Monopoles and 3-Manifolds

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PROPOSAL DMS-0303601PI: Liviu Nicolaescu, University of Notre DameTitle: Monopoles and 3-manifoldsABSTRACTThe topology of isolated complex two-dimensional singularities has abuilt-in rigidity which is not manifested in any other dimensions.For example, in a large and clearly delimited family oftwo dimensional singularities, one can determine fragile analyticinvariants of the singularity (such as the geometric genus, the signatureof the Milnor fibers, the Minor number etc.) by performing robusttopological manipulations. These are conveniently encoded by theSeiberg-Witten invariants. The investigator intends to investigate thereasons behind this surprising phenomenon by looking into the finerstructure of Seiberg-Witten monopoles relying on an adiabatic deformationof the Seiberg-Witten equations.The goal of the present proposal is to perform a micro-analysis ofsingularities, objects popularly known as `catastrophes'. One couldvisualise these as surfaces which are nice most everywhere except atfew places where one sees forming `cusps' and `spikes', mathematicallyreferred to as `singular points'. If an observer sits at a regular pointand looks around, the `horizon' he/she observes is round, spherical. The`horizon ' of a singular point is an object mathematicians referto as `the link of the singularity' and is shaped quite differentlythan the `horizon' of a regular point. In fact, the shape of this linkalone contains a wealth of information about how the space in thevicinity of the singular point twists, bends and folds. This kind ofinformation is carried in abundance by certain objects thephysicists refer to as `monopoles'. These are similar in many respects tothe electromagnetic waves, but they are more sensitive to the shape ofthe Universe they travel in. The investigator will analyze thestructure of these monopoles by `looking at these singularitiesthrough a high resolution microscope', a process mathematically know asadiabatic deformation of the space. This technique has already producedencouraging preliminary results.