Low-dimensional Manifolds and High-dimensional categories

History of Michael Freedman's Proof of the 4D Poincare Conjecture

Robion Kirby, Robert Edwards

Flexible and rigid Weinstein manifolds

Yakov Eliashberg

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 non-symplectomorphic 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 >4 . 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 >3.
Parts of this work are joint with K. Cieliebak and F. Bourgeois-T. Ekholm.

Smooth 4-manifolds: 2011

Ron Fintushel

This will be an expository lecture on the state ofaffairs with regard to smooth structures on simply connected4-manifolds with b^+ = 1. There has been considerable progresson this front in the last 5 to 10 years. I will concentrate ontechniques and ideas involved in making this progress and survey theresults that these techniques give as well as open questions andconjectures.




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Low-dimensional Morse 2-functions

David Gay

A Morse 2-function is just a generic (stable) smoothmapfrom a smooth n-manifold to a 2-manifold; here we will focus on maps to R^2 . We will look at some basic exampleswhen the domainis 2-dimensional, just to get our intuition on the right track, andthen look at the case where the domain is 3- or 4-dimensional. I willstate some results that are the result of joint work with Rob Kirbyand stress the underlying connection with Cerf theory (the study ofgeneric homotopies between ordinary Morse functions).

Exceptional Dehn filling

Cameron Gordon

It is rare for a hyperbolic knot in S^3 to have a non-trivial Dehn surgery that yields a non-hyperbolic 3-manifold, but it does happen. More generally, it is rare for a hyperbolic 3-manifold M to have two non-hyperbolic Dehn fillings M(a) and M(b)" along a given cusp. Consequently, one wonders if it might be possible to describe all such exceptional triples (M;a,b). We will give a survey of progress towards this goal, and some of the problems that remain.

The homological algebra of knots and BPS states

Sergei Gukov

A physical interpretation of knothomologies as spaces of the so-called "refined BPS states" leads tomanyinteresting and often surprising predictions, ranging from concreteexpressions for homological knot invariants associated with higherrepresentations of SL, SO, and Sp classical groups to intricate webs ofspectral sequences and anti-commuting differentials acting on thetriply-graded theories categorifying the HOMFLY and Kauffmanpolynomials. Inthis talk, I will review the enterprise of this connection between(enumerative) geometry and knot theory and, if time permits, describesomeof the latest developments, related to the interpretation of spectralsequences and differentials, and wall crossing for refined / motivicBPSinvariants.

HF=ECH via open book decompositions

Ko Honda

The goal of this talk is to sketch a proof of the equivalence of Heegaard Floer homology (due to Ozsvath-Szabo) and embedded contact homology (due to Hutchings). This is joint work with Vincent Colin and Paolo Ghiggini.




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Applications of embedded contact homology

Michael Hutchings

Embedded contact homology is an invariant of contact three-manifolds which is isomorphic to versions of Heegaard Floer homology and Seiberg-Witten 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 four-dimensional 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.




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Title: The A-B slice problem

Slava Krushkal

In his groundbreaking 1982 paper Michael Freedman proved a disk embedding theorem, leading to a classification of simply-connected topological 4-manifolds. 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 A-B slice problem is a reformulation (due to Freedman) of this question in terms of a generalized link-slicing problem. This talk will discuss the history of the problem, as well as recent developments and ideas for a solution.




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Fukaya category of the punctured torus

Yanki Lekili

In joint work with Tim Perutz, we give a complete characterization of the Fukaya category of the punctured torus, denoted by Fuk(To). This, in particular, means that one can write down an explicit minimal model for Fuk(To) in the form of an A-infinity algebra, denoted by A, and classify A-infinity structures on the relevant algebra. A result that we will discuss is that no associative algebra is quasi-equivalent to the model A of the Fukaya category of the punctured torus, i.e., A is non-formal Fuk(To). will be connected to many topics of interest:
1) It is the boundary category that we associate to a 3-manifold with torus boundary in our extension of Heegaard Floer theory to manifolds with boundary,
2) It is quasi-equivalent to the category of perfect complexes on an irreducible rational curve with a double point, an instance of homological mirror symmetry.

Khovanov homology for 4-manifolds.

Scott Morrison

I'll describe an invariant of smooth 4-manifolds defined in terms of Khovanov homology. It associates a doubly-graded vector space to each 4-manifold (optionally with a link in its boundary), generalizing the Khovanov homology of a link in the boundary of the standard 4-ball. 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.

Topology in the mystery dimension

Frank Quinn

Overview of 4-dimensional topology, with historical notes.

Fibered knots and potential counterexamples to the Property 2R and Slice-Ribbon Conjectures

Martin Scharlemann

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 two-component links that contain the square knot and which surger to (S^1xS^2)#(S^1xS^2). 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.


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Higher-order intersections in low-dimensional topology

Rob Schneiderman

The failure of the Whitney move in dimension 4 can be measured by constructing higher-order intersection invariants of Whitney towers built from iterated Whitney disks on immersed surfaces in 4--manifolds. For Whitney towers on immersed disks in the 4--ball, some of these invariants can be identified with previously known link invariants like Milnor, Sato-Levine and Arf invariants. This approach also leads to the definition of higher-order Sato-Levine 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 higher-order invariants classify the existence of (twisted) Whitney towers of increasing order in the 4--ball.




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Fractional Euler characteristics and completed Grothendieck groups

Catharina Stroppel

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 Reshetikhin-Turaev 3-manifold 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 p-adic completion of the Grothendieck groups and the notion of mixed categories with weight filtrations. I will illustrate the concept by categorified link invariants and 3j-symbols.

Higher categories for low-dimensional topologists

Peter Teichner

We'll motivate the definition of d-categories via the d-dimensional bordism category of manifolds (with geometric structures). Then we'll go through the low-dimensional 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, K-theory and (conjecturally) topological modular forms. This describes joint work with Stolz and partially Hohnhold, Kreck and others.

Categorification, Lie algebras and topology

Ben Webster

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 semi-simple 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.




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3-dimensional manifolds and symplectic categories

Katrin Wehrheim

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 2-category 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).




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Gauge Theory And Khovanov Homology

Edward Witten

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 five-dimensional manifold with boundary.




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Lecture Notes