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We extend the theory of Thom spectra and the associated obstruction theory for orientations in order to support the construction of the $E_{\infty }$ string orientation of $tmf$, the spectrum of topological modular forms. Specifically, we show that, for an $E_{\infty }$ ring spectrum $A$, the classical construction of $gl_{1} {A}$, the spectrum of units, is the right adjoint of the functor\[\Sigma ^{\infty }_{+ } \Omega ^{\infty } \colon \mathrm {ho} (\mbox {{connective spectra}}) \longrightarrow \mathrm {ho} ({\text {{$E_{\infty } $ ring spectra}}}).\]To a map of spectra\[f\colon b \longrightarrow bgl_{1} {A},\]we associate an $E_{\infty }$ $A$-algebra Thom spectrum $Mf$, which admits an $E_{\infty }$ $A$-algebra map to $R$ if and only if the composition\[b \longrightarrow bgl_{1} {A} \longrightarrow bgl_{1} {R}\]is null; the classical case developed by May, Quinn, Ray, and Tornehave arises when $A$ is the sphere spectrum. We develop the analogous theory for $A_{\infty }$ ring spectra: if $A$ is an $A_{\infty }$ ring spectrum, then to a map of spaces\[f\colon B \longrightarrow B{ GL}_{1} {A},\]we associate an $A$-module Thom spectrum $Mf,$ which admits an $R$-orientation if and only if\[B \longrightarrow B{ GL}_{1} {A} \longrightarrow B{ GL}_{1} {R}\]is null. Our work is based on a new model of the Thom spectrum as a derived smash product. [ABSTRACT FROM AUTHOR]

We develop a generalization of the theory of Thom spectra using the language of ∞-categories. This treatment exposes the conceptual underpinnings of the Thom spectrum functor: we use a new model of parameterized spectra, and our definition is motivated by the geometric definition of Thom spectra of May–Sigurdsson. For an A∞-ring spectrum R, we associate a Thom spectrum to a map of ∞-categories from the ∞-groupoid of a space X to the ∞-category of free rank one R-modules, which we show is a model for BGL1R; we show that BGL1R classifies homotopy sheaves of rank one R-modules, which we call R-line bundles. We use our R-module Thom spectrum to define the twisted R-homology and cohomology of R-line bundles over a space classified by a map X→BGL1R, and we recover the generalized theory of orientations in this context. In order to compare this approach to the classical theory, we characterize the Thom spectrum functor axiomatically, from the perspective of Morita theory. [ABSTRACT FROM AUTHOR]

In an earlier paper, the authors constructed a natural map, called the sigma orientation, from the Thom spectrum MU(6) to any elliptic spectrum. MU(6) is an H∞ ring spectrum, and in this paper we show that if (E, C, t) is the elliptic spectrum associated to the universal deformation of a supersingular elliptic curve over a perfect field of characteristic p > 0, then the sigma orientation is a map of H∞ ring spectra. [ABSTRACT FROM AUTHOR]

A code for carrying out stochastic simulations of the MPS econometric model, a simultaneous, nonlinear, dynamic system of equations involving approximately 600 variables, was adapted to run on the CYBER 205 vector processor. The effi ciency gained relative to performance on an IBM 3081-GX was significant: the CPU time re quired for a given simulation was reduced by a factor of 10 to 12. However, this is considerably smaller than the theoretical gain expected from the manufacturer's ratings. The basic causes of this smaller gain are the fairly small real memory of the CYBER 205 and the inefficiency of the CYBER 205 vector processor in performing ele mentary function look-up. The primitive nature of the operating system and some aspects of the FORTRAN compiler on the CYBER 205 required more reprogramming of our large code than such a conversion should demand. However, potential gains from such a conversion will be great once the deficiencies described above are eliminated. [ABSTRACT FROM PUBLISHER]