These are informal notes meant to briefly summarize some of the ideas surrounding the work of my thesis and my work with A. Bertoloni Meli. As an added bonus, I try to motivate the Langlands conjecture via the cohomology of Shimura varieties.

These are very old notes written for a seminar on automorphic representations. They discuss some examples of representations of reductive \(p\)-adic groups.

These are very old notes written for a seminar on automorphic representations. They discuss some examples of automorphic representations for \(\mathrm{GL}_2\).

These are very old notes written for a seminar on Galois representations. The goal is to essentially discuss some basic aspects of the Galois groups of local and global fields, give some geometric intuition for these objects, and explain why one would expect them to be difficult to study.

These are very old notes written for a seminar on Galois representations. The goal is to motivate Weil–Deligne representations (i.e., why they naturally show up), \(p\)-adic Hodge theory, and how the two notions connect.

These notes are a discussion on the theory of characteristic \(p\) representations of \(p\)-adic and \(t\)-adic reductive groups and their relationship with the characteristic 0 theory. There is, in particular, a discussion of Kazhdan's theorem which relates Hecke algebras of \(p\)-adic and \(t\)-adic groups.

These are notes written for two mentees I had in an independent study concerning the étale fundamental group. The goal was to motivate cohomology (in particular étale cohomology) via torsors. I think that the notes are well-intentioned and do genuinely have interesting didactic value buried deep inside them. Unfortunately, they are long-winded, meandering, and overly self-indulgent. One day I intend to go back and tighten them up.

My talk at the CARTOON Conference on my work with Bertoloni Meli on characterizing the local Langlands conjecture.

My talk at the University of Tokyo number theory seminar.