According to the general philosophy of "brave new algebra", the stable oo-category of spectra serves as a higher homotopical version of the usual category of abelian groups. In ordinary algebra one has the local-to-global paradigm of studying objects one prime at a time and then assembling the local information together. The chromatic picture affords an analogous paradigm for the oo-category of spectra. For each classical prime p, the Morava K-theories K(n) for 0 <= n <= oo interpolate between char 0 and char p. Thus, the categories of spectra localized with respect to these "intermediate" primes K(n) exhibit some intermediate characteristic behavior. A particularly remarkable property of the K(n)-local categories Sp_K(n) is that of "higher semiadditivity" (introduced and proved by Hopkins and Lurie) which allows integration of morphisms along homotopically finite spaces. In this talk, I will give an exposition of the theory of higher semiadditivity and describe a joint work with Shachar Carmeli and Tomer Schlank establishing higher semiadditivity for the telescopic localizations Sp_T(n) and some applications including the construction of Galois extensions of the telescopic spheres S_T(n).

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