By Zannier, Umberto (ed.)

Contributions via numerous authors equivalent to Michael G. Cowling.- Joseph A. Wolf.- Gisbert Wustholz and David Mumford

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G In this direct integral decomposition dχ could be replaced by any equivalent measure, so that decomposition is not as precise as the Fourier inversion formula. 2. The Heisenberg group Next, we see what happens when we weaken the commutativity condition. The ﬁrst case of that is the case of the Heisenberg group. There the Fourier transform and Fourier inversion are in some sense the same as in the classical case of a vector group, except that some of the integration occurs in the character formula and the rest in integration over the unitary dual.

After Lindemann’s discovery Hilbert asked in his famous address to the international congress of mathematicians at Paris in 1900 the question whether numbers of the form α β for algebraic numbers α and β are transcendental except that α is zero or one or β is rational. This problem was solved independently by A. O. Gel’fond and Th. Schneider. The result can be transformed into a question on rational integrals so to ﬁt into our general point of view: consider the afﬁne plane X with coordinates x and y, the differential form ω = (dx/x) − β(dy/y) and a path γ : [0, 1] → X with γ (0) = (1, 1) and γ (1) = (δ, α).

354 (1984), 164–174. ¨ , Zum Periodenproblem, Invent. Math. 78 (1984), [10] G. W USTHOLZ 381–391. [11] Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen, Annals of Math. 129 (1989), 501–517. ¨ , On Leibniz’ conjecture, periods and motives, in [12] G. W USTHOLZ preparation. The geometry and curvature of shape spaces David Mumford The idea that the set of all smooth submanifolds of a ﬁxed ambient ﬁnite dimensional differentiable manifold forms a manifold in its own right, albeit one of inﬁnite dimension, goes back to Riemann.