Modular forms and arithmetic geometry

把晶体想象成原子在空间中按照规则的重复模式排列的晶格。θ函数是一种数学对象，它编码了从一个固定点向外移动时晶格点（原子）的位置，具有许多显着的性质，自世纪以来一直在研究。最近，更复杂的θ函数被引入，例如在光锥Minkowski空间内部或外部的格点的研究中。更奇特的出现在算术几何的研究中--多项式方程的整数解理论。Theta函数是更广泛的一类函数、模形式和自守形式的特殊情况。本研究的目标是开发新类型的theta函数和自守形式，并研究它们在数论中的应用及其与物理学的联系，特别是与弦论中出现的计数函数的联系。

Imagine a crystal as a lattice of atoms arranged in space according to a regular repeating pattern. Theta functions, the mathematical objects which encode the locations of the lattice points (atoms) as one moves outward from a fixed point, have many remarkable properties and have been studied since the 19th century. Recently, much more complicated theta functions have been introduced, for example in the study of lattice points inside or outside the light cone Minkowski space. Still more exotic ones occur in the study of arithmetic geometry -- the theory of whole number solutions to polynomial equations. Theta functions are special cases of a wider class of functions, modular forms and automorphic forms. The goal of this research is to develop new types of theta functions and automorphic forms and to investigate their applications in number theory and their connections to physics, in particular, to the counting functions which arise in string theory.