Index theory for noncompact group actions with applications to Borsuk-Ulam theorems

1989 ◽  
pp. 52-68 ◽  
Author(s):  
E. Fadell ◽  
S. Y. Husseini
Keyword(s):  
Group Actions ◽  
Index Theory ◽  
Noncompact Group

Positivity of character subbundles and minimum principle for noncompact group actions

2016 ◽  
Vol 286 (1-2) ◽  
pp. 431-442 ◽  
Author(s):  
Fusheng Deng ◽  
Huiping Zhang ◽  
Xiangyu Zhou
Keyword(s):  
Minimum Principle ◽  
Group Actions ◽  
Noncompact Group

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.

Generating pairs and group actions

2014 ◽  
Vol 218 (5) ◽  
pp. 777-783
Author(s):  
Darryl McCullough
Keyword(s):  
Group Actions

scholarly journals Positive solutions of a discrete second-order boundary value problems with fully nonlinear term

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Liyun Jin ◽  
Hua Luo
Keyword(s):  
Positive Solutions ◽  
Fixed Point Index ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Index Theory ◽  
Second Order ◽  
Fully Nonlinear ◽  
Fixed Point Index Theory ◽  
Weaker Conditions ◽  
Lower Limits

Abstract In this paper, we mainly consider a kind of discrete second-order boundary value problem with fully nonlinear term. By using the fixed-point index theory, we obtain some existence results of positive solutions of this kind of problems. Instead of the upper and lower limits condition on f, we may only impose some weaker conditions on f.

scholarly journals The close relation between border and Pommaret marked bases

2021 ◽  
Author(s):  
Cristina Bertone ◽  
Francesca Cioffi
Keyword(s):  
Finite Order ◽  
Polynomial Ring ◽  
Group Actions ◽  
Close Relation ◽  
Open Covering ◽  
Order Ideal ◽  
Lower Dimension ◽  
Hilbert Schemes ◽  
Affine Spaces ◽  
The Ideal

AbstractGiven a finite order ideal $${\mathcal {O}}$$ O in the polynomial ring $$K[x_1,\ldots , x_n]$$ K [ x 1 , … , x n ] over a field K, let $$\partial {\mathcal {O}}$$ ∂ O be the border of $${\mathcal {O}}$$ O and $${\mathcal {P}}_{\mathcal {O}}$$ P O the Pommaret basis of the ideal generated by the terms outside $${\mathcal {O}}$$ O . In the framework of reduction structures introduced by Ceria, Mora, Roggero in 2019, we investigate relations among $$\partial {\mathcal {O}}$$ ∂ O -marked sets (resp. bases) and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked sets (resp. bases). We prove that a $$\partial {\mathcal {O}}$$ ∂ O -marked set B is a marked basis if and only if the $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked set P contained in B is a marked basis and generates the same ideal as B. Using a functorial description of these marked bases, as a byproduct we obtain that the affine schemes respectively parameterizing $$\partial {\mathcal {O}}$$ ∂ O -marked bases and $${\mathcal {P}}_{\mathcal {O}}$$ P O -marked bases are isomorphic. We are able to describe this isomorphism as a projection that can be explicitly constructed without the use of Gröbner elimination techniques. In particular, we obtain a straightforward embedding of border schemes in affine spaces of lower dimension. Furthermore, we observe that Pommaret marked schemes give an open covering of Hilbert schemes parameterizing 0-dimensional schemes without any group actions. Several examples are given throughout the paper.

scholarly journals Schottky groups acting on homogeneous rational manifolds

2019 ◽  
Vol 2019 (753) ◽  
pp. 23-56 ◽  
Author(s):  
Christian Miebach ◽  
Karl Oeljeklaus
Keyword(s):  
Deformation Theory ◽  
Group Actions ◽  
Picard Group ◽  
Schottky Group ◽  
Fundamental Groups ◽  
Complex Manifolds ◽  
Schottky Groups ◽  
Algebraic Dimension ◽  
Geometric Invariants ◽  
Compact Complex Manifolds

AbstractWe systematically study Schottky group actions on homogeneous rational manifolds and find two new families besides those given by Nori’s well-known construction. This yields new examples of non-Kähler compact complex manifolds having free fundamental groups. We then investigate their analytic and geometric invariants such as the Kodaira and algebraic dimension, the Picard group and the deformation theory, thus extending results due to Lárusson and to Seade and Verjovsky. As a byproduct, we see that the Schottky construction allows to recover examples of equivariant compactifications of {{\rm{SL}}(2,\mathbb{C})/\Gamma} for Γ a discrete free loxodromic subgroup of {{\rm{SL}}(2,\mathbb{C})}, previously obtained by A. Guillot.

scholarly journals Expected Genetic Contributions and Their Impact on Gene Flow and Genetic Gain

Genetics ◽  
1999 ◽  
Vol 153 (2) ◽  
pp. 1009-1020 ◽  
Author(s):  
J A Woolliams ◽  
P Bijma ◽  
B Villanueva
Keyword(s):  
Gene Flow ◽  
Genetic Gain ◽  
Overlapping Generations ◽  
Predictive Accuracy ◽  
Index Theory ◽  
Previous Theory ◽  
Selection Indices ◽  
Breeding Groups ◽  
Genetic Contributions

Abstract Long-term genetic contributions (ri) measure lasting gene flow from an individual i. By accounting for linkage disequilibrium generated by selection both within and between breeding groups (categories), assuming the infinitesimal model, a general formula was derived for the expected contribution of ancestor i in category q (μi(q)), given its selective advantages (si(q)). Results were applied to overlapping generations and to a variety of modes of inheritance and selection indices. Genetic gain was related to the covariance between ri and the Mendelian sampling deviation (ai), thereby linking gain to pedigree development. When si(q) includes ai, gain was related to E[μi(q)ai], decomposing it into components attributable to within and between families, within each category, for each element of si(q). The formula for μi(q) was consistent with previous index theory for predicting gain in discrete generations. For overlapping generations, accurate predictions of gene flow were obtained among and within categories in contrast to previous theory that gave qualitative errors among categories and no predictions within. The generation interval was defined as the period for which μi(q), summed over all ancestors born in that period, equaled 1. Predictive accuracy was supported by simulation results for gain and contributions with sib-indices, BLUP selection, and selection with imprinted variation.

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