The equivariant K-theory of isotropy actions

We compute the equivariant K-theory with integer coefficients of an equivariantly formal isotropy action, subject to natural hypotheses which cover the three major classes of known examples. The proof proceeds by constructing a map of spectral sequences from Hodgkin's K\"unneth spectral sequence in equivariant K-theory to that in Borel cohomology. A new characterization of equivariant formality appears as a consequence of this construction, and we are now able to show that weak equivariant formality in the sense of Harada--Landweber is equivalent with integer coefficients to surjectivity of the forgetful map under a standard hypothesis. The main structure theorem is formally similar to that for Borel equivariant cohomology, which appears in the author's dissertation/dormant book project and whose proof is finally made accessible in an appendix. The most generally applicable corollary of the main theorem for rational coefficients depends on a strengthening of the characterization of equivariant formality due to Shiga and Takahashi, which appears as a second appendix.Comment: 22 pages. Comments extremely welcome

The Borel equivariant cohomology of real Grassmannians

Recent work of Chen He has determined through GKM methods the Borel equivariant cohomology with rational coefficients of the isotropy action on a real Grassmannian and an real oriented Grassmannian through GKM methods. In this expository note, we propound a less involved approach, due essentially to Vitali Kapovitch, to computing equivariant cohomology rings $H^*_K(G/H)$ for $G,K,H$ connected Lie groups, and apply it to recover the equivariant cohomology of the Grassmannians. The bulk is setup and commentary; once one believes in the model, the proof itself is under a page.Comment: 10-page expository note. Comments welcom

Equivariant formality of isotropic torus actions

Considering the potential equivariant formality of the left action of a connected Lie group $K$ on the homogeneous space $G/K$, we arrive through a sequence of reductions at the case $G$ is compact and simply-connected and $K$ is a torus. We then classify all pairs $(G,S)$ such that $G$ is compact connected Lie and the embedded circular subgroup $S$ acts equivariantly formally on $G/S$. In the process we provide what seems to be the first published proof of the structure (known to Leray and Koszul) of the cohomology rings $H^*(G/S;\mathbb Q)$.Comment: Completely revised. Many proofs simplified, including reduction to toral isotropy and classification of reflected circles. An error in the reduction to the semisimple case is corrected. New: a reduction to the compact case; partial reductions if the groups are disconnected or compact but not Lie. Citations to literature improved. To be published in the Journal of Homotopy and Related Structure

Special Education in Catholic Schools Viewed from a Liberatory Hermeneutic

This study explores anew the issue of providing special education in Catholic schools by viewing the ethical implications from a liberatory hermeneutic. By utilizing an interdisciplinary perspective, the research draws upon liberation theology, liberation psychology, liberation pedagogy, and liberation ethics to support the moral mandate for providing education for all God’s children, including those persons with disabilities. The study challenges Catholic educational leaders to reimagine their positions on how schools might promote a more inclusive, liberatory approach to serving the special needs of children with disabilities. Finally, this research provides a Catholic, liberatory, ethical framework for inclusive Catholic education to assist school leaders in the development of appropriate pedagogy and programming to address the issue of inclusion of students with disabilities

Effects of movements in equities prices on M2 demand

Large swings in stock prices are sometimes associated with a redirection of household savings flows. Such changes can lead to transitory increases in M2 as investors temporarily “park” funds in depository assets while they determine the funds’ ultimate destination. The authors find that, although stock price changes are statistically significant as an explanation for M2 growth, they do not account for much of M2’s recent strength.Stock - Prices ; Demand for money

Products on Tor

In 1974 work establishing the collapse of certain Eilenberg-Moore spectral sequences, Munkholm constructs, in passing, a bilinear multiplication operation on Tor of a triple of $A_\infty$-algebras. In 2020, the present author, pursuing a multiplicative collapse result extending Munkholm's, studied a variant of this product, without actually showing it agrees with Munkholm's. In 2019, Franz had defined a weak product on the two-sided bar construction of a triple of $A_\infty$-algebras under similar hypotheses, with which this author proved a related collapse result, but without investigating the properties of the induced product on Tor. The present work demonstrates that the two products on Tor agree and are induced by the product of Franz.Comment: 19 pages, comments welcom

Assessment of Ozone Initiated Chemistry in Portable Classrooms

A model was developed to predict concentrations of homogeneous and heterogeneous ozone initiated reaction by-products in portable classrooms. The resulting estimates offer reference values for highly occupied indoor environments. Indoor concentrations of formaldehyde (HCHO), 4-oxopentanal (4-OPA), 6-methyl-5-hepten-2-one (6-MHO), and secondary organic aerosols (SOA) were predicted based on various parameters of specific portable classrooms in central Texas. The results were not significant to raise concern for formaldehyde, but they did yield relatively high average and maximum concentrations of 4-OPA, 6-MHO, and SOA. While the health implications of SOA are more well known, less has been done to determine the toxicological effects of 4-OPA and 6-MHO. The results of this report indicate that more research should be conducted to further understand the effects of these compounds on the indoor environment, as well as human health.Civil, Architectural, and Environmental Engineerin

Fixed points and semifree bordism

We apply fixed-point techniques to compute the coefficient ring of semifree geometric circle-equivariant complex cobordism with isolated fixed points, recovering a 2004 result of Sinha through 19th-century methods. This should be viewed as an initial proof-of-concept for a larger program employing the Atiyah-Bott/Berline-Vergne localization theorem and Chern numbers in Borel cohomology to compute equivariant complex cobordism.Comment: 5 pages; comments gratefully accepte

The K-theory of the conjugation action

In 1999, Brylinski and Zhang computed the complex equivariant K-theory of the conjugation self-action of a compact, connected Lie group with torsion-free fundamental group. In this note we show it is possible to do so in under a page.Comment: Same as the published version up to formatting. 2pp. plus reference