Project description
'Pieces of strings' may not exhibit the same invariance as the whole
Mathematics is a field that enables us to represent physical phenomena and interrelationships in concrete ways. Manifolds are based on the hypothesis that real-world high-dimensional data (for example, a digital image) lie in low-dimensional topological manifolds embedded in high-dimensional space. String topology is the study of the algebraic and differential properties of a topological space or manifold that are invariant or possessed by every other space that is mathematically homeomorphic to it (a one-to-one mapping with even more stringent requirements). The EU-funded StringFrob project is out to show that string topology at the level of chains (intuitively, a linear combination of cells in the space) is not invariant in the way that string topology considered as a whole is. The path to that goal will encompass several important mathematical descriptions.
Objective
The ultimate goal of this action is to establish that chain-level string topology is not a homotopy invariant. This is achieved by showing that chain-level string topological structures are induced by a homotopy Frobenius structure on the cochain algebra and by connecting the homotopy Frobenius structure with known invariants from quantum field theory. This is broken down into four independent work packages. The first goal is to show that from a Chern-Simons type partition function one can construct a homotopy Frobenius algebra and show that this is essentially an equivalence between the relevant deformation spaces. The second goal is to algebraically construct string topology operations on the Hochschild homology of a homotopy Frobenius algebra. The third goal compares the induced structure on the cyclic homology with the known homotopy involutive Lie bialgebra structure. And ultimately, the fourth goal is to compare the algebraically constructed operations with geometric ones on the loop space under the comparison map given by Chen's iterated integrals.
Fields of science
Keywords
Programme(s)
Funding Scheme
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinator
1165 Kobenhavn
Denmark