A differential graded (DG) algebra is an algebra equipped with a differential satisfying the graded Leibniz (product) rule. The existence of associative DG structures on minimal free resolutions implies many desirable properties of the module being resolved. Moreover, any such DG structure induces an algebra structure on so-called Tor modules. In this talk, we will examine the Tor algebra structure of certain classes of compressed rings. In particular, we will encounter a new class of counterexamples to a conjecture of Avramov and show that there exist ideals defining rings of arbitrarily large type whose Tor algebra has class G.

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Let R be a standard graded polynomial ring and let I be a homogenous prime ideal of R. Bhatt, Blickle, Lyubeznik, Singh, and Zhang examined the local cohomology of R/I^t as t grows arbitrarily large. I will discuss their results and give an explicit description of the transition maps between these local cohomology modules in a particular example.
I will also consider asymptotic structure in a different direction: as the number of variables of R grows. This study of families of modules over compatible varying rings hints at the existence of OI structures, which I will discuss if time permits.

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