CAREER: Methods and Outreach in Modern Combinatorics

职业：现代组合学的方法和推广

• 批准号: 0745185
• 负责人: Jozsef Balog
• 金额: $ 52.99万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-02-01 至 2015-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0745185&HistoricalAwards=false
• 关键词: CAREER; Methods; Outreach; Modern; Combinatorics

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.

PI建议进行与他在极端和概率组合学方面的研究兴趣有关的广泛学术和教育项目。在他的研究中，PI大量使用了Szemeredi的规律性引理来解决结构性和枚举问题。这种引理是现代组合技术中最强大的工具之一，它不仅在组合学中，而且在分析和添加数理论中都有应用。引理的原始形式涉及密集的图，但是在许多应用中，人们需要稀疏的引理版本。 PI计划发展稀疏规律的理论，以便将其应用于解决广泛的问题。此外，PI提议研究引导性渗透中的相变，这些渗透与统计物理学（例如ISING模型）中有关。除了说服自己的研究外，研究人员还提出了各种各样的教育和外展活动。在现代科学和技术的许多领域，对复杂离散和组合结构的理解至关重要。 PI坚信，由他的项目产生的新方法和技术将不仅在离散的数学上，而且在其他领域，例如信息理论，计算机科学和统计物理学等其他领域。由于他的地区相对可容纳更多的受众，因此PI提出了许多教育外展活动。他计划组织几个会议，其中包括一个参与者将主要是研究生的会议，以开发课程，为现代组合中的各种强大方法提供温和的介绍，其中包括一项针对本科生，并在自己的项目中指导学生，从而导致了大师和博士学位。