Modular forms and arithmetic geometry

把晶体想象成按照规则的重复图案在空间中排列的原子晶格。Theta函数是对格点(原子)的位置进行编码的数学对象，当一个人从一个固定点向外移动时，它具有许多显著的性质，自19世纪以来一直被研究。最近，在研究光锥Minkowski空间内外的格点时，引入了复杂得多的theta函数。在算术几何--多项式方程的整数解理论--的研究中，出现了更多的奇特之处。Theta函数是更广泛的函数类、模形式和自同构形式的特例。这项研究的目的是发展新类型的theta函数和自同构形，并研究它们在数论中的应用以及它们与物理的联系，特别是与弦理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.