Embeddings, Intersections and Symmetries

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This proposal involves various aspects of the homotopy theory ofmanifolds, all of which, in one way or another, have connections tothe theory of embeddings. The proposalhas four parts: 1) Embedding theory. The PI will use techniques arising from thehomotopy theory of Poincare complexes to study embedding questions. 2) Group Actions. The study of equivariant manifold classificationproblems will be approached in a new way using homotopy theory. 3) String Topology. This is emerging field, created by Chas andSullivan, studies algebraic structures associated with the free loopspace of a manifold. The PI proposesto study homotopy theoretic aspects of string topology. 4) Thickenings. A "thickening" of a space is a manifold model forits homotopy type. The PI proposes to analyse the homotopy type of the moduli space of thickenings of a finite complex.A "manifold" is a space which is locally Euclidean. Manifolds arisenaturally in physics, chemistry and biology as spaces of solutions ofa suitably "nice" set of algebraic equations modeling the scientificobject of study (space-time, atoms, dynamical systems, etc.) .Manifolds play a central role in mathematics. It is often the casethat mathematical questions about manifolds can be formulated interms of parametrized families of functions between spaces associatedwith manifolds. Homotopy theory is a subject designed to tacklequestions about such families of functions. The PI proposes to studycertain kinds of manifold questions which can be hopefully be analyzedfrom the homotopy theoretic toolbox. These involve problems relatedwith intersections, embeddings and symmetries.