'Yes, the century is an ashen sun': poem and subject in the philosophy of Alain Badiou

This thesis examines the relation between philosophy, the poem and the subject in the mature philosophy of Alain Badiou. It investigates Badiou’s decisive contribution to these questions primarily by means of comparison, especially to Martin Heidegger, Philippe Lacoue-Labarthe and Theodor Adorno, as well as by analysing Badiou’s readings of poems and prose by Paul Celan and Samuel Beckett respectively as sites of potential dialogue with his immediate predecessors. The thesis stresses the importance of French philosophy’s German heritage, emphasising not only Badiou’s radical departure from Heidegger and his legacy, but also the former’s wholesale rejection of philosophies that would, in the wake of twentieth-century violence and beyond, proclaim their own end or completion. The thesis argues Badiou’s innovative readings of Celan and Beckett to be crucial to understanding this endeavour: for Badiou, both writers use the poem to affirm novel conceptions of subjectivity capable of transcending the historical conditions of their presentation. The title quotation from Badiou’s The Century, ‘Yes, the century is an ashen sun’, anticipates both the affirmative nature of these subjective figures, and their presience, beyond the bounds of a twentieth-century ‘ashen sun’ pervaded by melancholy, for the ‘new suns’ of the twenty-first. The thesis is in four chapters. The first chapter unfolds the central concepts of Badiou’s departure from Heidegger using Paul Celan’s poems to focus the enquiry. It is guided by two of Badiou’s most condensed declarations about the poem, that, firstly, ‘the modern poem harbours a central silence’, and secondly, that ‘Celan completes Heidegger’. The second chapter exposes the political implications of Heidegger’s writings on Friedrich Hölderlin and the role of the subject therein, offering at its close some thoughts about what Badiou calls, following Philippe Lacoue-Labarthe, the poem’s ‘becoming-prose’. It concludes by drawing the poem and politics into relation by way of the philosophical category of the subject. The third chapter reads Badiou’s concept of ‘anabasis’ against Heidegger’s ‘homecoming’ in order to think the possibility of a collective political subject’s formation in the wake of Auschwitz. The final chapter examines the imbrication of the Two of love and the ‘latent poem’ in Badiou’s reading of Samuel Beckett’s late prose, contrasting this ‘affirmative’ reading of Beckett to Theodor Adorno’s earlier emphases on negation. Following its investigations of subjectivity, poem and prose throughout, the thesis concludes by returning to the title quotation in order to unfold the particular relations between subject, affirmation and negation Badiou’s philosophy enacts, and to offer further routes forward for research regarding Badiou’s philosophy and aesthetic figuration.

Alternating surgeries

This thesis is concerned with the question of when the double branched cover of an alternating knot can arise by Dehn surgery on a knot in S^3. We approach this problem using a surgery obstruction, first developed by Greene, which combines Donaldson's Diagonalization Theorem with the $d$-invariants of Ozsvath and Szabo's Heegaard Floer homology. This obstruction shows that if the double branched cover of an alternating knot or link L arises by surgery on S^3, then for any alternating diagram the lattice associated to the Goeritz matrix takes the form of a changemaker lattice. By analyzing the structure of changemaker lattices, we show that the double branched cover of L arises by non-integer surgery on S^3 if and only if L has an alternating diagram which can be obtained by rational tangle replacement on an almost-alternating diagram of the unknot. When one considers half-integer surgery the resulting tangle replacement is simply a crossing change. This allows us to show that an alternating knot has unknotting number one if and only if it has an unknotting crossing in every alternating diagram. These techniques also produce several other interesting results: they have applications to characterizing slopes of torus knots; they produce a new proof for a theorem of Tsukamoto on the structure of almost-alternating diagrams of the unknot; and they provide several bounds on surgeries producing the double branched covers of alternating knots which are direct generalizations of results previously known for lens space surgeries. Here, a rational number p/q is said to be characterizing slope for K in S^3 if the oriented homeomorphism type of the manifold obtained by p/q-surgery on K determines K uniquely. The thesis begins with an exposition of the changemaker surgery obstruction, giving an amalgamation of results due to Gibbons, Greene and the author. It then gives background material on alternating knots and changemaker lattices. The latter part of the thesis is then taken up with the applications of this theory.

Pricing derivatives with stochastic volatility

This Ph.D. thesis contains 4 essays in mathematical finance with a focus on pricing Asian option (Chapter 4), pricing futures and futures option (Chapter 5 and Chapter 6) and time dependent volatility in futures option (Chapter 7). In Chapter 4, the applicability of the Albrecher et al.(2005)'s comonotonicity approach was investigated in the context of various benchmark models for equities and com- modities. Instead of classical Levy models as in Albrecher et al.(2005), the focus is the Heston stochastic volatility model, the constant elasticity of variance (CEV) model and the Schwartz (1997) two-factor model. It is shown that the method delivers rather tight upper bounds for the prices of Asian Options in these models and as a by-product delivers super-hedging strategies which can be easily implemented. In Chapter 5, two types of three-factor models were studied to give the value of com- modities futures contracts, which allow volatility to be stochastic. Both these two models have closed-form solutions for futures contracts price. However, it is shown that Model 2 is better than Model 1 theoretically and also performs very well empiri- cally. Moreover, Model 2 can easily be implemented in practice. In comparison to the Schwartz (1997) two-factor model, it is shown that Model 2 has its unique advantages; hence, it is also a good choice to price the value of commodity futures contracts. Fur- thermore, if these two models are used at the same time, a more accurate price for commodity futures contracts can be obtained in most situations. In Chapter 6, the applicability of the asymptotic approach developed in Fouque et al.(2000b) was investigated for pricing commodity futures options in a Schwartz (1997) multi-factor model, featuring both stochastic convenience yield and stochastic volatility. It is shown that the zero-order term in the expansion coincides with the Schwartz (1997) two-factor term, with averaged volatility, and an explicit expression for the first-order correction term is provided. With empirical data from the natural gas futures market, it is also demonstrated that a significantly better calibration can be achieved by using the correction term as compared to the standard Schwartz (1997) two-factor expression, at virtually no extra effort. In Chapter 7, a new pricing formula is derived for futures options in the Schwartz (1997) two-factor model with time dependent spot volatility. The pricing formula can also be used to find the result of the time dependent spot volatility with futures options prices in the market. Furthermore, the limitations of the method that is used to find the time dependent spot volatility will be explained, and it is also shown how to make sure of its accuracy.