Leavitt path algebras: the first decade

The algebraic structures known as {\it Leavitt path algebras} were initially developed in 2004 by Ara, Moreno and Pardo, and almost simultaneously (using a different approach) by the author and Aranda Pino. During the intervening decade, these algebras have attracted significant interest and attention, not only from ring theorists, but from analysts working in C$^*$-algebras, group theorists, and symbolic dynamicists as well. The goal of this article is threefold: to introduce the notion of Leavitt path algebras to the general mathematical community; to present some of the important results in the subject; and to describe some of the field's currently unresolved questions.Comment: 53 pages. To appear, Bulletin of Mathematical Sciences. (page numbering in arXiv version will differ from page numbering in BMS published version; numbering of Theorems, etc ... will be the same in both versions

The twisted inverse image pseudofunctor over commutative DG rings and perfect base change

Let $K$ be a Gorenstein noetherian ring of finite Krull dimension, and consider the category of cohomologically noetherian commutative differential graded rings $A$ over $K$, such that $H^0(A)$ is essentially of finite type over $K$, and $A$ has finite flat dimension over $K$. We extend Grothendieck's twisted inverse image pseudofunctor to this category by generalizing the theory of rigid dualizing complexes to this setup. We prove functoriality results with respect to cohomologically finite and cohomologically essentially smooth maps, and prove a perfect base change result for $f^{!}$ in this setting. As application, we deduce a perfect derived base change result for the twisted inverse image of a map between ordinary commutative noetherian rings. Our results generalize and solve some recent conjectures of Yekutieli.Comment: 38 pages. Final version, to appear in Advances in Mathematic