Alternatively have a look at the program.

## Whitney towers and Milnor invariants, a survey of some recent results

## Quilted Floer homology of 3-manifolds

We introduce quilted Floer homology (QFH), a new invariant of 3-manifolds equipped with an indefinite circle valued Morse function (i.e. broken fibration). This is yet another localization of Seiberg-Witten theory and a natural extension of Perutz's 4-manifold invariants associated with broken Lefschetz fibrations, making it a (3+1) theory. We relate Perutz's theory to Heegaard Floer theory by giving an isomorphism between QFH and HF+ for extremal spin^c structures with respect to the fibre of the Morse function.

## A different view on Casson-Gordon invariants for knots

## Contact structures on product manifolds

Contact structures, by virtue of their nonintegrability, are better adapted to twisted products (fibre bundles) than cartesian products. In this talk I shall discuss a construction of contact structures on products of $S^1$ with manifolds admitting a suitable decomposition into exact symplectic pieces. Such a decomposition will be shown to exist for symplectic 4-manifolds. This is joint work with A. Stipsicz.

## An application of discrete Morse theory

## Splice diagrams and universal abelian covers

## Decompositions of low-dimensional objects

We develop a new version of the famous Diamond Lemma of M. H. A. Newman (1942) and use it to describe several results on decompositions of different geometric objects. 1. A spherical splitting theorem for knotted graphs in 3-manifolds. 2. Counterexamples to the folklore prime decomposition theorem for 3-orbifolds. 3. A theorem on annular splittings of 3-manifolds, which is independent of the JSJ-decomposition theorem. 4. Prime decomposition theorem for homologically trivial knots in thick surfaces.

## Volume of Representations: Definition and Applications

## Topological Classification of Bott manifolds

## The symplectic representation of the mapping class group is unique

The mapping class group, MCG, of a closed orientable surface of

genus g has classical symplectic representation coming from its action on

the first

homology of the surface. I will show that any 2g-dimensional nontrivial

representation of the group MCG is conjugate to this classical

representation. I will also prove that MCG has no faithful linear

representation in dimensions less than 3g-2. This contributes to the

well-known problem on the linearity of MCG. Some algebraic corollaries on

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