 4dimensional sphere embedding theorem. This is a joint
project with Mark Powell and Peter Teichner.
 Matthias
Nagel (Oxford University): 11:4012:40
Title: Slice disks in
stabilized 4balls.
Abstract: We consider
knots K which bound (nullhomotopic) slice disks in a stabilized
4ball, that is in D^4 # n S^2 \times S^2. We explain how to construct
examples of such disks, and discuss bounds on the minimal number n of
stabilizations needed. Then we compare this minimal number to the
4genus of K. This is joint work with Anthony Conway.
 Maciej
Borodzik (University of Warsaw): 13:5514:55
Title: Signatures of
linking forms.
Abstract: Given a
nondegenerate form over $\mathbb{F}[t,t^{1}]$, where
$\mathbb{F}=\mathbb{R},\mathbb{C}$ we discuss various possible
definitions of signatures. We discuss the behavior under Witt
equivalence. In case a form is represented by a square matrix, we show
establish a relation between the signatures of forms and the
signatures of the representing matrix, generalizing a classical result
by Matumoto. This is a joint work with Anthony Conway and Wojciech
Politarczyk.
 Celeste
Damiani (University of Leeds): 16:1517:15
Title: Alexander
invariants for ribbon tangles.
Abstract: Ribbon
tangles are proper embeddings of tori and annuli in the 4dimensional
ball, bounding 3manifolds with only ribbon singularities. We
construct an Alexander invariant for these objects that induces a
functorial generalisation of the Alexander polynomial. This functor is
an externsion of the Alexander functor for usual tangles defined by
BigelowCattabrigaFlorens and studied by FlorensMassuyeau. If
considered on braidlike ribbon tangles, this functor coincides with
the exterior powers of the BurauGassner representation. On one hand,
we observe that the action of cobordsms on ribbon tangles endows them
with a circuit algebra structure over the operad of cobordisms, and we
show that the Alexander invariant commutes with the circuit algebra's
composition. On the other hand, ribbon tangles can be represented by
welded tangle diagrams: this allows to give a combinatorial
description of the Alexander invariant.
Saturday
December 15:

Przemyslaw Grabowski (University of Warsaw): 10:0011:00
Title: Finite
orthogonal groups and periodicity of links.
Abstract: The talk
will be on an application of padic orthogonal groups in theory of
periodic links. This is a joint work with Maciej Borodzik, Adam Król
and Maria Marchwicka.

Wojciech Politarczyk (University of Warsaw): 11:2012:20
Title: Cabling
formula for twisted Blanchfield forms.
Abstract: Many
interesting examples of knots are obtained with the aid of the
satellite construction. Therefore it is crucial to understand how knot
invariants behave under this construction. In this talk we will
discuss a cabling formula for twisted Blanchfield forms and associated
twisted signatures. As an example we will rederive classical cabling
formulas of Litherland for CassonGordon signatures. This is a joint
work with Anthony Conway and Maciej Borodzik.