Lattice calculations on the spectrum of Dirac and Dirac-K\"ahler operators

We present a matrix technique to obtain the spectrum and the analytical index of some elliptic operators defined on compact Riemannian manifolds. The method uses matrix representations of the derivative which yield exact values for the derivative of a trigonometric polynomial. These matrices can be used to find the exact spectrum of an elliptic operator in particular cases and in general, to give insight into the properties of the solution of the spectral problem. As examples, the analytical index and the eigenvalues of the Dirac operator on the torus and on the sphere are obtained and as an application of this technique, the spectrum of the Dirac-Kahler operator on the sphere is explored.Comment: 11 page

Understanding Musicking on Social Media: Music Sharing, Sociality and Citizenship

Based on online and offline ethnographic fieldwork among the Spanish community in London, this thesis investigates why people circulate and share music on social media platforms, particularly within a context of migration. Written from the perspective of an insider-outsider, it combines research data from ethnographic participant observation and interviews to understand how music media circulates online, studying the different roles it fulfils within these contexts as different forms of musicking. The thesis responds to these questions by addressing six areas. Chapter 1 explores the uses of music to perform and articulate cultural and gender identity on social media profiles. Chapter 2 investigates how music sharing and compiling can be used to maintain relationships and social capital and to participate in transnational sociality. Chapter 3 addresses how the circulation of music parody as citizen engagement in contexts of political conflict such as the Brexit and Catalonian referendums gives rise to temporary political alliances in the form of citizen assemblages. Chapter 4 considers the circulation of music on social media from the perspective of fandom and how it is influenced by post-object ephemeral and reflexive practices. Chapter 5 analyses how the ubiquitous and imagined character of online music media fosters its circulation as a silent and visual element of online sociality, effectively generating practices of imagined listening. Chapter 6 examines ritualistic practices of music exchange on social media and their relationship with emerging moral economies of music circulation as foundational elements of online sociality and citizenship. I conclude by arguing that, in circulating music online through their social media and streaming profiles, users develop new forms of (im)material culture-making. Online musicking enables new ways of being in the world, turning the intangible, time-bound aural and visual experience of online music into something that has a materialised impact in the social lives of users. Through the circulation of music online users form, dissolve and inhabit temporary music-based alliances and expand their social worlds