Algebraic Structures in Topology and Geometry

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The goal of this project is to understand geometric space in terms of algebraic data and to develop algebraic theories that unlock effective analyses of quantitative and qualitative properties of geometric objects. The main tools come from algebraic topology, a general framework that allows one to reformulate questions about topology and geometry in terms of equivalent questions about algebraic structures such as vector spaces. Topological spaces will be encoded in terms of algebraic structures with particular operations. The PI will also use a similar algebraic framework to study string topology, a theory concerned with interactions of strings and loops in a geometric space. The study of topological and geometric spaces by means of algebraic structures is of fundamental importance in mathematics as well as in mathematical physics. The project aims to explain mathematically the sense in which the fields of topology, geometry, and algebra are equivalent, and the sense in which they are different. The computational tools and invariants that arise from studying the interplay between these fields are useful in the mathematical formulation of quantum field theory, string theory, and mirror symmetry in physics. The award provides funds for graduate students to be involved in parts of this research. The PI will build an inclusive and diverse research group and will promote initiatives directed towards groups that are currently underrepresented in mathematics research.In the first part of the project, the PI will characterize homotopy types through the algebraic concept of an E-infinity coalgebra viewed from the lens of Koszul duality theory. This viewpoint is motivated by a new observation of the PI and M. Zeinalian: the E-infinity coalgebra structure of the singular chains on a space determines the fundamental group in complete generality and this data is preserved under maps which become quasi-isomorphisms after applying the cobar functor. Once homotopy types are understood through this framework, the resulting algebraic structure will be enhanced with extra operations describing Poincaré duality at the chain level in order to characterize topological manifold structures in a homotopy type. The second part of the project is concerned with both foundational and computational questions regarding the string topology of manifolds. Some of the algebraic structures that arise in string topology, in particular the operations related to the Goresky-Hingston coproduct, are able to detect fine geometric information that go beyond the homotopy type in the non-simply connected context. String topology operations will be analyzed using the framework of Hochschild homology and Tate cohomology, as developed in previous work of the PI and Z. Wang. The PI aims to understand the full algebraic structure of string topology, its dependence on the background geometric space, and the new invariants for manifolds that arise.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.