We explore the generalization of cellular decomposition in chromatically localized stable categories suggested by Picard--graded homotopy groups. In particular, for K(d) a Morava K-theory, we show that the Eilenberg-Mac Lane space K(Z, d+1) has a K(d)-local cellular decomposition tightly analogous to the usual decomposition of infinite-dimensional complex projective space (alias K(Z, 2)) into affine complex cells. Additionally, we identify these generalized cells in terms of classical invariants --- i.e., we show that their associated line bundles over the Lubin-Tate stack are tensor powers of the determinant bundle. (In particular, these methods give the first choice--free construction of the determinantal sphere S[det].) Finally, we investigate the bottom attaching map in this exotic cellular decomposition, and we justify the sense in which it selects a particular K(d)-local homotopy class

Susp^{-1} S[det]^2 --> S[det]

generalizing the classical four "Hopf invariant 1" classes h = (2, eta, nu, sigma) in the cofiber sequences

Susp^{-1} S^(2n) --h-> P_k^1 --> P_k^2

associated to the four normed real division algebras k = (R, C, H, O) with real dimension n = (1, 2, 4, 8). We also include a lengthy introduction to the subject of chromatic homotopy theory, outlining all of the tools relevant to the statements of our original results.

In the 2010s, efforts to increase urban housing production to meet demand were frequently mired in controversies over questions of unaffordability. Even if new buildings were slated to include “below-market” housing, working class and residents of color were wary of this kind of development amid the specter of increased displacement. The opposition was sometimes ironic because gentrification came after decades of concerted efforts at “community control” by neighborhood-based nonprofit groups which had opposed outside government planning in their communities. Within the span of the decades between public housing’s defunding in the early 1970s and the fallout of 2008 economic crisis, the United States had developed a unique approach to the financing and local development of “affordable housing,” one which made it if not vulnerable to, then unable to address many of the effects of gentrification where they arose. This dissertation examines politics around low-income housing through key moments in the emergence of a nonprofit affordable housing system within the larger social and economic transformations of the United States from 1968–2018. In particular, I focus on the role of housing within racial capitalism in the United States which oversees vastly different outcomes in housing wealth for Black Americans in particular. I follow a focus especially on the rising emphasis of low-income homeownership, and Black homeownership, as the answer to a radicalized “urban crisis” signaled with urban uprising in the late 1960s. I argue that this emphasis on homeownership was an ideological response, fostered by the success of New Deal homeownership politics, which had the important effect of undermining the possibilities for a socially democratic form of municipal housing. At multiple moments in the creation of an alternative to public housing, homeownership programs were heralded as the solution, even while less attention was drawn to efforts to also increase financing to the more practical nonprofit management of mutli-family or rental housing. I argue the homeownership emphasis has had damning implications for the ability of tenants to the realize political power necessary to win demands such as rent control in majority-homeowner jurisdictions. To understand the development of a nonprofit controlled affordable housing industry, I examine two related developments in US political and economic life: the rise of community development, and the increasingly prominent role of a philanthropic sector in designing and administering social welfare programs. Both of these developments are in turn related to a need, on the part of public officials, to find or create alternative sources of funding in the wake of public austerity. I examine the early efforts of groups like Chicago’s The Woodlawn Organization, a prominent Black Power group, to contest urban renewal and create their own paradigm of community-controlled housing development. I argue that the contradictions of these efforts, which lacked capital resources, presaged larger issues with Community Development Corporations (CDCs) as they viewed them as a more appropriate alternative to government development of inner-city housing. This dissertation contextualizes the need to create a funding source for this nonprofit system within a neoliberal political climate which turned to novel sources in the tax code and using issuance of tax credits to solve political and fiscal concerns. Housing Capital brings housing policy into conversation with politics by utilizing the methods of scholars of social movement politics, racial capitalism, and labor history. Architectural and urban historians, by focusing on discrete places and events, can overlook the role of larger institutions, political tendencies, and ways of organizing politics. I look at the interaction of local groups and specific urban contexts with larger federal policies and national organizations, such as foundations, economists, and other policy advisors, to understand the impact of the later groups in shaping the political possibilities in local contexts. I argue that this approach is necessary to understand how certain modes of housing provision and certain market arrangements achieve dominance, and to understand their implications for community groups and tenants writ large.

Engineers use sketches in the early phases of design because their expressiveness and ease of creation facilitate creativity and efficient communication. Our goal is to build software that leverages these strengths and enables natural sketch-based human-computer interaction. Specifically, our work is focused on creating algorithms that group hand-drawn strokes into individual objects so that they can be recognized. Grouping strokes is a difficult problem; many previous approaches have required the user to manually group the strokes. Others have used a search process, resulting in a computational cost that rises exponentially with the number of strokes in the sketch. In this dissertation we present a novel method for grouping strokes into objects based on a two-step algorithm that has a polynomial computational cost. In the first step, strokes are classified according to the type of object to which they belong, thus helping to create artificial separation between objects. In the second step, strokes of the same class are grouped into individual objects. Both steps rely on general machine learning techniques which seamlessly integrate spatial and temporal information, and which can be extended to new domains with no hand-coding. The single-stroke classifier used to separate strokes is the first in literature to perform multi-way classification, and it performs as well as or better than previous classifiers on text vs. non-text classification. Our grouping algorithm correctly groups between 84% and 91% of the ink in diagrams from four different domains, with between 61% and 82% of shapes being perfectly clustered. Our method runs in O(n²) time, where n is the number of points in the sketch. Real-world performance is improved with a conservative filter to eliminate consideration of distant strokes, and computation occurs incrementally as the sketch is constructed. Even without the filter, the computation for a large sketch containing over 700 strokes took less than 12% of the time required to draw the sketch. Experimental evaluation of our technique has shown it to be accurate and effective in four domains.

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