Extremal graph theory, Ramsey theory and additive combinatorics

在我读博士的第一年，我研究了组合数学（也被称为离散数学）中的三个主题。其中一个是极值图论，它是关于在不包含给定子图的图中确定最大边数的问题。第二个是拉姆齐理论。粗略地说，这个领域的重点是寻找图或其补图中的结构。最后，第三个领域是加性组合学，它研究群的子集。（群是可以定义加法的对象，例如整数或实数。）最后一个项目在理论计算机科学中也具有重要意义。与我的导师蒂莫西·高尔斯一起，我们在这三个领域都取得了可发表的成果。在这些项目中大量使用的一种常见技术是所谓的概率方法，这是一种强大的工具，允许我们通过从所有可用对象中随机选择一个对象来查找具有特定属性的对象。在未来，我将致力于解决上述三个领域的进一步开放性问题。

In the first year of my PhD I was working on three topics in combinatorics (also known as discrete mathematics). One of them is extremal graph theory which is concerned with determining the maximal number of edges in a graph which does not contain a given subgraph. The second one is Ramsey theory. Roughly speaking, this field focuses on finding structures in a graph or its complement. Finally, the third area is additive combinatorics, which investigates subsets of groups. (Groups are objects on which addition can be defined, such as the integers or real numbers.) This last project also has significance in theoretical computer science.Together with my supervisor, Timothy Gowers, we have managed to establish publishable results in all three of these areas. A common technique which was used heavily in these projects is the so-called probabilistic method, which is a powerful tool allowing us to find objects with certain properties by randomly picking one object of all the available ones.In the future I will be aiming to solve further open problems in the three areas mentioned above.