Tropical Singularities

在热带几何中，代数簇退化为称为热带簇的多面体复合体。尽管有退化，代数簇的许多性质可以从热带对应的代数簇中读出，许多定理继续适用于热带方面。因此，热带几何提供了一种通过离散数学和组合学来研究代数几何问题的方法。热带几何已被成功地应用于计算问题和计数几何中的问题，特别是实计数问题。要真正利用好这一新工具，必须回答许多关于翻译性质的问题。这个项目的中心问题是：对热带世界来说，奇点概念的正确翻译是什么？这个问题是很自然地提出的，因此最近引起了几位作者的兴趣([3]、[4]、[13])。这个问题很难笼统地回答，这让它变得更加耐人寻味。这个项目的目的是显著提高对热带奇点的理解，将结果应用于计数几何中的问题，并调查这一主题与组合学中的问题的联系。

In tropical geometry, algebraic varieties are degenerated to polyhedral complexes called tropical varieties. In spite of the degeneration, many properties of an algebraic variety can be read off its tropical counterpart, and many theorems continue to hold on the tropical side. Tropical geometry thus provides an approach to study questions in algebraic geometry by means of discrete mathematics and combinatorics. Tropical geometry has been successfully applied to computational problems and problems in enumerative geometry, in particular real enumerative questions. To really make good use of the new tool, many questions of translating nature have to be answered. The central question for this project is: what is the correct translation of the concept of a singularity to the tropical world? This question is quite natural to ask and has consequently interested several authors recently ([3], [4], [13]). The fact that it is hard to answer in general makes it even more intriguing. The aim of this project is to considerably improve the understanding of tropical singularities, to apply the results to questions in enumerative geometry, and to investigate the connections of this subject to questions in combinatorics.