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Abstracts
The symplectic homology ring is a very efficient
tool in distinguishing affine (Weinstein) symplectic manifolds. On the
other hand, if symplectic homology is trivial the distinction is
difficult. Though there are known some examples of nonsymplectomorphic
Weinstein manifolds with trivial symplectic homology, is not at all
clear how much symplectic life exists beyond symplectic homology.
The result discussed in the talk indicates that not much if dimension .
It turns out that a typical operation which kills symplectic homology
transforms a Weinstein manifold into a pure topological object which
abides a certain principle. A key ingredient in the
proof is a recent result of my student Max Murphy who proved a
surprising principle for Legendrian knots in
contact manifolds of dimension .
Parts of this work are joint with K. Cieliebak and F. BourgeoisT.
Ekholm. 
This will be an expository lecture on the state of
affairs with regard to smooth structures on simply connected
4manifolds with . There has been considerable progress
on this front in the last 5 to 10 years. I will concentrate on
techniques and ideas involved in making this progress and survey the
results that these techniques give as well as open questions and
conjectures. 
A Morse 2function is just a generic (stable) smooth
map
from a smooth nmanifold to a 2manifold; here we will focus on maps
to . We will look at some basic examples
when the domain
is 2dimensional, just to get our intuition on the right track, and
then look at the case where the domain is 3 or 4dimensional. I will
state some results that are the result of joint work with Rob Kirby
and stress the underlying connection with Cerf theory (the study of
generic homotopies between ordinary Morse functions). 
It is rare for a hyperbolic knot in to have a nontrivial Dehn surgery
that yields a nonhyperbolic 3manifold, but it does happen. More
generally, it is rare for a hyperbolic 3manifold to
have two nonhyperbolic Dehn fillings and along a given cusp. Consequently, one
wonders if it might be possible to describe all such exceptional
triples . We will give a survey of progress
towards this goal, and some of the problems that remain. 
A physical interpretation of knot
homologies as spaces of the socalled "refined BPS states" leads to
many
interesting and often surprising predictions, ranging from concrete
expressions for homological knot invariants associated with higher
representations of SL, SO, and Sp classical groups to intricate webs of
spectral sequences and anticommuting differentials acting on the
triplygraded theories categorifying the HOMFLY and Kauffman
polynomials. In
this talk, I will review the enterprise of this connection between
(enumerative) geometry and knot theory and, if time permits, describe
some
of the latest developments, related to the interpretation of spectral
sequences and differentials, and wall crossing for refined / motivic
BPS
invariants. 
The goal of this talk is to sketch a proof of the
equivalence of Heegaard Floer homology (due to OzsvathSzabo) and
embedded contact homology (due to Hutchings). This is joint work with
Vincent Colin and Paolo Ghiggini. 
Embedded contact homology is an invariant of contact
threemanifolds which
is isomorphic to versions of Heegaard Floer homology and SeibergWitten
Floer
homology. I will give an introduction to what ECH is good for, aside
from being
isomorphic to other Floer theories. For example, the detailed structure
of ECH
leads to extensions of the Weinstein conjecture (joint work with Cliff
Taubes). In
addition, maps on ECH induced by fourdimensional symplectic cobordisms
lead to a
proof of the Arnold chord conjecture in three dimensions (joint with
Taubes), and to
new symplectic embedding obstructions in four dimensions which are
sharp in some
cases. 
In his groundbreaking 1982 paper Michael Freedman
proved a disk embedding theorem, leading to a classification of
simplyconnected topological 4manifolds. Although the class of
fundamental groups for which such geometric classification techniques
work has since been extended, the central case of free groups remains
open. The AB slice problem is a reformulation (due to Freedman) of
this question in terms of a generalized linkslicing problem. This talk
will discuss the history of the problem, as well as recent developments
and ideas for a solution. 
In joint work with Tim Perutz, we give a complete
characterization of the Fukaya category of the punctured torus, denoted
by . This, in particular, means that one
can write down an explicit minimal model for in the form of an infinity
algebra, denoted by , and classify infinity
structures on the relevant algebra. A result that we will discuss is
that no associative algebra is quasiequivalent to the model of
the Fukaya category of the punctured torus, i.e., is
nonformal. will be connected to many topics of
interest:
1) It is the boundary category that we associate to a 3manifold with
torus boundary in our extension of Heegaard Floer theory to manifolds
with boundary,
2) It is quasiequivalent to the category of perfect complexes on an
irreducible rational curve with a double point, an instance of
homological mirror symmetry. 
