Combinatorial aspects of representation theory and algebraic geometry

代数几何的基本方法是通过使用多项式方程对几何对象进行建模来研究几何对象。使用这种方法，可以使用代数中的技术来研究几何对象。相反，用几何思想来研究代数也是可能的。 如果一个人有一些复杂的代数对象，这可能很难理解，我们可以研究它的“表示”。人们可以把研究代数的表示法比作研究人类的X射线。如果你有一个人的一张X光照片，那么你就不能获得关于被X光检查的人的完整信息，但如果你从多个不同的角度进行X光检查，那么你可以开始对这个人形成一个准确的印象。同样，如果我们对一个代数感兴趣，我们会研究它的所有表示，或它们的一些重要子集，以找到关于该代数的更多信息。 在代数几何和表示理论中，涉及的对象可能非常复杂，很难处理。我在这个项目中的计划是使用组合学中的工具来帮助分析这些对象。

The fundamental approach in algebraic geometry is to study geometrical objects by modelling them using polynomial equations. Using this approach, techniques from algebra can be used to study geometrical objects. Conversely, it is also possible to use geometrical ideas to study algebra. If one has some complicated algebraic object, which could be hard to understand, we can study what are called its "representations". One could compare studying the representations of an algebra to studying the X-rays of a human being. If you have a single X-ray of a person, then you cannot obtain complete information about the person who was X-rayed, but if you have X-rays from a number of different angles, then you can begin to form an accurate impression of the human being. In the same way, if we are interested in an algebra, we study all its representations, or some significant subset of them, to find out more about the algebra. In both algebraic geometry and representation theory, the objects involved can be quite complicated and hard to work with. My plan in this project is to use tools from combinatorics to help analyze these objects.