格点上的最小化问题起源于解析数论学家在数论上的纯粹兴趣，近年来人们注意到其在物理中的诸多应用。本项目拟建立几类格点上最小化问题的最小化子的结果，并应用到波色爱因斯坦凝聚、重整化的Ginzburg-Landau泛函中涡旋结构和粒子系统的最优结构等物理问题中。具体拟考虑四类问题：(1) 大数目两分量波色爱因斯坦凝聚中涡旋结构问题(数学上证明Mueller-Ho[29]中的实验和数值模拟结果)。(2) Riesz作用下两分量二维周期竞争系统的最优构型以及正方形格点的万有界问题。(3) 重整化两分量Ginzburg-Landau理论中涡旋结构问题。(4) Lennard-Jones位势下单粒子系统的最优结构问题(Blanc-Lewin[10]等公开问题)。本项目拟糅合数论中椭圆函数和模形式、单调性方法和变分与椭圆微分方程的理论，推动物理驱动的格点上最小化问题的进展，同时用来解决上述提及的问题。

The minimization problems on lattices originate from analytical number theorists by the pure and inner interest of number theory. It turns out that these results have significant application to many physical problems. The objective of this project is to establish the existence and locate exactly the minimizers of several minimization problems on lattices, and then apply these results to analyze the vortices structure from two-component Bose-Einstein theory with large numbers, vortices in renormalized Ginzburg-Landau theory and optimal lattices structure in particle systems. It is planned to work on four kinds of problems: (1) Vortices in two-component Bose-Einstein condensates (rigorous prove the numerical/experimental results in Mueller-Ho [29]). (2) The universality of square lattice and optimal lattice structure in a competitive system under Riesz potential. (3) Vortices in two-component renormalized Ginzburg-Landau theory. (4) The optimal lattice structure in single-particle systems under Lennard -Jones potential(an open problem in Betermin-Petrache[8,9], Blanc-Lewin[10], etc). Among the tools involved, we integrate the elliptic function and modular form in number theory, the monotonicity method, and the calculus of variation and PDE theory.