The full C*-algebra associated to a free group on countably many generators is universal in the sense that it surjects onto any C*-algebra generated by as many or fewer unitaries. This algebra can be found at the heart of one of the most famous open problems in operator algebras. Still, despite its intractability, it has several properties that make it seem highly approachable, such as residual finite dimensionality and certain lifting properties. 

But this is not an ode to the full free group C*-algebra. There are several other universal algebras, which have gotten a fair bit of attention in recent years, that capture both its approachability and its role in the aforementioned open problem. In this talk, we will highlight four: the universal C*-algebras generated, respectively, by a contraction, a partial isometry, a row contraction, and a so-called Pythagorean tuple. Part of this talk is based on joint work with David Sherman. 

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