The topology of isolated complex two-dimensional singularities has a built-in rigidity which is not manifested in any other dimensions. For example, in a large and clearly delimited family of two dimensional singularities, one can determine fragile analytic invariants of the singularity (such as the geometric genus, the signature of the Milnor fibers, the Minor number etc.) by performing robust topological manipulations. These are conveniently encoded by the Seiberg-Witten invariants. The investigator intends to investigate the reasons behind this surprising phenomenon by looking into the finer structure of Seiberg-Witten monopoles relying on an adiabatic deformation of the Seiberg-Witten equations.
The goal of the present proposal is to perform a micro-analysis of singularities, objects popularly known as `catastrophes'. One could visualise these as surfaces which are nice most everywhere except at few places where one sees forming `cusps' and `spikes', mathematically referred to as `singular points'. If an observer sits at a regular point and looks around, the `horizon' he/she observes is round, spherical. The `horizon ' of a singular point is an object mathematicians refer to as `the link of the singularity' and is shaped quite differently than the `horizon' of a regular point. In fact, the shape of this link alone contains a wealth of information about how the space in the vicinity of the singular point twists, bends and folds. This kind of information is carried in abundance by certain objects the physicists refer to as `monopoles'. These are similar in many respects to the electromagnetic waves, but they are more sensitive to the shape of the Universe they travel in. The investigator will analyze the structure of these monopoles by `looking at these singularities through a high resolution microscope', a process mathematically know as adiabatic deformation of the space. This technique has already produced encouraging preliminary results.
