Morse Theory without Non-Degeneracy

Kirwan, Frances; Penington, Geoffrey

doi:10.1093/qmath/haaa064

“…Index pairs for isolated critical points of smooth functions have been thoroughly explored by Gromoll and Meyer [13]; the work of Chang and Ghoussoub [3] provides a convenient dictionary between Conley's index pairs and a generalised version of these Gromoll-Meyer pairs. Also close in spirit and generality to our (N , N − ) are the systems of Morse neighbourhoods around arbitrary isolated critical sets in the recent work of Kirwan and Penington [16].…”

Section: Introductionsupporting

confidence: 53%

Topological Inference of the Conley Index

Yim¹,

Nanda²

2022

Preprint

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The Conley index of an isolated invariant set is a fundamental object in the study of dynamical systems. Here we consider smooth functions on closed submanifolds of Euclidean space and describe a framework for inferring the Conley index of any compact, connected isolated critical set of such a function with high confidence from a sufficiently large finite point sample. The main construction of this paper is a specific index pair which is local to the critical set in question. We establish that these index pairs have positive reach and hence admit a sampling theory for robust homology inference. This allows us to estimate the Conley index, and as a direct consequence, we are also able to estimate the Morse index of any critical point of a Morse function using finitely many local evaluations.

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“…Isomorphisms of quiver representations can be used to characterise, for example, the Jordan normal form of matrices and the Kronecker normal form of matrix pencils. They also play an important role in various other fields, including the study of associative algebras [3], Gromov-Witten invariants [22], representations of Kac-Moody algebras [39], moduli stacks [48], Morse theory [35], persistent homology [41], and perverse sheaves [19], among others.…”

Section: Introductionmentioning

confidence: 99%

Principal Components Along Quiver Representations

Seigal

¹

,

Harrington

²

,

Nanda

³

2022

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Quiver representations arise naturally in many areas across mathematics. Here we describe an algorithm for calculating the vector space of sections, or compatible assignments of vectors to vertices, of any finite-dimensional representation of a finite quiver. Consequently, we are able to define and compute principal components with respect to quiver representations. These principal components are solutions to constrained optimisation problems defined over the space of sections and are eigenvectors of an associated matrix pencil.

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Topological Inference of the Conley Index

Yim,

Nanda

2023

J Dyn Diff Equat

0

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The Conley index of an isolated invariant set is a fundamental object in the study of dynamical systems. Here we consider smooth functions on closed submanifolds of Euclidean space and describe a framework for inferring the Conley index of any compact, connected isolated critical set of such a function with high confidence from a sufficiently large finite point sample. The main construction of this paper is a specific index pair which is local to the critical set in question. We establish that these index pairs have positive reach and hence admit a sampling theory for robust homology inference. This allows us to estimate the Conley index, and as a direct consequence, we are also able to estimate the Morse index of any critical point of a Morse function using finitely many local evaluations.

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Morse Theory without Non-Degeneracy

Abstract: We describe an extension of Morse theory to smooth functions on compact Riemannian manifolds, without any non-degeneracy assumptions except that the critical locus must have only finitely many connected components.

Cited by 4 publications

References 70 publications

Topological Inference of the Conley Index

Topological Inference of the Conley Index

Principal Components Along Quiver Representations

Topological Inference of the Conley Index

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