Group actions in geometric/arithmetic Combinatorics

这个项目寻求将有限群增长理论的结果和工具与算术和几何组合学的最新方法进一步结合起来。这是一个现代研究领域，处于纯数学的十字路口，与计算机科学、编码和复杂性理论有关，由伪随机性这一总主题统一起来。这一领域的重大进展始于本世纪头十年，在赫尔夫戈特的基础性工作之后，随后是布尔加、甘伯德、萨纳克和其他人。这种增长现象似乎与鄂尔多斯和Szemerédi著名的和积猜想有内在联系，在过去15年里，这一猜想已经取得了很大进展。更具体地说，该项目旨在研究特定群体家庭，如上三角矩阵的家庭，发现并分类其中阻碍增长的结构，并在没有这些结构的情况下建立增长的量化估计。这些障碍的性质在很大程度上取决于领域，矩阵元素从何而来：分析各种影响这一影响的情景是该项目的一个具体的新特点。部分地，这一问题范围进一步推进了Breuillard、Green和Tao、Gill和Helfget、Murphy和Petridis等人的早期结果。群的增长，特别是在它的研究中产生的能量的概念，直接与几何关联理论估计有关，这些估计与这些群在齐次空间上的作用有关。这是该项目的另一条主线，目前的重点是莫比乌斯双曲线。特别是，目的是通过使用群增长和几何关联理论中的一组特殊工具来改进Bourain、Solymosi和Tardos、Shkredov等人的早期结果。

This project seeks to further and combine results and tools from the theory of growth in finite groups with state of the art methods of arithmetic and geometric combinatorics. This is a modern area of research at the crossroads of pure mathematics, with connections to computer science and coding and complexity theory, unified by the general theme of pseudorandomness. A significant progress in this area began in the 2000s after foundational work of Helfgott, followed by Bourgain, Gamburd, Sarnak, and others. The growth phenomenon appears to be inherently connected with the renown Sum-Product conjecture of Erdos and Szemerédi, towards which there has been a lot of progress in the past 15 years. More specifically the project aims to look at specific groups families, such as those of upper-triangular matrices, uncover and categorise the structures therein that pose obstruction to growth and establish quantitative estimates for growth in their absence. The nature of these obstructions much depends on the field, where the matrix elements come from: analysing various scenarios to this effect is a specific novel feature of this project. Partially this scope of questions furthers the earlier results by Breuillard, Green and Tao, Gill and Helfgot, Murphy and Petridis and others. Growth in groups, and especially the concept of energy arising in its study are immediately related to geometric incidence theory estimates, arising in connection of these groups' action son homogeneous spaces. This constitutes the other thread of the project, currently focusing on the Mobius hyperbolae. The aim, in particular, is to improve on earlier results due to to Bourgain, Solymosi and Tardos, Shkredov and others by using a special set of tools both from growth in groups and geometric incidence theory.