Morse Theory and Floer Homology

Audin, Michèle; Damian, Mihai

doi:10.1007/978-1-4471-5496-9

Cited by 88 publications

(146 citation statements)

References 0 publications

Supporting

Mentioning

145

Contrasting

Order By: Relevance

“…What we shall call the "Novikov decomposition" is a generalization of the Morse decomposition [59][60][61], which was introduced in [19,20] (see also [18,57]) and used extensively in [21]- [29]; the name is motivated by analogy with "Morse decomposition", due to the role which this decomposition plays in the modern study of the topology of closed oneforms [58]. Define C max to be the union of all compact leaves and C min to be the union of all non-compactifiable leaves of F ω ; it is clear that these two subsets of M are disjoint.…”

Section: The Novikov Decomposition Of Mmentioning

confidence: 99%

See 1 more Smart Citation

Singular foliations for M-theory compactification

Babalic

Lazaroiu

2015

J. High Energ. Phys.

View full text Add to dashboard Cite

Abstract:We use the theory of singular foliations to study N = 1 compactifications of eleven-dimensional supergravity on eight-manifolds M down to AdS 3 spaces, allowing for the possibility that the internal part ξ of the supersymmetry generator is chiral on some locus W which does not coincide with M . We show that the complement M \ W must be a dense open subset of M and that M admits a singular foliationF endowed with a longitudinal G 2 structure and defined by a closed one-form ω, whose geometry is determined by the supersymmetry conditions. The singular leaves are those leaves which meet W. When ω is a Morse form, the chiral locus is a finite set of points, consisting of isolated zero-dimensional leaves and of conical singularities of seven-dimensional leaves. In that case, we describe the topology ofF using results from Novikov theory. We also show how this description fits in with previous formulas which were extracted by exploiting the Spin(7) ± structures which exist on the complement of W.

show abstract

Section: The Novikov Decomposition Of Mmentioning

confidence: 99%

“…The singular points belonging to S 1 and S N are centers, while the remaining critical points are saddle points. The geometry of such foliations is a classical subject in Morse theory [59][60][61]. In this case, the form ω is generic iff h is generic in the sense of Morse theory, i.e.…”

Section: E6 Exact Formsmentioning

confidence: 99%

Singular foliations for M-theory compactification

Babalic

Lazaroiu

2015

J. High Energ. Phys.

View full text Add to dashboard Cite

show abstract

“…A gap in the topological proof lies with the degenerate stagnation points of Type IV baroclinic flows, as Theorem 2 only deals with Morse function. Meanwhile a basic result of Morse theory says that almost all functions are Morse functions, and every function can be approximated uniformly by a Morse function (Audin and Damian 2014). This result is corroborated by the PIF solutions, as explained below.…”

Section: Discussionmentioning

confidence: 61%

Vertical alignment of stagnation points in pseudo-plane ideal flows

Sun

2017

AIP Advances

View full text Add to dashboard Cite

Recent studies of pseudo-plane ideal flow (PIF) reveal a ubiquitous presence of vortex alignment in both homogeneous and stratified fluids, and in both inertial and rotating reference frames as well. The exact solutions of a steady-state PIF model suggest that stagnation points tend to be vertically aligned and the concentric structure represents a fixed-point phenomenon of the Euler equations. Exception occurs in the rotating frame when a flow holds inertial period and skew center becomes possible. Properties of stagnation points based on Morse theory are obtained, leading to a topological explanation of vertical alignment via pressure Hessian. The study thus uncovers a new aspect of vortex behavior in ideal fluid that requires vortex center to align with the direction of gravity when vortex evolution reaches a laminar end state characterized by steady pseudo-plane velocities. Though the phenomenon arises from the constraint of the Euler equations, under specific conditions the topological theory is applicable to viscous fluid and explains the curvilinear tilting of von Kármán swirling vortex.

show abstract

“…The theory of Lusternik and Schnirelman will be presented with a geometric language, introducing the minimal amount of auxiliary technical tools from relative cohomology; a largely exhaustive presentation of the subject from the homological point of view can be found in the beautiful textbook [11]. There exists a corresponding analytical-topological presentation, known as 'category theory'.…”

Section: Notes On Lusternik-schnirelman and Morse Theoriesmentioning

confidence: 99%

“…Finally, we recognize that the (degenerate) critical point in A D B D C D D is a 'monkey saddle', like g.x; y/ D x 3 3xy 2 in R 2 . 11 That is, with non degenerate Hessian f 00ˇf 0 D0 .…”

Section: A Dramatic Doubtmentioning

confidence: 99%

Elementary Symplectic Topology and Mechanics

Cardin

2015

Lecture Notes of the Unione Matematica Italiana

View full text Add to dashboard Cite

Morse Theory and Floer Homology

Cited by 88 publications

References 0 publications

Singular foliations for M-theory compactification

Singular foliations for M-theory compactification

Vertical alignment of stagnation points in pseudo-plane ideal flows

Elementary Symplectic Topology and Mechanics

Contact Info

Product

Resources

About