Eisenstein series for G₂ and the symmetric cube Bloch--Kato conjecture

The purpose of this thesis is to construct nontrivial elements in the Bloch--Kato Selmer group of the symmetric cube of the Galois representation attached to a cuspidal holomorphic eigenform of level 1. The existence of such elements is predicted by the Bloch--Kato conjecture. This construction is carried out under certain standard conjectures related to Langlands functoriality. The broad method used to construct these elements is the one pioneered by Skinner and Urban in [SU06a] and [SU06b]. The construction has three steps, corresponding to the three chapters of this thesis. The first step is to use parabolic induction to construct a functorial lift of to an automorphic representation π of the exceptional group G₂ and then locate every instance of this functorial lift in the cohomology of G₂. In Eisenstein cohomology, this is done using the decomposition of Franke--Schwermer [FS98]. In cuspidal cohomology, this is done assuming Arthur's conjectures in order to classify certain CAP representations of G₂ which are nearly equivalent to π, and also using the work of Adams--Johnson [AJ87] to describe the Archimedean components of these CAP representations. This step works for of any level, even weight ≥ 4, and trivial nebentypus, as long as the symmetric cube -function of vanishes at its central value. This last hypothesis is necessary because only then will the Bloch--Kato conjecture predict the existence of nontrivial elements in the symmetric cube Bloch--Kato Selmer group. Here this hypothesis is used in the case of Eisenstein cohomology to show the holomorphicity of certain Eisenstein series via the Langlands--Shahidi method, and in the case of cuspidal cohomology it is used to ensure that relevant discrete representations classified by Arthur's conjecture are cuspidal and not residual. The second step is to use the knowledge obtained in the first step to -adically deform a certain critical -stabilization π of π in a generically cuspidal family of automorphic representations of G₂. This is done using the machinery of Urban's eigenvariety [Urb11]. This machinery operates on the multiplicities of automorphic representations in certain cohomology groups; in particular, it can relate the location of π in cohomology to the location of π in an overconvergent analogue of cohomology and, under favorable circumstances, use this information to -adically deform π in a generically cuspidal family. We show that these circumstances are indeed favorable when the sign of the symmetric functional equation for is -1 either under certain conditions on the slope of π, or in general when has level 1. The third and final step is to, under the assumption of a global Langlands correspondence for cohomological automorphic representations of G₂, carry over to the Galois side the generically cuspidal family of automorphic representations obtained in the second step to obtain a family of Galois representations which factors through G₂ and which specializes to the Galois representation attached to π. We then show this family is generically irreducible and make a Ribet-style construction of a particular lattice in this family. Specializing this lattice at the point corresponding to π gives a three step reducible Galois representation into GL₇, which we show must factor through, not only G₂, but a certain parabolic subgroup of G₂. Using this, we are able to construct the desired element of the symmetric cube Bloch--Kato Selmer group as an extension appearing in this reducible representation. The fact that this representation factors through the aforementioned parabolic subgroup of G₂ puts restrictions on the extension we obtain and guarantees that it lands in the symmetric cube Selmer group and not the Selmer group of itself. This step uses that is level 1 to control ramification at places different from , and to ensure that is not CM so as to guarantee that the Galois representation attached to π has three irreducible pieces instead of four

From axiomatization to generalizatrion of set theory

The thesis examines the philosophical and foundational significance of Cohen's Independence results. A distinction is made between the mathematical and logical analyses of the "set" concept. It is argued that topos theory is the natural generalization of the mathematical theory of sets and is the appropriate foundational response to the problems raised by Cohen's results. The thesis is divided into three parts. The first is a discussion of the relationship between "informal" mathematical theories and their formal axiomatic realizations this relationship being singularly problematic in the case of set theory. The second part deals with the development of the set concept within the mathemtical approach. In particular Skolem's reformulation of Zermlelo's notion of "definite properties". In the third part an account is given of the emergence and development of topos theory. Then the considerations of the first two parts are applied to demonstrate that the shift to topos theory, specifically in its guise of LST (local set theory), is the appropriate next step in the evolution of the concept of set, within the mathematical approach, in the light of the significance of Cohen's Independence results

Transcendent experience or the transcendence of experience? An analysis of transcendent realization in Shankara, Ibn Arabi and Meister Eckhart

This research aims at investigating the nature, meaning and implications of 'transcendent realization', that which is held to be the summit of spiritual realization by three renowned and highly influential mystics; Shankara. from the Hindu tradition, Ibn Arabi, from the Islamic tradition and Meister Eckhart, from the Christian tradition. The central methodological principle of the analysis is interntionalityl the opening chapter situates and discusses this principle in relation to the phenomenological method, while also highlighting the importance of the concept of transcendence for the contemporary discussion in comparative mysticism between the 'contextualist' school of Steven Katz and the 'Pure Consciousness' school of Robert Forman. Three chapters follow, dealing in turn with each of the three mystics, analyzing in some depth their respective pronouncements on transcendence; this theme is explored in both doctrinal and realizational terms: what transcendence means objectively, and how it is assimilated, realized or attained subjectively, with what pre-conditions and with what ramifications. The penultimate chapter brings together those features of transcendence shared in common bu the three mystics; differences as well as similarities are analyzed here. The final chapter consists in a critique of recent scholarly approaches to mysticism. In the light of the conclusions presented in this thesis, the reductive aspect of these approaches - their failure to take into account fully the nature and implications of trascendence with regard to mystical experience - is clearly discerned. The central conclusion of the thesis is that transcendent realization consists in the realization of identity with the Absolute, an identity which strictly transcends the individual, and by that very token transcends all possible 'experience' defined in relation to the individual; it also necessarily transcends all contextual factors that presuppose the individual as the ground of their mediating influence. The realization of this transcendent identity is incommunicable as regards its intrinsic nature but can be extrinsically described as the realization of the unique and undifferentiable Essence of 'Being-Consciousness-Bliss'

Social work with airports passengers

Social work at the airport is in to offer to passengers social services. The main methodological position is that people are under stress, which characterized by a particular set of characteristics in appearance and behavior. In such circumstances passenger attracts in his actions some attention. Only person whom he trusts can help him with the documents or psychologically

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum