Combinatorial Approaches to Deformation and Degeneration in Algebraic Geometry

代数几何是研究多项式方程组的解集。这样的集合，称为代数簇，出现在从理论物理和计算生物学到计算机科学和统计学的许多领域。一般来说，了解品种的结构是一个极具挑战性的问题。代数几何中的一个常见技术是将一个变量退化或变形为一个结构更容易理解的变量。我的研究计划的首要目标是通过变形和退化的研究获得新的数学见解代数几何。在这个程序中，我将广泛使用一类特殊的几何对象，称为复曲面簇。它们是用特别简单的方程定义的，使它们更容易详细理解。我建议的研究的第一个主题领域是研究复曲面退化。通过将任意簇退化为复曲面簇，人们可以利用从复曲面簇获得的容易获得的信息来获得关于原始簇的知识。然而，这样的退化一般可能不存在。我将研究如何明确地构建这种复曲面退化，以及在什么情况下人们可以期望它们存在。这是一个非常活跃的话题，有许多微妙的问题和特征。我提出的研究的第二个主题领域是组合变形理论。我将研究复曲面变种和其他具有类似组合结构的变种如何变形为其他变种。这与第一个专题领域的观点相反。变形问题的显式解往往很难得到。这激发了对具有额外结构的变形问题的研究。由于与代数几何中的其他问题的联系，这些问题本身往往具有重要的意义。该研究计划将提供纯数学，特别是代数几何的基本见解。拟议的研究将提供工具来解决数学中的基本问题，如镜像对称，代数几何中的分类问题，以及极值度量的存在。我的研究将进一步推进应用代数几何领域，该领域在生物学，化学和统计学等各种学科中都有应用。最后，拟议研究的影响将有助于加强加拿大在纯数学研究中的领导地位，并将有助于培养加拿大新一代的数学家。

Algebraic geometry is the study of solution sets of systems of polynomial equations. Such sets, called algebraic varieties, appear in connection to numerous fields ranging from theoretical physics and computational biology to computer science and statistics. In general, it is an extremely challenging problem to understand the structure of varieties. A common technique in algebraic geometry is to degenerate or deform a variety to one whose structure is easier to understand. The overarching goal of my program of research is to gain new mathematical insight into algebraic geometry through the study of deformation and degeneration. In this program, I will make extensive use of a special class of geometric objects called toric varieties. They are defined using especially simple equations, making them much easier to understand in detail. The first thematic area of my proposed research is the study of toric degenerations. By degenerating an arbitrary variety to a toric variety, one can use the easily accessible information obtained from the toric variety to gain knowledge about the original variety. However, such a degeneration may not exist in general. I will investigate how to explicitly construct such toric degenerations, and under what circumstances one can expect them to exist. This is a highly active topic, with many subtle questions and features. The second thematic area of my proposed research is combinatorial deformation theory. I will study how toric varieties, and other varieties with similar combinatorial structure, can be deformed into other varieties. This takes the opposite perspective of the first thematic area. Explicit solutions to deformation problems are often difficult to obtain. This motivates the study of deformation problems with extra structure. Such problems are often of significant interest in their own right, due to connections to other problems in algebraic geometry. This program of research will provide fundamental insights in pure mathematics, in particular, algebraic geometry. The proposed research will provide tools to address fundamental problems in mathematics, such as mirror symmetry, classification problems in algebraic geometry, and the existence of extremal metrics. My research will additionally advance the field of applied algebraic geometry, which has applications in a variety of disciplines such as biology, chemistry, and statistics. Finally, the impact of the proposed research will contribute to strengthening Canada's leadership in the study of pure mathematics, and will serve to help train a new generation of mathematicians in Canada.