Denis Auroux's research project aims to use Lefschetz fibrations, branched coverings, and their monodromy invariants (mapping class group or braid group factorizations) to study the topology of symplectic 4-manifolds. In particular, Auroux is studying isotopy and non-isotopy phenomena for singular symplectic curves, the relationship between complex projective surfaces and symplectic 4-manifolds, and the role of Luttinger surgery along Lagrangian tori in this context. This leads him to explore some algorithmic aspects of monodromy invariants, most notably algorithms for manipulating braids and braid factorizations, and the Hurwitz problem. He also plans to investigate enumerative invariants for Lefschetz fibrations over the disc, and the relation between the contact homology of a contact manifold equipped with an open book structure and the Floer homology of its monodromy. In a different direction, Auroux is exploring Kontsevich's homological mirror symmetry conjecture and some of its generalizations, building upon recent joint work with L. Katzarkov and D. Orlov. The main ingredient is the study of Landau-Ginzburg models and their symplectic geometry, in order to understand mirror symmetry for some examples of varieties of general type and explore various constructions in algebraic geometry from the perspective of homological mirror symmetry.
Symplectic manifolds are geometric spaces with special structures (allowing area measurements, but not distance measurements). While they first arose in the Hamiltonian formulation of classical mechanics, mathematicians have recently become very interested in their geometry and topology (their intrinsic "shape"), in part due to motivating questions from theoretical physics (string theory). This project aims to study the topology of symplectic manifolds using an approach developped first by S. Donaldson and subsequently by Auroux, which consists in projecting them onto simpler manifolds and studying the points where this projection is "folded". This yields a complete description by combinatorial data, reducing much of the geometry to purely algorithmic considerations. One of the main goals of the project is to relate the topological features of symplectic manifolds with those of complex algebraic manifolds (a more special, much better understood class of geometric spaces). In addition, Auroux is also investigating the phenomenon of mirror symmetry, by studying the symplectic geometry of spaces that are "mirror" to some well-understood families of complex manifolds; this is an important question at the interface between mathematics and theoretical physics.
