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Most of modern geometry studies some kind of space. The spaces considered in differential geometry are called manifolds , spaces which locally look like n-dimensional Euclidean space but globally have an interesting shape. A manifold is compact if it is closed up, with no edges. The surface of a doughnut is a compact 2-dimensional manifold. A submanifold N of a manifold M is a subset of M which is itself a manifold, usually of smaller dimension than M. There are two kinds: embedded submanifolds, which may not intersect (cross) themselves, and immersed submanifolds, which may. One usually considers manifolds with some extra geometric structure, such as a Riemannian metric , which tells you the lengths of paths in the manifold, or a symplectic structure , which tells you the areas of 2-dimensional submanifolds of the manifold. Symplectic manifolds are the foundation of the mathematical formulation of mechanics, and so of much of classical physics. They are also very interesting in their own right. Mathematicians like them because they are one of very few structures with an infinite-dimensional amount of local symmetry, which gives symplectic geometry an unusual, entirely global flavour. Lagrangian submanifolds are a special kind of submanifold of a symplectic manifold. Given two compact, embedded Lagrangian submanifolds N, N* of a symplectic manifold M, one can under certain conditions define the Floer homology groups HF(N,N*), which are roughly speaking finite-dimensional vector spaces. The definition is very difficult. To do it, one chooses an auxiliary complex structure J on M and counts J-holomorphic 2-dimensional discs D in M with boundary (edge) in the union of N and N*. The remarkable thing about HF(N,N*) is that it is independent of the choice of J, and is also unchanged by moving N and N* around amongst Lagrangian submanifolds. It encodes some mysterious, nontrivial information about Lagrangian submanifolds one cannot get at in any other known way. It is a powerful tool in symplectic geometry. The main aim of the research is to extend the definition of Floer homology groups HF(N,N*) to immersed Lagrangian submanifolds N, N*, and to understand the conditions under which they can be defined ( obstructions to their definition). The new technical problems this involves have to do with J-holomorphic discs D whose boundary passes through self-intersection points in N or N*, and what is the right algebraic set-up for including and counting these to get well-behaved groups HF(N,N*). We also want to understand the allowed motions of N and N* amongst immersed Lagrangian submanifolds which do not change HF(N,N*). As well as being interesting to symplectic geometers, we believe these results will have important applications to several major conjectures about special Lagrangian geometry and Calabi-Yau manifolds, which are of interest to physicists working in String Theory. The point is that these conjectures can only be true if one works in the right class of Lagrangians, and embedded nonsingular Lagrangians are not a large enough class. There is good evidence that the right class to consider may be immersed Lagrangians whose Floer homology is unobstructed, but to understand what this means we first need a theory of Floer homology for immersed Lagrangians, which we hope to develop.
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