Topological Index and Analytical Index: Reformulation of Index Theory

2019 ◽  
pp. 329-342
Author(s):  
Neculai S. Teleman
Keyword(s):  
Topological Index ◽  
Index Theory ◽  
Analytical Index

Bifurcations, symmetries and the notion of fixed subspace

2021 ◽  
pp. 2130006
Author(s):  
Giampaolo Cicogna
Keyword(s):  
Topological Index ◽  
Ad Hoc ◽  
Index Theory ◽  
Typical Case ◽  
One Dimensional ◽  
Dimension Formula ◽  
Symmetry Properties ◽  
Equivariant Bifurcation ◽  
Stationary Bifurcation ◽  
Bifurcation Problems

In the context of stationary bifurcation problems admitting a symmetry, this paper is focused on the key notion of Fixed Subspace (FS), and provides a review of some applications aimed at detecting bifurcating solutions in various situations. We start recalling, in its commonly used simplified version, the old Equivariant Bifurcation Lemma (EBL), where the FS is one-dimensional; then we provide a first generalization in a typical case of non-semisimple critical eigenvalues, where the presence of the symmetry produces a non-trivial situation. Next, we consider the case of FSs of dimension [Formula: see text] in very different contexts. First, relying on the topological index theory and in particular on the Krasnosel’skii theorem, we provide a largely applicable statement of an extension of the EBL. Second, we propose a completely different and new application which combines symmetry properties with the notion of stability of bifurcating solutions. We also provide some simple examples, constructed ad hoc to illustrate the various situations.

Recent results on the majorization theory of graph spectrum and topological index theory

2015 ◽  
Vol 30 ◽  
Author(s):  
Muhuo Liu ◽  
Bolian Liu ◽  
Kinkar Das
Keyword(s):  
Spectral Radius ◽  
Topological Index ◽  
Index Theory ◽  
Degree Sequences ◽  
Extremal Graphs ◽  
Graph Spectrum ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian ◽  
Majorization Theorem

Suppose π = (d_1,d_2,...,d_n) and π′ = (d′_1,d′_2,...,d′_n) are two positive non- increasing degree sequences, write π ⊳ π′ if and only if π \neq π′, \sum_{i=1}^n d_i = \sum_{i=1}^n d′_i, and \sum_{i=1}^j d_i ≤ \sum_{i=1}^j d′_i for all j = 1, 2, . . . , n. Let ρ(G) and μ(G) be the spectral radius and signless Laplacian spectral radius of G, respectively. Also let G and G′ be the extremal graphs with the maximal (signless Laplacian) spectral radii in the class of connected graphs with π and π′ as their degree sequences, respectively. If π ⊳ π′ can deduce that ρ(G) < ρ(G′) (respectively, μ(G) < μ(G′)), then it is said that the spectral radii (respectively, signless Laplacian spectral radii) of G and G′ satisfy the majorization theorem. This paper presents a survey to the recent results on the theory and application of the majorization theorem in graph spectrum and topological index theory.

scholarly journals A topological index theorem for manifolds with corners

2011 ◽  
Vol 148 (2) ◽  
pp. 640-668 ◽  
Author(s):  
Bertrand Monthubert ◽  
Victor Nistor
Keyword(s):  
Compact Manifold ◽  
Principal Symbol ◽  
Topological Index ◽  
Index Theorem ◽  
Subsequent Paper ◽  
Fredholm Index ◽  
Analytic Index ◽  
Manifolds With Corners ◽  
Analytical Index

AbstractWe define an analytic index and prove a topological index theorem for a non-compact manifold M0 with poly-cylindrical ends. Our topological index theorem depends only on the principal symbol, and establishes the equality of the topological and analytical index in the group K0(C*(M)), where C*(M) is a canonical C*-algebra associated to the canonical compactification M of M0. Our topological index is thus, in general, not an integer, unlike the usual Fredholm index appearing in the Atiyah–Singer theorem, which is an integer. This will lead, as an application in a subsequent paper, to the determination of the K-theory groups K0(C*(M)) of the groupoid C*-algebra of the manifolds with corners M. We also prove that an elliptic operator P on M0 has an invertible perturbation P+R by a lower-order operator if and only if its analytic index vanishes.

scholarly journals Topological Index Theory for surfaces in 3–manifolds

2010 ◽  
Vol 14 (1) ◽  
pp. 585-609 ◽  
Author(s):  
David Bachman
Keyword(s):  
Topological Index ◽  
Index Theory

Higher Index Theory

2020 ◽  
Author(s):  
Rufus Willett ◽  
Guoliang Yu
Keyword(s):  
Index Theory

Hypoelliptic Laplacian and Orbital Integrals (AM-177)

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Fixed Point ◽  
Symmetric Space ◽  
Heat Kernel ◽  
Geodesic Flow ◽  
Casimir Operator ◽  
Index Theory ◽  
Weighted Sum ◽  
Orbital Integrals ◽  
Hypoelliptic Heat Kernel ◽  
Wave Kernel

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.

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