New examples for logarithmic ring spectra

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Rings play a fundamental role in many areas of pure mathematics, for example as number rings in number theory or as building blocks of geometric objects in algebraic geometry. There are several generalizations of rings. Two of them play an important role in this project: On the one hand these are the differential graded algebras of homological algebra, which are given by chain complexes with compatible ring structures. They are a typical source of homology algebras. On the other hand, we are interested in the structured ring spectra of algebraic topology. These objects represent multiplicative cohomology theories and cover differential graded algebras as special cases. In both situations, we require that the multiplication is commutative up to coherent homotopy.The aim of this project is to find a suitable definition of differential graded algebras with a so called logarithmic structure, and to develop examples and structural results about these new objects. Logarithmic structures on ordinary rings were originally introduced in algebraic geometry, amongst others to extend the notion of a smooth map. In recent years, the concept of a logarithmic structure was successfully generalized to structured ring spectra, where it can be used for the study of arithmetic properties.The differential graded algebras with logarithmic structures considered in this project are interesting since they provide new examples of structured ring spectra with logarithmic structures. Thus they will help to gain a better understanding of the latter objects. Moreover, in examples we would like to use logarithmic structures to analyze arithmetic properties of differential graded algebras and to obtain new results about the passage from differential graded algebras to structured ring spectra. Altogether, we expect that the mutual transfer of concepts from homotopy theory and algebraic geometry will lead to new insights about the structured ring spectra of algebraic topology and the differential graded algebras of homological algebra.