Group actions in geometric/arithmetic Combinatorics

该项目旨在将有限群增长理论的结果和工具与最先进的算术和几何组合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联系，由伪随机性的一般主题统一。在Helfgott、Bourgain、Gamburd、Sarnak等人的基础工作之后，这一领域在2000年代开始取得重大进展。这种增长现象似乎与著名的Erdos和szemerzedi的和积猜想有着内在的联系，在过去的15年里，这个猜想已经取得了很大的进展。更具体地说，该项目旨在研究特定的群体家族，例如上三角矩阵，揭示并分类其中构成生长障碍的结构，并在没有它们的情况下建立生长的定量估计。这些障碍物的性质很大程度上取决于场地，矩阵元素来自哪里：分析各种场景来达到这种效果是这个项目的一个特定的新颖特征。在一定程度上，这一问题范围进一步推动了布鲁拉德、格林和陶、吉尔和赫尔夫戈特、墨菲和彼得里迪斯等人早期的研究结果。群的增长，特别是在其研究中产生的能量概念，与几何关联理论的估计直接相关，与这些群在齐次空间中的作用有关。这构成了项目的另一个主线，目前关注的是莫比乌斯双曲线。特别是，其目的是通过使用一套特殊的工具，即群体增长和几何关联理论，来改进布尔金、索利莫西和塔尔多斯、什克雷多夫等人的早期结果。

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.