New examples for logarithmic ring spectra

环在纯数学的许多领域中扮演着重要的角色，例如在数论中作为数环，或者在代数几何中作为几何物体的积木。有几种环的推广。其中两个在本课题中起着重要的作用：一方面，它们是同调代数的微分梯度代数，它们是由具有相容环结构的链配合物给出的。它们是同调代数的典型来源。另一方面，我们对代数拓扑结构的环谱很感兴趣。这些对象表示乘法上同调理论，并涵盖微分梯度代数作为特殊情况。在这两种情况下，我们都要求乘法是可交换的直到相干同伦。本项目的目的是寻找具有所谓对数结构的微分梯度代数的合适定义，并开发有关这些新对象的示例和结构结果。普通环上的对数结构最初是在代数几何中引入的，其中包括扩展光滑映射的概念。近年来，对数结构的概念被成功地推广到结构环谱中，它可以用于研究算术性质。本课题中考虑的具有对数结构的微分梯度代数是有趣的，因为它们提供了具有对数结构的结构环谱的新例子。因此，它们将有助于更好地理解后一个对象。此外，在实例中，我们想用对数结构来分析微分梯度代数的算术性质，并得到微分梯度代数向结构环谱过渡的新结果。总之，我们期望同伦理论和代数几何概念的相互转移将导致对代数拓扑的结构环谱和同伦代数的微分梯度代数的新见解。

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.