Periodic Reporting for period 3 - Loops and groups (Loops and groups: Geodesics, moduli spaces, and infinite discrete groups via string topology and homological stability)
Okres sprawozdawczy: 2021-09-01 do 2023-02-28
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. Particular attention is given to Thompson-like groups, building on a recent breakthrough of the PI with Szymik, but classical groups, such as unitary groups are also studied. 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, which we propose to use to address counting problems for geodesics on manifolds. An unexpected connection between string topology and surgery theory is given. 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.
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 a stability result for unitary groups, in a very general sense of the word. 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 advanced our understanding of fusion 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. Particular attention is given to Thompson-like groups, building on a recent breakthrough of the PI with Szymik, but classical groups, such as unitary groups are also studied. 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, which we propose to use to address counting problems for geodesics on manifolds. An unexpected connection between string topology and surgery theory is given. 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.
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 a stability result for unitary groups, in a very general sense of the word. 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 advanced our understanding of fusion categories and their associated field theories.
We have proved a stability result for unitary groups, in a very general sense of the word. 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 advanced our understanding of fusion categories and their associated field theories.
We expect to push 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, including stability for groups, spaces, or algebras. We expect to be able to show that a certain string topology operation detects Whitehead torsion, answering a long awaited quest for string topology in its basic form to detect more than then homotopy type of a manifold. Finally we expect to gain new knowledge on moduli spaces of graphs.