I'll describe an invariant of smooth 4manifolds
defined in terms of
Khovanov homology. It associates a doublygraded vector space to each
4manifold (optionally with a link in its boundary), generalizing the
Khovanov homology of a link in the boundary of the standard 4ball.
For now, we can only make the construction in characteristic two. I'll
finish by talking about relations to TQFT, and the prospects for
concrete calculations. This is joint work with Chris Douglas and Kevin
Walker. 
Overview of 4dimensional topology, with historical
notes. 
If there are any two component counterexamples to
the Generalized Property R Conjecture, a least genus component of all
such counterexamples cannot be a fibered knot. Furthermore, the
monodromy of a fibered component of any such counterexample has
unexpected restrictions.
The simplest plausible counterexample to the Generalized Property R
Conjecture could be a two component link containing the square knot. We
characterize all twocomponent links that contain the square knot and
which surger to . We exhibit a family of such links
that are probably counterexamples to Generalized Property R. These
links can be used to generate slice knots that do not seem to be
ribbon. 
The failure of the Whitney move in dimension 4 can
be measured by constructing
higherorder intersection invariants of Whitney towers built
from iterated
Whitney disks on immersed surfaces in 4manifolds. For Whitney towers
on immersed
disks in the 4ball, some of these invariants can be identified with
previously
known link invariants like Milnor, SatoLevine and Arf invariants. This
approach
also leads to the definition of higherorder SatoLevine and Arf
invariants which
detect the obstructions to framing a twisted Whitney tower, and
appear to be
new invariants. Recent joint work with Jim Conant and Peter Teichner
has shown that,
together with Milnor invariants, these higherorder invariants classify
the
existence of (twisted) Whitney towers of increasing order in the
4ball. 
Khovanov homology is a categorification of the Jones
polynomial. The
fact that the Jones polynomial can be normalized such that it is
defined over the integers is used hereby in a crucial way here. When
constructing the colored Jones polynomial and ReshetikhinTuraev
3manifold invariants naturally
polynomials over rational numbers or rational functions in a variable q
occur. In this talk I want to address the problem of categorifying such
functions.
The main idea here is a padic completion of the Grothendieck groups
and
the notion of mixed categories with weight filtrations. I will
illustrate the concept by categorified link invariants and 3jsymbols. 
We'll motivate the definition of dcategories via
the ddimensional
bordism category of manifolds (with geometric structures). Then we'll
go through the
lowdimensional cases d=0,1 and 2 and rediscover some well known
notions With the
addition of one odd direction, the spaces of super symmetric Euclidean
field
theories (aka representations of the super Euclidean bordism category)
turn into
classifying spaces of de Rham cohomology, Ktheory and (conjecturally)
topological
modular forms. This describes joint work with Stolz and partially
Hohnhold, Kreck
and others. 
It's a long established principle that an
interesting way to think about numbers as the sizes of sets or
dimensions of vector spaces, or better yet, the Euler characteristic of
complexes. You can't have a map between numbers, but you can have one
between sets or vector spaces. For example, Euler characteristic of
topological spaces is not functorial, but homology is.
One can try to extend this idea to a bigger stage, by, say, taking a
vector space, and trying to make a category by defining morphisms
between its vectors. This approach (interpreted suitably) has been a
remarkable success with the representation theory of semisimple Lie
algebras (and their associated quantum groups). I'll give an
introduction to this area, with a view toward applications in topology;
in particular to replacing polynomial invariants of knots that come
from representation theory with vector space valued invariants that
reduce to knot polynomials under Euler characteristic. 
A Lagrangian correspondence is a Lagrangian
submanifold in the product of two symplectic manifolds. This
generalizes the notion of a symplectomorphism and was introduced by
Weinstein in an attempt to build a symplectic category. In joint work
with Chris Woodward we define such a category in which all Lagrangian
correspondences are composable morphisms.
We extend it to a 2category by extending Floer homology to cyclic
sequences of Lagrangian correspondences. This is based on counts of
'holomorphic quilts'  a collection of holomorphic curves in different
manifolds with 'seam values' in the Lagrangian correspondences. A
fundamental isomorphism of Floer homologies ensures that our
constructions are compatible with the geometric composition of
Lagrangian correspondences.
This provides a general prescription for constructing topological
invariants by decomposition into simple pieces and a partial functor
into the symplectic category (which need only be defined on simple
pieces; with moves corresponding to geometric composition). 
I will describe how the Jones polynomial
and Khovanov homology can be computed by counting the
solutions of certain elliptic differential equations on a
four or fivedimensional manifold with boundary. 
