In previous work, the principal investigator constructed a highly nontrivial filtration of the knot concordance group, indexed by the height of gropes embedded in the 4-ball, and partially detected by von Neumann's continuous dimension. The resulting graded group remains unknown, and the principal investigator proposes various approaches to uncover the structure of this group. Related questions about link concordance and embedding problems of 2-spheres into 4-manifolds can also be studied by these methods. In the second part of the project, the principal investigator is attempting to give a geometric definition of elliptic cohomology in terms of a modification of Segal's "elliptic objects". These are conformal field theories parametrized by a topological space X, in particular to each circle in X they associate a Hilbert space. For more than 15 years, Segal's approach could not be turned into a cohomology theory because of the failure of the Mayer-Vietoris principle. The new idea is to apply the fusion of bimodules of von Neumann algebras, developed by Connes, to make the conformal field theory "local in X". Fusion is used to decompose the Hilbert space whenever the corresponding circle in X is decomposed. Such a local theory should then satisfy all the axioms of a cohomology theory.
Both parts of the project relate notions from theoretical physics to mathematics. Historically, the converse relation was more common, where a mathematical notion (like Riemannian geometry or functional analysis) was used to explain a physical theory (like relativity or quantum mechanics). In the last decades, surprising mathematical predictions (provable only in very rare cases) came out of considerations in theoretical physics (like quantum gravity or conformal field theory). It is thus of the ultimate importance for mathematical research to incorporate such considerations into the body of well understood theories. In the first part of this project, the principal investigator proposes to continue his successful study of 4-dimensional manifolds (most relevant in relativity) via techniques originally proposed by von Neumann for the study of quantum mechanics. In the second part, the principal investigator proposes to refine the notion of a conformal field theory so that it leads to a geometrical definition of "elliptic cohomology". This cohomology is an enormously successful tool in mathematics and the proposed refinement has the potential to lead to a topological understanding of all conformal field theories.
