Related papers: Morse field theory
In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by constructing a moduli space of graph…
By studying spaces of flow graphs in a closed oriented manifold, we construct operations on its cohomology, parametrized by the homology of the moduli spaces of compact Riemann surfaces with boundary marked points. We show that the…
This thesis is concerned with the application of operadic methods, particularly modular operads, to questions arising in the study of moduli spaces of surfaces as well as applications to the study of homotopy algebras and new constructions…
We construct topological invariants, called abstract weak orbit spaces, of flows and homeomorphisms on topological spaces, to describe both gradient dynamics and recurrent dynamics. In particular, the abstract weak orbit spaces of flows on…
In various situations in Floer theory, one extracts homological invariants from "Morse-Bott" data in which the "critical set" is a union of manifolds, and the moduli spaces of "flow lines" have evaluation maps taking values in the critical…
For Morse-Smale pairs on a smooth, closed manifold the Morse-Smale-Witten chain complex can be defined. The associated Morse homology is isomorphic to the singular homology of the manifold and yields the classical Morse relations for Morse…
The goal of this article is to motivate and describe how Gromov-Witten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from Gromov-Witten theory have led to both conjectures and…
We provide a new approach to studying the moduli space of curves via Morse theory and hyperbolic geometry, by introducing a family of Morse functions on the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable curves of genus $g$ with $n$…
We consider the path space of a curved manifold on which a point particle is introduced in a conservative physical system with constant total energy to formulate its action functional and geodesic equation together with breaks on the path.…
We construct a deformed Morse complex computing the equivariant cohomology of a manifold M endowed with a smooth S^1-action. The deformation of the coboundary operator is given by counting gradient flow lines of a Morse function f that are…
Stable fold maps are fundamental tools in a generalization of the theory of Morse functions on smooth manifolds and its application to studies of topological properties of smooth manifolds. Round fold maps were introduced as stable fold…
Given a smooth closed manifold M, the Morse-Witten complex associated to a Morse function f and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines…
We study moduli spaces of twisted maps to a smooth pair in arbitrary genus, and give geometric explanations for previously known comparisons between orbifold and logarithmic Gromov--Witten invariants. Namely, we study the space of twisted…
We consider different notions of equivalence for Morse functions on the sphere in the context of persistent homology, and introduce new invariants to study these equivalence classes. These new invariants are as simple, but more discerning…
{\it Fold maps} are fundamental tools in generalizing the theory of Morse functions and its application to studies of geometric properties of manifolds. One of the fundamental and important problems in the theory of fold maps is to…
This note is about invariants of moduli spaces of curves. It includes their intersection theory and cohomology. Our main focus in on the distinguished piece containing the so called tautological classes. These are the most natural classes…
In this expository article we give a categorical definition of the integral cohomology ring of a stack. We show that for quotient stacks the categorical cohomology may be identified with equivariant cohomology. Via this identification we…
Fold maps are smooth maps at each singular point of which it is represented as the product map of a Morse function and the identity map. Round fold maps are, in short, such maps the sets of all singular points of which are embedded…
Stable fold maps are fundamental tools in a generalization of the theory of Morse functions on smooth manifolds and its application to studies of geometric properties of smooth manifolds. Round fold maps were introduced as stable fold maps…
Stable fold maps are fundamental tools in studying a generalized theory of the theory of Morse functions on smooth manifolds and its application to geometry of the manifolds. It is important to construct explicit fold maps systematically to…