Structural Graph Theory and Additive Combinatorics

我的主要研究领域是结构图论。这是一门深奥的数学学科，与现实世界的问题有着重要的联系。例如，这门学科的一个基本定理是库拉托夫斯基-瓦格纳定理，它精确地描述了哪些图可以在平面上画而不相交。这不仅是一个美丽的数学定理，而且在创建电路板时也是一个重要的概念：可以在电路板的一侧构建的网络正是那些具有这样的绘图的网络。更一般地说，网络是现代世界中无处不在的对象（互联网，社交网络，道路网络等），抽象定理提供了关于其结构的全局信息，可以成为强大的工具。在过去的半个世纪里，结构图理论得到了巨大的发展，该理论的某些方面现在已经得到了很好的确立和理解。值得注意的是，Robertson和Seymour对Kuratowski-Wagner定理进行了一个宏大的（尽管是粗略的）推广，该推广给出了所有不包含特定子结构的图的全局描述，称为禁止子结构。这个深刻的定理在图论中具有深远的影响，与数学中的其他学科有重要的联系，以及以新的图算法的形式应用。虽然带禁小线的图的结构现在已经确立了，但仍有许多重要的途径有待探索。我的研究计划的主要长期目标是建立强大的新结构理论，以改变我们对图，有向图和其他组合对象的理解。一个目标是确定不包含另一种称为禁止浸入的子结构的图的结构。这是罗伯逊-西摩理论的自然类比，可以产生同样广泛的影响。另一个目标是提高我们对小产品集的结构性理解。这是加性组合学中的一门学科，与图论密切相关，但也与数学的许多其他部分有重要的联系。我的短期目标是在三种情况下研究结构理论。首先，我希望证明任意k边连通图G不包含最大k次的固定图H作为浸入式的一个全局结构定理。这是实现禁浸有效结构理论长期目标的重要一步。第二个目标是为群中的集合A，B对建立一个强结构定理，其中|AB| < |A| + |B| + c对于固定常数c。这样的定理将在加性组合学中有很大的影响。最后但并非最不重要的是，我计划通过使用组合群论中的一些工具来研究图的展开。这是对具有小增长的有向图的结构理解的第一步。

My primary research area is structural graph theory. This is a deep mathematical subject with important ties to real world problems. For instance, one of the foundational theorems of this subject is the Kuratowski-Wagner Theorem which characterizes exactly which graphs can be drawn in the plane without crossings. Not only is this a beautiful mathematical theorem in its own right, it is also a vital concept when creating circuit boards: The networks that can be constructed on one side of a circuit board are precisely those which have such a drawing. More generally, networks are ubiquitous objects in the modern world (the internet, social networks, road networks, etc.) and abstract theorems which give global information about their structure can be powerful tools. Structural graph theory has developed tremendously over the past half century, and some aspects of this theory are now quite well established and understood. Notably, there is a grand (albeit rough) generalization of the Kuratowski-Wagner theorem due to Robertson and Seymour which gives a global description of all graphs not containing a certain substructure called a forbidden minor. This deep theorem has profound consequences in graph theory, significant connections to other subjects in mathematics, as well as applications in the form of new graph algorithms. Although the structure of graphs with a forbidden minor is now well-established, there are many important avenues still to explore. The primary long-term goal of my research program is to establish powerful new structure theory to transform our understanding of graphs, directed graphs, and other combinatorial objects. One aim is to determine the structure of a graph not containing another type of substructure called a forbidden immersion. This is a natural analogue of the Robertson-Seymour theory that could have similarly broad based impact. Another aim is to improve our structural understanding of small product sets. This is a subject in additive combinatorics which is intimately related to graph theory, but also has important connections to many other parts of mathematics. My short-term objectives are to pursue structure theory in three settings. First, I hope to prove a global structure theorem for any k-edge-connected graph G not containing a fixed graph H of maximum degree k as an immersion. This is an important step in realizing the long-term goal of an effective structure theory for forbidden immersions. A second goal is to establish a strong structure theorem for pairs of sets A,B in a group for which |AB| < |A| + |B| + c for a fixed constant c. Such a theorem would be highly influential within additive combinatorics. Last but not least, I plan to study expansion in graphs by adapting some tools from combinatorial group theory. This is a first step toward a structural understanding of directed graphs with small growth.