Doctor of Philosophy (PhD)
For a simple complex algebraic group $G$, M. Kamgarpour and D. Sage have shown that the adjoint irregularity of an irregular singular flat $G$-bundle on the formal punctured disc is bounded from below by the rank of $G$, moreover the rank is realized by the formal Frenkel-Gross connection. This is a geometric analog of a conjecture of Gross and Reeder on the swan conductor of arithmetic local Langlands parameters. In this work, we explore an interesting combinatorial problem which arises when trying to consider the minimal value of the irregularity function with respect to an arbitrary representation of $G$.
Alaniz, Andrew, "A Conjecture on the Irregularity Function for Local Geometric Langlands Parameters and the Formal Frenkel-Gross Connection" (2021). LSU Doctoral Dissertations. 5445.