In 1989, Mazur defined the deformation functor associated to a residual Galois representation, which played an important role in the proof by Wiles of the modularity theorem. This was used as a basis over which many mathematicians constructed variations both to further specify it or to expand the contexts where it can be applied. These variations proved to be powerful tools to obtain many strong theorems, in particular of modular nature. In this talk I will give an overview of the deformation theory of Galois representations and describe two variants of Mazur's functor that allow one to properly deform reducible residual representations (which is one of the shortcomings of Mazur's original functor). Namely, I will present the theory of determinant-laws initiated by Bellaïche-Chenevier on the one hand, and an idea developed by Calegari-Emerton on the other.
If time permits, I will also describe results that seem to indicate a possible comparison between the two seemingly unrelated constructions.

Named after mathematical physicists Kubo, Martin, and Schwinger, KMS states are a special class of states on any C$^*$-algebra admitting a continuous action of the real numbers. Unlike in the case of von Neumann algebras, where each modular flow has a unique KMS state, the collection of KMS states for a given flow on a C$^*$-algebra can be quite intricate. In this talk, I will explain what abstract properties these simplices have and show how one can realise such a simplex on various classes of simple C$^*$-algebras.

Let G be a reductive group defined over a local field of characteristic 0 (real or p-adic). By Harish-Chandra’s regularity theorem, the character Θ_π of an irreducible representation π of G is given by a locally integrable function f_π on G. It turns out that f_π has even better integrability properties, namely, it is locally L^{1+r}-integrable for some r>0. This gives rise to a new singularity invariant of representations \e_π by considering the largest such r.

We explore \e_π, show it is bounded below only in terms of the group G, and calculate it in the case of a p-adic GL(n). To do so, we relate \e_π to the integrability of Fourier transforms of nilpotent orbital integrals appearing in the local character expansion of Θ_π. As a main technical tool, we use explicit resolutions of singularities of certain hyperplane arrangements. We obtain bounds on the multiplicities of K-types in irreducible representations of G for a p-adic G and a compact open subgroup K.

Based on a joint work with Itay Glazer and Julia Gordon.