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Let f:N --> R be a Morse function and let g be a Riemannian metric such that (f,g) is Morse-Smale. I will explain a construction which, under certain restrictions, produces a Lefschetz fibration π: T*N --> C extending f and having the same critical points. In addition I will explain a relation between the directed Fukaya category of π and the flow category of (f,g). If time permits, I will describe an application to the study of exact Lagrangian submanifolds of T*N, using a spectral s equence of Seidel.

Holomorphic curves are powerfull tools in symplectic topology. In many cases, information about holomorphic curves is obtained by considering a degenerating family of complex structures. The category of exploded torus fibrations is an extension of the category of smooth manifolds in which there is a good theory of holomorphic curves and some of these degenerations are well behaved. I will give some examples and explain the relationship of this with symplectic field theory and tropical geometry.

In his work on symplectic Lefschetz pencils, Donaldson introduced the notion of estimated transversality for a sequence of sections of a bundle. Together with asymptotic holomorphicity, it is the key ingredient allowing the construction of symplectic submanifolds. Despite its importance in the area, estimated transversality has remained a mysterious property. The aim of this talk is to shed some light into this notion by studying it in the simplest possible case namely that of S². We state some new results about high degree rational maps on the 2-sphere that can be seen as consequences of Donaldson’s existence theorem for pencils, and explain how one might go about answering a question of Donaldson: what is the best estimate for transversality that can be obtained? We also show how the methods applied to S² can be further generalized.

The Cremona transform is a classically studied rational map on projective space Pⁿ. It admits a resolution on X, an iterated toric blowup of Pⁿ. The space X possesses a symmetry which descends to the theory of the blowup of Pⁿ along only points, and hence to Pⁿ itself. This symmetry expresses some difficult to compute invariants in terms of others that are easy to compute, and provides a new technique for the computation of these invariants of Pⁿ. Also, this recovers interesting enumerative results.

We explain the equivalence of the three theories described in the title for resolved A_n singularities and discuss related questions and applications. This is joint work with A. Oblomkov.