Lefschetz fibrations in symplectic topology and applications to mirror symmetry

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DMS-060148Denis AurouxDenis Auroux's research project aims to use Lefschetz fibrations, branchedcoverings, and their monodromy invariants (mapping class group or braidgroup factorizations) to study the topology of symplectic 4-manifolds.In particular, Auroux is studying isotopy and non-isotopy phenomena forsingular symplectic curves, the relationship between complex projectivesurfaces and symplectic 4-manifolds, and the role of Luttinger surgeryalong Lagrangian tori in this context. This leads him to explore somealgorithmic aspects of monodromy invariants, most notably algorithms formanipulating braids and braid factorizations, and the Hurwitz problem.He also plans to investigate enumerative invariants for Lefschetzfibrations over the disc, and the relation between the contact homologyof a contact manifold equipped with an open book structure and the Floerhomology of its monodromy. In a different direction, Auroux is exploringKontsevich's homological mirror symmetry conjecture and some of itsgeneralizations, building upon recent joint work with L. Katzarkov andD. Orlov. The main ingredient is the study of Landau-Ginzburg models andtheir symplectic geometry, in order to understand mirror symmetry for someexamples of varieties of general type and explore various constructionsin algebraic geometry from the perspective of homological mirror symmetry. Symplectic manifolds are geometric spaces with special structures (allowingarea measurements, but not distance measurements). While they first arosein the Hamiltonian formulation of classical mechanics, mathematicians haverecently become very interested in their geometry and topology (theirintrinsic "shape"), in part due to motivating questions from theoreticalphysics (string theory). This project aims to study the topology ofsymplectic manifolds using an approach developped first by S. Donaldsonand subsequently by Auroux, which consists in projecting them onto simplermanifolds and studying the points where this projection is "folded".This yields a complete description by combinatorial data, reducing muchof the geometry to purely algorithmic considerations. One of the maingoals of the project is to relate the topological features of symplecticmanifolds with those of complex algebraic manifolds (a more special, muchbetter understood class of geometric spaces). In addition, Auroux is alsoinvestigating the phenomenon of mirror symmetry, by studying the symplecticgeometry of spaces that are "mirror" to some well-understood families ofcomplex manifolds; this is an important question at the interface betweenmathematics and theoretical physics.