Carla Farsi

Generalized orbifold Euler characteristics for general orbifolds and wreath products

We introduce the A ‐Euler‐Satake characteristics of a general orbifold Q presented by an orbifold groupoid G , extending to orbifolds that are not global quotients the generalized orbifold Euler characteristics of Bryan‐Fulman and Tamanoi. Each of these Euler characteristics is defined as the Euler‐Satake characteristic of the space of A ‐sectors of the orbifold where A is a finitely generated discrete group. We study the behavior of these Euler characteristics under product operations applied to the group A as well as the orbifold and establish their relationships to existing Euler characteristics for orbifolds. As applications, we generalize formulas of Tamanoi, Wang and Zhou for the Euler characteristics and Hodge numbers of wreath symmetric products of global quotient orbifolds to the case of quotients by compact, connected Lie groups acting locally freely, in particular including all closed, effective orbifolds. 22A22, 55S15; 58E40, 55N91

GENERALIZED TWISTED SECTORS OF ORBIFOLDS

For a finitely generated discrete group Γ, the Γ-sectors of an orbifold Q are a disjoint union of orbifolds corresponding to homomorphisms from Γ into a groupoid presenting Q. Here, we show that the inertia orbifold and k-multisectors are special cases of the Γ-sectors, and that the Γ-sectors are orbifold covers of Leidas fixed-point sectors. In the case of a global quotient, we show that the Γ-sectors correspond to orbifolds considered by other authors for global quotient orbifolds, as well as their direct generalization to the case of an orbifold given by a quotient by a Lie group. Furthermore, we develop a model for the Γ-sectors corresponding to a generalized loop space.

NONVANISHING VECTOR FIELDS ON ORBIFOLDS

We introduce a complete obstruction to the existence of nonvanishing vector fields on a closed orbifold Q. Motivated by the inertia orbifold, the space of multi-sectors, and the generalized orbifold Euler characteristics, we construct for each finitely generated group Γ an orbifold called the space of Γ-sectors of Q. The obstruction occurs as the Euler-Satake characteristics of the Γ-sectors for an appropriate choice of Γ; in the case that Q is oriented, this obstruction is expressed as a cohomology class, the Γ-Euler-Satake class. We also acquire a complete obstruction in the case that Q is compact with boundary and in the case that Q is an open suborbifold of a closed orbifold.

Wavelets and Graph C∗-Algebras

Here we give an overview on the connection between wavelet theory and representation theory for graph C∗-algebras, including the higher-rank graph C∗-algebras of A. Kumjian and D. Pask. Many authors have studied different aspects of this connection over the last 20 years, and we begin this paper with a survey of the known results. We then discuss several new ways to generalize these results and obtain wavelets associated to representations of higher-rank graphs. In Farsi et al. (J Math Anal Appl 425:241–270, 2015), we introduced the “cubical wavelets” associated to a higher-rank graph. Here, we generalize this construction to build wavelets of arbitrary shapes. We also present a different but related construction of wavelets associated to a higher-rank graph, which we anticipate will have applications to traffic analysis on networks. Finally, we generalize the spectral graph wavelets of Hammond et al. (Appl Comput Harmon Anal 30:129–150, 2011) to higher-rank graphs, giving a third family of wavelets associated to higher-rank graphs.

On Orbifold Criteria for Symplectic Toric Quotients

We introduce the notion of regular symplectomorphism and graded regular sym- plectomorphism between singular phase spaces. Our main concern is to exhibit examples of unitary torus representations whose symplectic quotients cannot be graded regularly sym- plectomorphic to the quotient of a symplectic representation of a finite group, while the corresponding GIT quotients are smooth. Additionally, we relate the question of simplicial- ness of a torus representation to Gaussian elimination.

An orbifold relative index theorem

Abstract In this paper we prove a relative index theorem for pairs of generalized Dirac operators on orbifolds which are the same at infinity. This generalizes to orbifolds a celebrated theorem of Gromov and Lawson.

Stratifications of Inertia Spaces of Compact Lie Group Actions

We study the topology of the inertia space of a smooth

Monic representations of finite higher-rank graphs

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SOFT C * -ALGEBRAS

-manifold

Equivariant spin bordism in low dimensions

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