There exists an interesting family of finite-dimensional representations called the Kirillov-Reshetikhin modules over the quantum affine algebra $U_q(widehat{mathfrak{g}})$. The isotypic decomposition of theses modules or their tensor products as $U_q(mathfrak{g})$-modules is given by the fermionic formula which can be regarded as a representation theoretic version of completeness of the Bethe ansatz.

In spite of its elegance, it quickly becomes impractical as the rank of $mathfrak{g}$ increases due to its complicated combinatorial nature. Thus it is advantageous to have a more explicit description of this decomposition for practical purposes. Such a formula is well-known in classical types, but remains largely conjectural in exceptional types.

In this talk, I will talk about linear recurrence relations satisfied by the sequence ${Q_m^{(a)}}_{m=0}^{infty}$ of the characters of the Kirillov-Reshetikhin modules and how they shed light on the above problem. The key idea is to regard this decomposition as a summation over the lattices points in a suitable polyhedron.

Kollar—Shepherd-Barron—Alexeev (KSBA) have given a general construction that provides a geometric compactification for the moduli space of varieties of general type. Unfortunately, even in relatively simple cases (e.g. surfaces of general type with small invariants) it is difficult to understand the boundary points and the structure of this KSBA compactification. Thus, it is natural to try to compare the KSBA construction with other constructions, in particular with Hodge theoretic constructions of the moduli space. The Hodge theoretic construction has the advantage of having a lot of structure (of arithmetic and representation theoretic nature), but except a few cases (essentially abelian varieties and K3s) it is highly transcendental. In this talk, I will report on joint work with P. Griffiths, M. Green and C. Robles on the study of the moduli and periods of H-surfaces (Horikawa surfaces). The H-surfaces are surfaces of general type with p_g=2, q=0, K^2=2. They are essentially the simplest case where both the KSBA and Hodge theoretic construction are non-trivial. Considering and comparing the two approaches gives a rich picture which suggests an important role for the period map in the study of moduli spaces beyond the classical cases of abelian varieties and K3s.