Geometric Aspects of Algebraic Topology

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DMS-0204080Po HuIn this project, the investigator intends to consider a circle of ideas in algebraic topology and its interactions with other areas of mathematics. One point of entry to these ideas is Real-oriented homotopy theory, which uses the complex conjugation action on the complex cobordism spectrum to approach homotopy groups of spheres. The method used here is a case of descent, which appears also in Voevodsky's homotopy theory of algebraic varieties. In this direction, the investigator is interested in algebraic cobordism, the algebro-geometric analogue of complex cobordism. Another closely related idea is that of Verdier and Grothendieck dualities, both in algebraic geometry and in equivariant homotopy theory. Yet another related but different kind of duality is Koszul duality. Using this duality, the investigator, jointly with collaborators, proved a version of Kontsevich's conjecture on Hochschild cohomology of k-algebras. This in turn is related to the question of deformation quantazation in mathematcial physics. Another area into which Koszul duality enters is the string topology of Chas and Sullivan, which gives another connection between algebra and homotopy theory. The investigator is also involved with another project related to physics, namely constructing geometric models of elliptic cohomology.Topology is the study of spaces that can be deformedcontinuously. In part, this proposal seeks to better understand maps between such objects by ``stable'' methods, i. e. considering a sequence of objects of all higher dimensions at once, and by using certain summetries upon them. Similar methods can also be used to study more rigid geometric objects, for example in algebraic geometry the solution sets of systems of algebraic equations. This leads in turn to the study of algebraic structures on an abstract level.Finally, the related idea of string topology considers not just a space itself, but structures on the space consisting of loops in it. In particular, this is important to string theory in physics, which sees the universe as composed not of point-like particles but of loop-like ``strings''.