This lecture series is based on the 6 lectures I gave at the instructional workshop "Iwasawa Theory over function fields" at ICMAT (Madrid, Spain). The aim of this lecture series is to explain the formulation of the equivariant BSD conjecture over global function fields, following the joint work with D. Burns and M. Kakde. In the first two talks, I will explain the backgrounds on Selmer groups and flat cohomology.

The celebrated formula of Otter \emph{[Ann. of Math. (2) 49 (1948), 583--599]} asserts that the complete graph contains exponentially many non-isomorphic spanning trees. In this paper, we show that every connected almost regular graph with sufficiently large degree already contains exponentially many non-isomorphic spanning trees. Indeed, we prove a stronger statement: for every fixed $n$-vertex tree $T$, $$\Pr\bigl[\mathcal{T} \simeq_{\mathrm{iso}} T\bigr] = e^{-\Omega(n)},$$ where $\mathcal{T}$ is a uniformly random spanning tree of a connected $n$-vertex almost regular graph with sufficiently large degree. To prove this, we introduce a graph-theoretic variant of the classical balls--into--bins model, which may be of independent interest.

Many complete Riemannian manifolds of constant curvature, including complete hyperbolic manifolds, can be realized as quotients of convex domains in real projective spaces by discrete linear group actions. Among these discrete groups, an important and broad class is given by linear reflection groups (also called linear Coxeter groups). In dimension 3, many such linear reflection groups arise from hyperbolic reflection groups, using Andreev’s theorem, through small deformations and gluing constructions. In this talk, I will introduce these ideas and concepts, explain the basic picture, and discuss some recent developments related to the projective Andreev theorem.