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

研究与各种积分结构相关的theta和zeta函数使我们能够发展格论计数问题、凸几何和模形式理论之间的原始联系。具体地说，我们以一种新颖的方式将经典的Ehrhart凸多面体理论与Hecke辛p-进李群的Hecke级数相结合。我们使用代数、组合学和几何学的方法来解决晶体计数问题。最后，我们的目标是将模形式理论的经典方面从格推广到几乎格。

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.