Complex Analysis and Geometry

PI将研究辛几何中的一些问题（奇异环面空间的模与退化），Akhiezer-Gindikin域（有界实现问题），霍奇理论中Beauville类的可实现性，渐近复双曲流形的刚性以及旗簇中极值循环的一些新例子。PI将在复杂几何和分析中的数学密切相关的领域进行几个研究项目。这些领域是直接后裔的工作笛卡尔的几何或形状的解决方案套方程。这些研究的动机主要是几何，大多数采用分析，微积分的后代。一个项目涉及与经典物理学中的特殊力学系统（哈密顿动力学和可积系统）相关的几何学，另一个项目涉及与量子物理学相关的复杂几何学，特别是弦理论，这是一种高度投机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，因为人们在其中向地平线旅行。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。这些项目指的是一种刚性：如果对空间的增长或复杂性进行适度的控制，因为它延伸到它的地平线，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。他们处理特殊的空间是削减了较简单的空间的方程，并有最大的交集与固定的参考空间。人们能从这个最大交性质得出精确的几何吗？关键是要使用几何微分方程，这与力学中的微分方程没有什么不同。这些项目中的大多数将与年轻的研究人员，同事或PI的前学生一起进行。希望他们将有助于发展这些年轻数学家的研究兴趣。在另一个方向上，PI也希望进一步提高他对分子生物学和遗传学的兴趣，与学生一起开展项目，以了解遗传分子所携带信息的数学性质，尽管这项拨款申请并不寻求资助这些活动，除了几个本科生，他们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.