Complex Analysis and Geometry

PI将研究辛几何中的一些问题（奇异环空间和退化的模），Akhiezer-Gindikin域（有界实现问题），Hodge理论中Beauville类的可实现性，以及渐近复双曲流形的刚性和一些新的极值环的例子。不那么技术性地说，PI将在复杂几何和分析中密切相关的数学领域开展几个研究项目。这些领域直接继承了笛卡尔关于方程解集的几何或形状的研究。这些研究的动机主要是几何的，并且大多数使用分析，微积分的后裔。一个项目涉及与经典物理中的特殊机械系统（哈密顿动力学和可积系统）相关的几何，另一个项目间接涉及与量子物理相关的复杂几何，特别是弦理论，这是一种高度推测但在最小尺度上具有几何吸引力的基本力理论。其中两个子项目涉及所谓的空间增长，当一个人在空间中向地平线移动时。在这个尺度上可见的性质被称为渐近性质，PI将研究某些局部几何条件是否会导致这些空间的遥远区域的规则模式行为。这些项目涉及一种刚性：如果对空间的增长或复杂性进行适度的控制，当它延伸到它的地平线时，那么它将“冻结”成一个非常具体和确定的形式。最后，其中一个子项目涉及更多的代数问题。这意味着最终它们是关于多项式的问题，而不是更复杂的函数。它们处理的是由方程从简单空间中分割出来的特殊空间，这些空间与固定参考空间有极大交集。我们能从这个极大交性质中得出确切的几何形状吗？关键是要使用几何微分方程，与源自力学的微分方程没有什么不同。这些项目大部分将由年轻的研究人员、PI的同事或以前的学生进行。希望它们将有助于培养这些年轻数学家的研究兴趣。在另一个方向上，PI也希望进一步发展他在分子生物学和遗传学方面的兴趣，与学生一起研究项目，以了解遗传分子所携带信息的数学本质，尽管这项拨款申请并不为这些活动寻求资助，除了几个本科生可能在数学和生物学的这些方面有暑期研究经验。

The PI will work on some problems in symplectic geometry (moduli of singular toric spaces and degenerations), Akhiezer-Gindikin domains (bounded realization problem), realizability of the Beauville class in Hodge theory, and the rigidity of asymptotically complex hyperbolic manifolds and of some new examples of extremal cycles in flag varieties.Put less technically, the PI will carry out several research projects in closely related areas of mathematics in complex geometry and analysis. These are areas which are direct descendants of the work of Descartes on the geometry or shape of solution sets of equations. The motivations for these studies are primarily geometric, and most employ analysis, the descendent of the calculus. One project deals with geometry related to that of special mechanical systems in classical physics (Hamiltonian dynamics and integrable systems), and one deals obliquely with complex geometry related to quantum physics, specifically string theory, a highly speculative but geometrically attractive theory of elementary forces on the smallest scales. Two of the subprojects deal with what is called the growth of spaces as one travels far out in them towards the horizon. Properties that are visible at this scale are known as asymptotic properties, and the PI will study whether certain local geometric conditions can lead to regular patterned behavior in distant regions of these spaces. These projects refer to a kind of rigidity: if a modest amount of control over the growth or complexity of a space as it stretches out to its horizon, then it will ``freeze'' into a very specific and determined form. Finally, one of the subprojects deals with more algebraic problems. This means that ultimately they are questions about polynomials rather than more complicated functions. They deal with special spaces which are cut out of simpler spaces by equations, and which have maximal intersection with fixed reference spaces. Can one conclude the exact geometry from this maximal intersection property? The point is to use geometric differential equations, not dissimilar to the ones originating in mechanics. Most of these projects will be carried out with young researchers, colleagues or former students of the PI. Hopefully, they will serve to develop the research interests of these young mathematicians. In another direction, the PI hopes as well to further his interests in molecular biology and genetics, working with students on projects to understand the mathematical nature of the information carried by genetic molecules, though this grant application does not seek funding for such activities beyond a couple of undergraduate students who might have a summer research experience on such aspects of math and biology.