Inhibitory Long Range Interaction in Pattern Forming Physical and Biological Systems

抑制体系，如嵌段共聚物，以其形态相的结构模式而闻名。对这些图案的研究揭示了其机械、光学、电学、离子、势垒和其他应用性质。例如，嵌段共聚物在商业上用作热塑性弹性体：光缆外壳、粘合剂、沥青改性剂或人造器官技术。研究人员将要研究的数学理论将为这些材料的性质提供洞察力，并最终为工业设备的建造提供信息。这个项目还将为本科生和研究生提供实验室经验和研究指导机会。抑制相互作用出现在几何变分问题中，与其他数学分支中的现象有关：几何测度论中的气泡团，解析数论中的椭圆曲线，以及几何中的球面填充。负分数和负分数屏蔽拉普拉斯算子提供了抑制相互作用的准确描述，并与其他数学理论相关。这个项目的第一个任务是确定这些算子具有抑制性质的分数指数的范围。第二种是构造抑制系统中周期结构的积木，例如非标准双泡，并利用它们来寻找整个空间的周期驻点。一个相关的问题是找到一个能产生自由能密度最小的周期构型的最佳晶格。积木也可以用来在一般有界域上锻造接近定期的固定集会。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Inhibitory systems, such as block copolymers, are known for their structured patterns of morphological phases. Studying these patterns reveals the mechanical, optical, electrical, ionic, barrier and other properties for applications. For instance, block copolymers are commercially used as thermoplastic elastomers: outer coverings for optical fiber cables, adhesives, bitumen modifiers, or in artificial organ technology. The mathematical theory to be studied by the investigator will offer insights into the nature of these materials and ultimately inform the building of industrial devices. This project will also provide laboratory experience and research mentoring opportunities for undergraduate and graduate students.Inhibitory interaction appears in geometric variational problems with connections to phenomena in other branches of mathematics: bubble clusters in geometric measure theory, elliptic curves in analytic number theory, and sphere packing in geometry. The negative fractional and the negative fractional screened Laplace operators provide accurate descriptions of inhibitory interaction and are relevant to other mathematical theories. The first task in this project is to identify the ranges for the fractional exponents in which these operators possess inhibition properties. The second is to construct building blocks for periodic structures in inhibitory systems, such as the non-standard double bubble, and to use them to find periodic stationary points for the whole space. A related question is to find an optimal lattice that gives rise to a periodic configuration with the least free energy density. Building blocks can also be used to forge near periodic stationary assemblies on a general bounded domain.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.