By Claudio Marmo

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Commonly considered as a vintage of recent arithmetic, this extended model of Felix Klein's celebrated 1894 lectures makes use of modern innovations to envision 3 recognized difficulties of antiquity: doubling the amount of a dice, trisecting an perspective, and squaring a circle. cutting-edge scholars will locate this quantity of specific curiosity in its solutions to such questions as: less than what conditions is a geometrical development attainable?

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E ∞ Ring Spaces and E ∞ Ring Spectra. With contributions by Frank Quinn, Nigel Ray, and Jørgen Tornehave, Lecture Notes in Mathematics, Vol. 577. Springer-Verlag, Berlin, 1977. [21] McDuff, D. and G. Segal. Homology fibrations and the ‘group-completion’ theorem. Invent. , 31(3): 279–284, 1975/76. [22] Quillen, Daniel. Higher algebraic K -theory. I. In Algebraic K -theory, I: Higher K -theories (Proc. , 1972), pages 85–147. , Vol. 341. Springer, Berlin, 1973. [23] Ravenel, Douglas C. Complex Cobordism and Stable Homotopy Groups of Spheres, volume 121 of Pure and Applied Mathematics.

We first introduce a filtration of the class of spectra that is related to the chromatic filtration given by the property of being Bousfield local with respect to some Johnson–Wilson theory E(n) (cf. Ravenel [24, §7]), but is more appropriate for the connective spectra that arise from algebraic K -theory. Our Two-vector bundles and forms of elliptic cohomology 39 notion is also more closely linked to aspects of vn -periodicity than to being E(n)-local. Let p be a prime, K (n) the nth Morava K -theory at p and F a p-local finite CW spectrum.

3. Let E be a represented 2-vector bundle with gluing maps g αβ and coherence maps h αβγ . There is an associated charted 2-vector bundle with gluing bundles E αβ = (g αβ )∗ (E) over Uαβ and coherence isomorphisms ∼ = φ αβγ = (h αβγ )∗ (φ) : E αβ · E βγ = (g αβ ·g βγ )∗ (E) −−−−→ (g αγ )∗ (E) = E αγ over Uαβγ . This association induces a bijection between the equivalence classes of represented 2-vector bundles and the equivalence classes of charted 2-vector bundles of rank n over X . 4. Let 2-Vectn (X ) be the set of equivalence classes of 2-vector bundles of rank n over X .