The structure of (almost) lattices – algebra, analysis, and arithmetic (A01)

研究与各种积分结构相关的θ和ζ函数使我们能够在格论枚举问题、凸几何和模形式理论之间建立原始联系。具体地，我们以一种新颖的方式将经典的凸多面体的Ehrhart理论与辛p进李群的Hecke级数结合起来。晶体计数问题，我们处理的方法从代数组合和几何。最后，我们将模形式理论的经典方面从格推广到概格。

Studying theta and zeta functions associated with various integral structures allows us to develop original connections between lattice-theoretic enumeration problems, convex geometry, and the theory of modular forms. Concretely we combine, in a novel way, classical Ehrhart theory of convex polytopes with Hecke series of symplectic p-adic Lie groups. Crystallographic counting problems we tackle using methods from algebraic combinatorics and geometry. Lastly we aim at a comprehensive generalization of classical aspects of the theory of modular forms from lattices to almost lattices.