Periodic Reporting for period 4 - Loops and groups (Loops and groups: Geodesics, moduli spaces, and infinite discrete groups via string topology and homological stability)
Reporting period: 2023-03-01 to 2024-02-29
The research project lies at the intersection of algebra, topology, and geometry, with the scientific goal of answering central questions about homological stability, geodesics on manifolds, and the moduli space of Riemann surfaces.
Homological stability is a subject that has seen spectacular progress in recent years, and the project builds on this progress to go further in several directions. The idea of homological stability is that it is sometimes easier to compute the homology of an object if the object is part of a family with a certain structure. In recent work of the PI gave a way to canonically associate a family of spaces to any stability problem.The first two goals of the proposal are to give conditions under which this family of spaces is highly connected, and to use this to prove homological and representation stability theorems, with determination of the stable homology. New results were proved for mappin class groups, linear groups, and unitary groups. The last two goals of the project concern geodesics and moduli spaces via string topology: The third goal seeks a geometric construction of compactified string topology, and proposes to understand the geometric meaning of vanishing and non-vanishing of string operations, with long term goal to address counting problems for geodesics on manifolds. From work of Naef, it is now known that the "good coproduct" is not a homotopy invariant of the manifold, and we give a geometric obstruction for invariance under homotopy equivalences. Finally our fourth goal is to use compactified string topology and field theories to study the harmonic compactification itself, and give a new approach to finding families of unstable homology classes in the moduli space of Riemann surfaces. Different flavours of field theories were classified, and a new method to approach the moduli space was given.
The project combines methods from homotopy theory with methods from algebraic, differential and geometric topology.
The project is one of fundamental research, but the objects studied are relevant to other sciences such as physics, in particular through field theories, chemistry and biology, in particular through graph complexes, or the study of big data, in particular through simplicial complexes, while geodesics are shortest paths, and an important example of a minimisation problem.
We have proved new stability results and given a new simple proof of stability for mapping class groups of surfaces that opens a new line of attack in stability questions. We have proved connectivity results for a number of simplicial complexes associated to homological stability questions, with applications outside the subject. We have obtained a much closer understanding of the string topology coproduct, an operation on the homology of the free loop space of a manifold that has turned out more subtle than expected. We have also advanced our understanding of fusion categories, representation categories and their associated field theories.
Homological stability is a subject that has seen spectacular progress in recent years, and the project builds on this progress to go further in several directions. The idea of homological stability is that it is sometimes easier to compute the homology of an object if the object is part of a family with a certain structure. In recent work of the PI gave a way to canonically associate a family of spaces to any stability problem.The first two goals of the proposal are to give conditions under which this family of spaces is highly connected, and to use this to prove homological and representation stability theorems, with determination of the stable homology. New results were proved for mappin class groups, linear groups, and unitary groups. The last two goals of the project concern geodesics and moduli spaces via string topology: The third goal seeks a geometric construction of compactified string topology, and proposes to understand the geometric meaning of vanishing and non-vanishing of string operations, with long term goal to address counting problems for geodesics on manifolds. From work of Naef, it is now known that the "good coproduct" is not a homotopy invariant of the manifold, and we give a geometric obstruction for invariance under homotopy equivalences. Finally our fourth goal is to use compactified string topology and field theories to study the harmonic compactification itself, and give a new approach to finding families of unstable homology classes in the moduli space of Riemann surfaces. Different flavours of field theories were classified, and a new method to approach the moduli space was given.
The project combines methods from homotopy theory with methods from algebraic, differential and geometric topology.
The project is one of fundamental research, but the objects studied are relevant to other sciences such as physics, in particular through field theories, chemistry and biology, in particular through graph complexes, or the study of big data, in particular through simplicial complexes, while geodesics are shortest paths, and an important example of a minimisation problem.
We have proved new stability results and given a new simple proof of stability for mapping class groups of surfaces that opens a new line of attack in stability questions. We have proved connectivity results for a number of simplicial complexes associated to homological stability questions, with applications outside the subject. We have obtained a much closer understanding of the string topology coproduct, an operation on the homology of the free loop space of a manifold that has turned out more subtle than expected. We have also advanced our understanding of fusion categories, representation categories and their associated field theories.
We have proved new stability results and given a new simple proof of stability for mapping class groups of surfaces that opens a new line of attack in stability questions. We have proved connectivity results for a number of simplicial complexes associated to homological stability questions, with applications outside the subject. We have obtained a much closer understanding of the string topology coproduct, an operation on the homology of the free loop space of a manifold that has turned out more subtle than expected. We have also advanced our understanding of fusion categories, representation categories and their associated field theories.
We have pushed our understanding of the phenomenon of homological stability to a new level, in particular through a better understanding of the simplicial complexes associated to stability problems. We have used string topology to define a new invariant of maps that is expected to be closely related to Whitehead torsion. This fits into a new, maybe unexpected, development of string topology, that shows that compactified string operations on the loop space of a manifold can detect more than then the homotopy type of the manifold.