Singularities and symplectic topology

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Singularities are everywhere we look: from the cuspy caustic curve that forms when you shine light into your coffee cup to the microscopic black holes that we expect to form in extra-dimensions in the theory of strings and branes.The mathematical study of singularities involves algebraic geometry. This is a branch of geometry which involves writing equations for the geometrical spaces of interest. For example, the cusp curve is described by the equation y^2=x^3 in the plane. For algebraic geometers, singularities appear naturally when you try to classify all the possible spaces you can write down via equations (the efforts to carry out this classification go by the name of "the minimal model program").I am proposing a new way to study singularities. If you deform the equation (for example you study y^2=x^3+t for some constant t) you can sometimes smooth out the singularity, but the smoothed space can have highly nontrivial topology (you can see this happen if you tilt your coffee cup into the light). This new piece of topology, which is crushed back down to the singular point as t goes to zero, is called a vanishing cycle. Let's suppose you want to show that two different singularities cannot form at the same time. I will try to do this by showing that the vanishing cycles cannot be moved apart from one another ("displaced"). If the vanishing cycles are nondisplaceable then one or other singularity can form, but both singularities cannot form at the same time.The difficulty in making this kind of argument rigorous is that the vanishing cycles can themselves be singular! To prove such results, I will need to develop existing techniques for proving nondisplaceability ("Floer theory") to the situation where the vanishing cycles have singularities. These techniques should then have applications in other parts of mathematics where singular cycles play a role, for example the "singular SYZ fibres" which arise in studying the mysterious geometric duality called Mirror Symmetry (predicted by string theorists in the early 1990s and still not fully understood) or the singular limits of Lagrangian mean curvature flows in geometric analysis.