CAREER: Methods and Outreach in Modern Combinatorics

他建议开展广泛的学术和教育项目，与他在极值和概率组合学方面的研究兴趣有关。在他的研究中，PI大量使用了Szmeredi的正则性引理来处理结构和计数问题。这个引理是现代组合学中最有力的工具之一，它不仅在组合学中有应用，而且在分析和加法数论中也有应用。引理的原始形式涉及稠密图，但在许多应用中，人们需要引理的稀疏版本。PI计划发展稀疏正则性理论，以便它可以应用于解决广泛的问题。此外，PI建议研究Bootstrap渗流中的相变，这与统计物理中的模型有关，如Ising模型。除了进行自己的研究外，这位研究人员还提出了各种各样的教育和推广活动。对复杂的离散和组合结构的理解在现代科学技术的许多领域都是至关重要的。国际和平研究所相信，从他的项目中产生的新方法和新技术将导致进一步的发展，不仅在离散数学，而且在其他领域，如信息论，计算机科学和统计物理。由于他所在的地区对更广泛的受众来说是相对容易接近的，PI提出了许多教育外展活动。他计划组织几次会议，其中一次的参与者将主要是研究生，以开发课程，提供对现代组合学中各种强大方法的温和介绍，包括为本科生举办的一次会议，并指导学生完成自己的项目，最终获得硕士和博士学位。

The PI proposes to carry out a wide range of scholarly and educational projects related to his research interests in extremal and probabilistic combinatorics. In his research, the PI makes substantial use of Szemeredi's Regularity Lemma to tackle structural and enumerative problems. This Lemma is one of the most powerful tools in modern combinatorics, it has applications not only in combinatorics but in analysis and additive number theory as well. The original form of the Lemma concerns dense graphs, but in many applications one would need a sparse version of the Lemma. The PI plans to develop the theory of sparse regularity, so that it might be applied to solve a wide range of problems. Furthermore, the PI proposes to study phase transitions in bootstrap percolation, which are related to models in statistical physics, such as the Ising model. In addition to persuing his own research, the investigator proposes a great variety of educational and outreach activities.The understanding of complex discrete and combinatorial structures is crucial in many areas of modern science and technology. The PI is convinced that the new approaches and techniques resulting from his project will lead to further developments, not only in Discrete Mathematics, but also in other fields such as Information Theory, Computer Science and Statistical Physics. Since his area is relatively accessible to a wider audience, the PI proposes numerous educational outreach activities. He plans to organize several conferences, including one whose participants will mainly be graduate students, to develop courses providing a gentle introduction to a variety of powerful methods in modern Combinatorics, including one for undergraduates, and mentoring students on their own projects, leading to Masters and Ph.D degrees.