Algebraic Geometry Seminar

The moduli of (possibly irregular) formal connections in one variable (up to gauge transformations) is an infinite-dimensional space that "feels finite-dimensional," e.g., it has finite-dimensional tangent spaces. However, it is not so clear how this perception is actually reflected in the global geometry of this space. Previous works have focused on explicit parametrization of this space. As we will recall during the talk, this approach has significant limitations, and is insufficient to say anything about the global geometry. But we will instead find that this space appears much kinder through the lens of homological (or more poetically, noncommutative) geometry, exhibiting much better features than all its close relatives. In particular, we will give a first global sense in which it appears finite-dimensional. Finally, we will discuss how these results lend credence to the existence of a de Rham Langlands program incorporating arbitrary singularities (the usual story is unramified, or at worst has Iwahori ramification). Moreover, we will be able to formulate a precise conjecture that is a first approximation to a local geometric Langlands conjecture.