Structural Graph Theory and Additive Combinatorics

我的主要研究领域是结构图理论。这是一门深奥的数学科目，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Kuratowski-Wagner定理，它准确地描述了哪些图可以在没有交叉点的平面上绘制。这不仅本身就是一个美丽的数学定理，而且在创建电路板时也是一个至关重要的概念：可以在电路板的一侧构建的网络就是那些拥有这样一张图的网络。更广泛地说，网络是现代世界中无处不在的对象(互联网、社交网络、公路网等)。而抽象定理可以提供关于它们结构的全局信息，这可能是强大的工具。结构图理论在过去的半个世纪中得到了巨大的发展，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，由于Robertson和Seymour，Kuratowski-Wagner定理得到了一个宏大的(尽管粗略的)推广，它给出了所有不包含称为禁止子结构的特定子结构的图的全局描述。这一深刻的定理在图论中有深远的影响，与数学中的其他学科有重要的联系，以及以新的图形算法的形式应用。尽管带有禁止子图的图的结构现在已经很好地建立起来了，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。一个目的是确定一个图的结构，该图不包含另一种称为禁止浸没的子结构。这是罗伯逊-西摩理论的自然类比，也可能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。这是加法组合学中的一个主题，它与图论密切相关，但也与数学的许多其他部分有重要联系。我的短期目标是在三个背景下追求结构理论。首先，我希望证明任何不包含最大度为k的固定图H的k边连通图G的整体结构定理。这是实现禁止沉浸的有效结构理论长期目标的重要一步。第二个目标是建立|AB|&lt；|A||B|c为固定常数的群中集合A，B对的强结构定理。这样的定理在加性组合学中具有很大的影响力。最后但并非最不重要的一点是，我打算采用组合群论中的一些工具来研究图的扩张。这是向具有小增长的有向图的结构理解迈出的第一步。

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.