# PDF Gauge Theory and the Topology of Four-Manifolds

In this talk, I will describe how string topology arises in the formulation of rational SFT as well as recent results on a suitable model of string topology in dimension 2. Abstract: The classical problem of enumerating rational curves in projective spaces is solved using a recursion formula for Gromov-Witten invariants. In this talk, I will describe a similar relation for real Gromov-Witten invariants with conjugate pairs of constraints.

An application of this relation provides a complete recursion for counts of real rational curves with such constraints in odd-dimensional projective spaces.

## a collection of literature that the course may refer to

I will outline the proof and discuss some vanishing and non-vanishing results. This is joint work with A. Abstract: Heegaard Floer homology is a powerful 3-manifold invariant, but computing it in general is difficult. I will describe a method for computing HF-hat in the case of graph manifolds. A graph manifold decomposes nicely into pieces which are circle bundles over surfaces.

The key machinery for computing HF-hat is bordered Heegaard Floer homology, an extension of the Heegaard Floer package to manifolds with boundary. By computing the bordered invariants of the circle bundles over surfaces and piecing them together in an appropriate way, we can compute HF-hat of any graph manifold.

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## Research in Geometry/Topology | Department of Mathematics at University of California Berkeley

Abstract: We consider the problem of defining cylindrical contact homology, in the absence of contractible Reeb orbits, using "classical" methods. The main technical difficulty is failure of transversality of multiply covered cylinders. We outline how fixing the latter difficulty ultimately leads to a different theory, an analogue of positive symplectic homology.

This talk is intended to be part of a series of expository talks on the foundations of contact homology, but prerequisites should be minimal. Abstract: In , Khovanov defined a TQFT which categorifies the Kuperberg bracket this is a flat version of the sl3 polynomial for links. The Khovanov-Kuperberg algebras are central in this construction.

### Gauge Theory Foundations

By contrast with the sl2 case, the natural candidates the so called web-modules for being a complete family of indecomposable projective modules, may decompose. However I'll explain that that one can characterise the indecomposability with a very simple criteria. The proof is based on some combinatorial objects called red graphs.

- Passar bra ihop.
- Geometry & Topology Editors Interests;
- Latent Variable Analysis and Signal Separation: 9th International Conference, LVA/ICA 2010, St. Malo, France, September 27-30, 2010. Proceedings.
- Max Planck Institute for Mathematics.
- Gauge Theory and Four-Manifold Topology;

Abstract: There are a number of ways to define a Floer homology for three-manifolds using Seiberg-Witten theory. Two such examples are Kronheimer and Mrowka's monopole Floer homology and Manolescu's Seiberg-Witten Floer spectrum, each of which has its own advantages and applications. We will define these and discuss the relationship between these two objects, after describing an analogue in Morse theory.

This is work in progress with Ciprian Manolescu. Abstract: Cochran and Gompf defined a notion of positivity for concordance classes of knots that simultaneously generalizes the usual notions of sliceness and positivity of knots.

## Floer Homology, Gauge Theory, and Low Dimensional Topology

Abstract: The lectures in this volume provide a perspective on how 4-manifold theory was studied before the discovery of modern-day Seiberg-Witten theory. Volume: 4. Publication Month and Year: Copyright Year: Page Count: Cover Type: Hardcover. Print ISBN Online ISBN Print ISSN: Online ISSN: Primary MSC: 14 ; 32 ; 53 ; 57 ; Applied Math?

MAA Book? Electronic Media? The interaction between gauge theory, low—dimensional topology, and symplectic geometry has led to a number of striking new developments in these fields. The aim of this volume is to introduce graduate students and researchers in other fields to some of these exciting developments, with a special emphasis on the very fruitful interplay between disciplines. Several of the authors have added a considerable amount of additional material to that presented at the school, and the resulting volume provides a state-of-the-art introduction to current research, covering material from Heegaard Floer homology, contact geometry, smooth four—manifold topology, and symplectic four—manifolds.