Ramified extensions of commutative ring spectra

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The theory of ramified extensions of rings of integers in number fields is a classical topic in algebraic number theory. In stable homotopy theory, ramified extensions of structured ring spectra occur for instance as connective covers of Galois extensions of commutative ring spectra. A systematic approach for studying ramification in stable homotopy theory is missing. The distinction of tame and wild ramification is rather ad hoc so far. The examples that were studied until now are mostly of chromatic type less or equal to one, i.e. these are extensions that concern singular homology and topological K-theory and variations of these. In this project I will study ramified extensions of chromatic type two or higher and using these examples I aim to develop a suitable notion of tamely ramified extensions. Important examples of such maps come from function spectra and spectra of topological modular forms. Besides ramification at prime numbers there might be ramification at higher chromatic primes. Technical means for studying ramified maps are homology theories such as topological Hochschild homology, topological Andre-Quillen homology together with their logarithmic versions.