Hida theory provides a p-adic interpolation of modular forms that have a property known as ordinary. Hida’s interpolation inspired Mazur to formulate the deformation theory of Galois representations, which Wiles used to prove (among other things) that p-adically interpolated modular form correspond closely, via an “R=T theorem”, to p-adically deformed ordinary 2-dimensional Galois representations. This notion of “ordinary” 2-dimensional Galois representation means that the representation becomes reducible when restricted to a decomposition subgroup at the prime p. But which ordinary modular forms have a Galois representation that is not only reducible at p but also decomposable at p? After explaining some arithmetic-geometric motivations for this question, I will explain some joint work with Francesc Castella in which we construct a length 1 “bi-ordinary complex” of modular forms that has a Hida-type interpolation property and whose associated Galois representations are reducible decomposable at p. This construction builds upon on Coleman’s work on presenting de Rham cohomology of modular curves as a quotient of differentials of the second kind, as well as Boxer and Pilloni’s work on higher Hida theory.

Gravitational instantons are non-compact Calabi--Yau metrics with L^2 bounded curvature and are categorized into six types. I will describe three projects on gravitational instantons including: (a) Fredholm theory and deformation of the ALH* type; (b) non-collapsing degeneration limits of ALH* and ALH types; (c) existence of stable non-holomorphic minimal spheres in some ALF types. These three projects utilize geometric microlocal analysis in different singular settings. Based on works joint with Rafe Mazzeo, Yu-Shen Lin and Sidharth Soundararajan.