In the study of connections between Galois representations and modular forms, one often seeks an R=T theorem, which asserts that the deformation ring R is isomorphic to a (localized) Hecke algebra T. However, sometimes only the framed deformation ring exists. With the framing variables, it is obviously larger than the Hecke algebra. Pseudorepresentations, which is a generalization of the notion of traces of representations, was invented to get around this issue. We will introduce the notion of pseudorepresentations and discuss the criteria for pseudorepresentations to arise from genuine representations. Next, we will introduce the algorithm used to explicitly compute usual deformation rings and pseudodeformation rings for finitely presented groups, which leads to the discovery of a counterexample.

When does an algebraic differential equation admit an algebraic solution? This has been an animating question behind much mathematics since the latter half of the 19th century. It is now understood (due to work of Siegel, Grothendieck, Katz, and others) to be closely related to number theory. I'll survey some of the conjectures and results on this topic, and explain some recent progress in understanding what happens in the case of non-linear algebraic differential equations--for example, the Painlevé VI equation and Schlesinger system--in joint work with Josh Lam.