• Journal of the Korean Mathematical Society
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• v.46 no.2
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• pp.363-447
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• 2009

The author previously defined the spectral invariants, denoted by $\rho(H;\;a)$, of a Hamiltonian function H as the mini-max value of the action functional ${\cal{A}}_H$ over the Novikov Floer cycles in the Floer homology class dual to the quantum cohomology class a. The spectrality axiom of the invariant $\rho(H;\;a)$ states that the mini-max value is a critical value of the action functional ${\cal{A}}_H$. The main purpose of the present paper is to prove this axiom for nondegenerate Hamiltonian functions in irrational symplectic manifolds (M, $\omega$). We also prove that the spectral invariant function ${\rho}_a$ : $H\;{\mapsto}\;\rho(H;\;a)$ can be pushed down to a continuous function defined on the universal (${\acute{e}}tale$) covering space $\widetilde{HAM}$(M, $\omega$) of the group Ham((M, $\omega$) of Hamiltonian diffeomorphisms on general (M, $\omega$). For a certain generic homotopy, which we call a Cerf homotopy ${\cal{H}}\;=\;\{H^s\}_{0{\leq}s{\leq}1}$ of Hamiltonians, the function ${\rho}_a\;{\circ}\;{\cal{H}}$ : $s\;{\mapsto}\;{\rho}(H^s;\;a)$ is piecewise smooth away from a countable subset of [0, 1] for each non-zero quantum cohomology class a. The proof of this nondegenerate spectrality relies on several new ingredients in the chain level Floer theory, which have their own independent interest: a structure theorem on the Cerf bifurcation diagram of the critical values of the action functionals associated to a generic one-parameter family of Hamiltonian functions, a general structure theorem and the handle sliding lemma of Novikov Floer cycles over such a family and a family version of new transversality statements involving the Floer chain map, and many others. We call this chain level Floer theory as a whole the Floer mini-max theory.

• Journal of English Language & Literature
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• v.58 no.6
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• pp.1143-1165
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• 2012

Tinted with ontological concern, Ted Hughes passes through an existential climate, eventually confirms death( or nothingness) as the new foundation of his poetry, and explores the various paradoxical effects of nothingness. Nihilism, fraught with rather negative and traumatic themes such as death, melancholy, and despair can, however, generate being (even in multiple modes), animalistic vitality, and insubstantial specters. Among these new functions of nothingness animality and spectrality are the most notable in Hughes's poetry. A considerable number of animals and bioorganisms that Hughes introduces exhibit the enormous energy derived from the dignity of death, from subversive challenges against the established hierarchy, and from new and dynamic multifaceted sources of nothingness. In other words, Hughes's animals, yield surplus power beyond themselves, as if they are demi-gods; in short, they feature the sublime as unidentified terrifying effects of nothingness. In a sense, animality means allowing some level of violence without legal sanction. Hughes inaugurates this kind of all bigotry-eradicating violence and attempts to subvert higher beings such as humans and gods, and existing doctrines: thrushes rise up against the animal and human worlds; a rush of ghostly crabs at night press through the human world. Hughes also resists the highest being, God, employing the technique of rewriting God's theology. Dirty, anomalous crows attack, subvert, and dismember the delicate, indurate, and thorough system of logos. Hughes, of course, does not place the animals merely in lofty regard, aware of the ulterior deprivation of the sublime animality, the trace of existential negativity. Thus, a seemingly omnipotent crow can become a mere beggar guzzling ice cream from the garbage bin on the beach. In addition, the violent and dignified aspects of nothingness can be transformed to reveal the thin and trivial traits as unreliable specters. Dark, heavy, and terrible nullity lessens its own volume and mass, and exposes the airy waves of shadows or specters. However, owing to nullity's untraceable track, the scarcity and unfamiliarity of the phantoms inversely display their foreign gigantic effects such as fantasy and violence.