One aim of the project is to complete the quasi-isometric classification of 3-manifold groups, on which the PI and Behrstock made very significant progress over the last five years. Another is to use new methods to study singularities of complex surfaces. In particular, the important role of bilipschitz geometry in revealing fine invariants has become clear through work of the PI, Birbrair, and Pichon. The PI will extend their recent bilipschitz classification of inner metric of complex surface germs to the outer metric, and apply this to such long-standing problems as the relationship between Lipschitz and Zariski equisingularity and the conjectured dualities between intersection homology and analytically based cohomology theories. A start will be also be made on extending the approach also to higher dimensions and sets in o-minimal structures. The PI will also continue studies related to the Casson invariant conjecture, a concrete formulation of a looser question originally asked by Sir Michael Atiyah, postulating links, motivated out of physics, between topological invariants related to topological quantum field theories and analytical invariants in algebraic geometry. At the other end of the spectrum, for hyperbolic manifolds there are also postulated connections between geometric, arithmetic, representation-theoretic and quantum based invariants of manifolds, which the PI and his students have made progress on and on which the PI will continue to work.
Quasi-isometric and bilipschitz geometry, which allows a limited amount of deformation, provides a framework in which geometries which would otherwise depend on choices become unique. This framework is useful both in the large, e.g., for the geometric study of universal covers of compact manifolds, where it connects geometry/topology with geometric group theory, and in the small, where it is the natural framework for the study of the local geometry of algebraic sets. In both situations three-dimensional manifolds play an important role. Moreover, through invariants of manifolds one has interactions of low dimensional topology with algebra, number theory, and theoretical physics. Using these approaches the project addresses long-term open theoretical questions that span different areas of mathematics. Links between disparate areas provide much of the power of mathematics, and strengthening these links increases the power.
