This presentation will begin with an introduction to noncommutative rings by way of several examples. I will then discuss two research programs that share the common motivation of producing methods to view noncommutative pehnomena as an extension of commutative algebra, as follows:

1. The search for a functor that extends the Zariski spectrum from commutative to noncommutative rings.
2. Finding ways to infer good ring-theoretic properties from good homological properties.

Zoom link: https://uci.zoom.us/j/95978949132?pwd=QzliZ0RWc2RyWURtSi9XS0FwRmZSZz09