CAREER: Methods and Outreach in Modern Combinatorics

PI建议开展与他在极值和概率组合学方面的研究兴趣相关的广泛的学术和教育项目。在他的研究中，PI大量使用了Szemeredi的正则引理来解决结构和枚举问题。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，而且在分析和加性数论中也有应用。引理的原始形式涉及密集图，但在许多应用中需要引理的稀疏版本。PI计划发展稀疏规则理论，以便它可以应用于解决广泛的问题。此外，PI建议研究自举渗流中的相变，这与统计物理模型（如Ising模型）有关。除了进行自己的研究外，研究者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。PI相信，从他的项目中产生的新方法和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。由于他的领域相对容易接触到更广泛的受众，PI提出了许多教育推广活动。他计划组织几次会议，其中一次与会者将主要是研究生，以开发课程，对现代组合学中各种强大的方法进行温和的介绍，其中包括针对本科生的课程，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.