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This appendix is dedicated to the body of results about structured ring spectra which are relevant to the String–orientation of elliptic spectra. We include four main topics: a discussion of the stable homotopy theoretic analog of character theory; the compatibility of the results of the preceding chapter on orientations in the homotopy category with power operations for Lubin–Tate spectra; the sheaf of ring spectra enriching the moduli stack of elliptic curves and the spectrum of topological modular forms; and orientations of the spectrum of topological modular forms by maps of structured rings.
Our goal in this chapter is to prove Quillen’s complex analog of Thom’s calculation of the bordism ring. We approach this using cyclic power operations in bordism homology, giving a brief tour of the general theory of power operations along the way, and ultimately employing Quillen’s comparison formula with stable Landweber–Novikov operations. In order to gain a handle on the stable operations, we investigate the formal schemes associated to classifying spaces for complex vector bundles and the algebro–geometric interpretation of the cohomology of Thom spectra. As we develop the topological results, we begin to re-prove in greater generality the specialized algebraic results from the preceding chapter as we find it necessary.
This chapter introduces the basic results of chromatic homotopy theory while adhering to the language of algebraic geometry. Our approach centers on the study of descent in the setting of stable homotopy theory; Quillen’s theorem from the preceding chapter gives an example of a ring spectrum whose unit map is of effective descent and whose algebraic properties are amenable to algebro–geometric study. Following these ideas to their conclusion leads us in turn to the study of the moduli of formal groups, and we dedicate several sections to the description of the geometry of this moduli stack. We produce reflections in the stable homotopy category of our main algebraic results, emphasizing especially the periodicity theorems of Devinatz, Hopkins, and Smith, which simultaneously give shape to the structure of the “prime ideals” of the stable homotopy category as well as organize the stable stems into identifiable families.
We recast Thom’s classical calculation of the unoriented bordism ring in the language of formal geometry, introducing foundational concepts as we go along, especially: Thom spectra and the Thom isomorphism, formal schemes, and the formal-geometric perspectives on the Steenrod algebra and on the Adams spectral sequence. This chapter is designed primarily with pedagogy in mind, as this calculation introduces all of the main structures used in the rest of the book, restricted to a particularly simple context.
This chapter approaches aspects of unstable homotopy theory from the perspective of the preceding chapter: We study the unstable analog of effective descent and thereby embark on an analysis of the algebro–geometric objects that arise from these considerations. This lends a modern perspective on a variety of classical topics: Hopf rings, the Ravenel–Wilson results on the infinite loopspaces associated to BP, Dieudonné theory, Brown–Gitler spectra, and the formal schemes associated to Eilenberg–Mac Lane spaces. We also include a novel discussion of the functorial algebraic geometry of Hopf rings, which gives a reinterpretation of the central Ravenel–Wilson relation and exerts an overall influence on our presentation of the other materials.
In this chapter, we bring the results of the preceding four chapters to bear on the problem of understanding String bordism, first using its lower–connectivity analogues and its complex analogue to paint a landscape into which these results naturally fit. Our overall strategy is again guided by the structure of the moduli of formal groups, which leads us to pursue a calculation of the Morava E–theory of complex bordism spectra before bringing us to the general topological results we seek. We conclude with a summary of unpublished results of Ando, Hopkins, and Strickland on the String–orientation of elliptic field spectra. We also include an extended discussion of elliptic curves, specialized to the context most relevant to us, for the reader’s convenience.
This text organizes a range of results in chromatic homotopy theory, running a single thread through theorems in bordism and a detailed understanding of the moduli of formal groups. It emphasizes the naturally occurring algebro-geometric models that presage the topological results, taking the reader through a pedagogical development of the field. In addition to forming the backbone of the stable homotopy category, these ideas have found application in other fields: the daughter subject 'elliptic cohomology' abuts mathematical physics, manifold geometry, topological analysis, and the representation theory of loop groups. The common language employed when discussing these subjects showcases their unity and guides the reader breezily from one domain to the next, ultimately culminating in the construction of Witten's genus for String manifolds. This text is an expansion of a set of lecture notes for a topics course delivered at Harvard University during the spring term of 2016.
Objectives: To evaluate prospective and retrospective memory abilities in Operation Enduring Freedom (OEF), Operation Iraqi Freedom (OIF), and Operation New Dawn (OND) Veterans with and without a self-reported history of blast-related mild traumatic brain injury (mTBI). Methods: Sixty-one OEF/OIF/OND Veterans, including Veterans with a self-reported history of blast-related mTBI (mTBI group; n=42) and Veterans without a self-reported history of TBI (control group; n=19) completed the Memory for Intentions Test, a measure of prospective memory (PM), and two measures of retrospective memory (RM), the California Verbal Learning Test-II and the Brief Visuospatial Memory Test-Revised. Results: Veterans in the mTBI group exhibited significantly lower PM performance than the control group, but the groups did not differ in their performance on RM measures. Further analysis revealed that Veterans in the mTBI group with current PTSD (mTBI/PTSD+) demonstrated significantly lower performance on the PM measure than Veterans in the control group. PM performance by Veterans in the mTBI group without current PTSD (mTBI/PTSD-) was intermediate between the mTBI/PTSD+ and control groups, and results for the mTBI/PTSD- group were not significantly different from either of the other two groups. Conclusions: Results suggest that PM performance may be a sensitive marker of cognitive dysfunction among OEF/OIF/OND Veterans with a history of self-reported blast-related mTBI and comorbid PTSD. Reduced PM may account, in part, for complaints of cognitive difficulties in this Veteran cohort, even years post-injury. (JINS, 2018, 24, 324–334)