Algebra and combinatorics

我的研究是关于被称为“环”的代数结构。我使用图和其他类似的几何结构来寻找环的代数性质，相反，我对环的代数如何影响我使用的对象的几何感兴趣。我经常应用的一种技术是将图关联到某个环，即一组点，其中一些点通过线连接，或者超图，这是图的更高维对应。使用这一技术，我已经能够从代数的角度对与具有所需性质的环相对应的超图进行部分分类。我的目标是在这个方向上进行更深入的研究，并找到更强大的分类技术。这个项目的重要之处在于，对于环来说，许多难以验证的代数性质一旦转化为超图的性质，可能更容易验证。

My research is on algebraic structures called "rings". I use graphs and other such geometric structures to find algebraic properties of rings, and conversely I am interested in how the algebra of rings can affect the geometry of the objects I use. One technique that I apply often is to associate to a certain ring a graph, that is, a set of points some of which are connected by lines, or a hypergraph, which is a higher-dimensional counterpart of a graph. Using this technique I have been able to partially classify hypergraphs that correspond to rings with desirable properties from an algebraic point of view. My goal is to conduct deeper investigations in this direction and to find stronger classification techniques. The importance of this project is that many algebraic properties that are difficult to verify for a ring may be easier to verify once they are translated into properties of hypergraphs.