Obstructions to positive curvature and symmetry

We discuss new obstructions to positive sectional curvature and symmetry. The main result asserts that the index of the Dirac operator twisted with the tangent bundle vanishes on a 2-connected manifold of dimension ≠8 if the manifold admits a metric of positive sectional curvature and isometric effective S¹-action. The proof relies on the rigidity theorem for elliptic genera and properties of totally geodesic submanifolds.     

Unconditional integrability for dual actions

The dual action of a locally compact abelian group, in the context of C^*-algebraic bundles, is shown to satisfy an integrability property, similar to Rieffel's proper actions. The tools developed include a generalization of Bochner's integral as well as a Fourier inversion formula for operator valued maps.Partially supported by CNPq, Brazil.     

On the Geometry of the Orbits of Hermann Actions

We investigate the submanifold geometry of the orbits of Hermann actions on Riemannian symmetric spaces. After proving that the curvature and shape operators of these orbits commute, we calculate the eigenvalues of the shape operators in terms of the restricted roots. As applications, we get a formula for the volumes of the orbits and a new proof of a Weyl-type integration formula for Hermann actions.      

Polar actions on Hermitian and quaternion-Kähler symmetric spaces

We analyze polar actions on Hermitian and quaternion-Kähler symmetric spaces of compact type. For complex integrable polar actions on Hermitian symmetric spaces of compact type we prove a reduction theorem and several corollaries concerning the geometry of these actions. The results are independent of the classification of polar actions on Hermitian symmetric spaces. In the second part we prove that polar actions on Wolf spaces are quaternion-coisotropic and that isometric actions on these spaces admit an orbit of special type, analogous to the existence of a complex orbit for an isometric action on a compact homogeneous simply connected Kähler manifold.     

On the Orbit Sizes of Permutation Groups on the Power Set

We study the action of finite permutation groups of odd order on the power set of a set on which they act naturally, and establish a theorem that guarantees the existence of a lot of distinct orbit sizes in this action. 1991 Mathematics Subject Classification: 20B05     

Low Cohomogeneity Representations and Orbit Maximal Actions

An orthogonal representation of a compact Lie group is called polar if thereexists a linear subspace which meets all orbits orthogonally.It has been shown by Conlon that one can associate a Coxeter groupto such a representation.From this, an upper bound for the cohomogeneity of an irreduciblepolar representation can be derived.Another property of irreducible polar representations isthat the action restricted to the unit spherehas maximal orbits in the sense that any action having largerorbits is transitive.We give a classification of orbit maximal actions on spheresand use it to show that irreducible polar representations arecharacterized by these two properties.     

Equivariant Ehrhart theory

Motivated by representation theory and geometry, we introduce and develop an equivariant generalization of Ehrhart theory, the study of lattice points in dilations of lattice polytopes. We prove representation-theoretic analogues of numerous classical results, and give applications to the Ehrhart theory of rational polytopes and centrally symmetric polytopes. We also recover a character formula of Procesi, Dolgachev, Lunts and Stembridge for the action of a Weyl group on the cohomology of a toric variety associated to a root system.     

奇数维spin~c流形和S~1作用(英文)

本文研究了奇数维spin~c流形上的等边指标定理.用这些指标定理和一些复分析的讨论,得到了两个奇数维spin~c流形上的Atiyah-Hirzebruch类型消灭定理,结果推广了刘-王的消灭定理.     

The conformal anomaly and the neutral currents sector of the Standard Model

We elaborate on the structure of the graviton–gauge–gauge vertex in the electroweak theory, obtained by the insertion of the complete energy–momentum tensor (T  ) on 2-point functions of neutral gauge currents (VV^′$V{V}^{\prime }$). The vertex defines the leading contribution to the effective action which accounts for the conformal anomaly and related interaction between the Standard Model and gravity. The energy–momentum tensor is derived from the curved spacetime Lagrangian in the linearized gravitational limit, and with the inclusion of the term of improvement of a conformally coupled Higgs sector. As in the previous cases of QED and QCD, we find that the conformal anomaly induces an effective massless scalar interaction between gravity and the neutral currents in each gauge invariant component of the vertex. This is described by the exchange of an anomaly pole. We show that for a spontaneously broken theory the anomaly can be entirely attributed to the poles only for a conformally coupled Higgs scalar. In the exchange of a graviton, the trace part of the corresponding interaction can be interpreted as due to an effective dilaton, using a local version of the effective action. We discuss the implications of the anomalous Ward identity for the TVV^′$TV{V}^{\prime }$ correlator for the structure of the gauge/gauge/effective dilaton vertex in the effective action. The analogy between these effective interactions and those related to the radion in theories with large extra dimensions is pointed out.     

An interplay model for authorities’ actions and rumor spreading in emergency event

Rumor spreading influences how rational individuals assess risks and evaluate needs, especially, it affects authorities to make decisions in an emergency-affected environments. Conversely, authorities’ response to emergency will induct public opinions as well. In this paper, we present a simple model to describe the interplay between rumor spreading and authorities’ actions in emergency situation based on utility theory. By drawing from differential equations we found that it is possible to minimize negative social utility of rumor spreading in the control of situation. At the same time, authorities’ proactive actions can improve rumor management in emergency situation and yield positive social utility. Finally, we outline strategies for authorities that can contribute to rumor management in an emergency event.