Low Dimensional Geometry and Monopoles

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PROPOSAL DMS-0305130LOW-DIMENSIONAL GEOMETRY AND MONOPOLESP.Is: Yann Rollin, Gang TianABSTRACTThe aim of the proposed project is to study a manifold with boundary, andsome additional structure, in relation with the boundary values of thestructure. This generic framework is a very natural approach to manyimportant questions of differential geometry. Some typical cases are thoseof contact structures fillable by symplectic forms, complex geometry andCauchy-Riemann boundary, Einstein geometry and conformal infinity. It isinteresting to study how deformation theory of the boundary inducesdeformations of a filling. This problem is often related to some positivefrequency condition, which is of particular interest for physicists. Thiswill be used to study singular Yang-Mills connections of (a conjectural)Donaldson theory on Calabi-Yau 3-manifolds. More generally, we look at thepicture of a moduli space on the manifold which projects on acorresponding moduli space defined on the boundary. Whenever it can beshown that the projection has a finite nonzero degree in some sense, weobtain a powerfull tool to prove the existence of fillings. Another way totackle the questions of existence is to develop gluing theorems betweenmoduli spaces. Some applications of gluing for Seiberg-Witten equationsare already being developped thus giving new perspectives on contactgeometry.A physical theory is defined by a configuration space, together withobjects (for example a metric) verifying particular equations.The state of a physical system is constrained by its configuration onthe boundary of the space, or, at infinity. Therefore, it is anatural question to ask wether a theory is rich or empty: is itpossible to find a physical system with a given behavior on theboundary, and if so, do we have many solutions? In other words is thesystem soft or rigid? Beyond the physical flavor, this approach, knownas a field theory, has deep implications in mathematics. First, it isrequired to elaborate original tools in analysis, called moduli spacetheories, that have their own beauty. Secondly, the method relatesvery different problems. To illustrate that, we mention the followingsituation: we consider a 3-dimensional space (it could be the3-dimensional sphere) bounding a 4-dimensional space (like the ballinside the sphere). Then, we look at knots on the boundary (they couldbe strands of DNA). These knots can be thought of as the boundary ofa surface lying into the 4-dimensional space. Then, it can be shownthat there is a subtle relation between the knot theory in dimension 3and the theory of surfaces lying in a 4-dimensional space.