Algebraic Geometry and Geometric Representation Theory

代数几何与几何表示论

• 批准号: 1000229452-2013
• 负责人: Cautis, Sabin
• 金额: $ 8.74万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Canada Research Chairs
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=646993
• 关键词: Algebraic; Geometry; Geometric; Representation; Theory

Categorification is a philosophy in mathematics that suggests that much of the theory developed so far in a variety of areas is just a shadow of something deeper and more complex. In my research over the last few years I have been following this line of thinking to study several closely related fields: algebraic geometry (which is the study of polynomials), representation theory (roughly, the study of matrices) and topology (the study of surfaces, knots and other more complicated shapes).*By thinking one level higher this philosophy has found surprising applications in the fields mentioned above. A shadow gives you only a rough (and possibly wrong) depiction of the real nature of things. Discovering the actual object can reveal amazing and unexpected detail. In the same way, categorification has revealed new, remarkable, deep structures in mathematics.

范畴化是数学中的一种哲学，它表明迄今为止在各个领域发展起来的大部分理论只是更深层次和更复杂的东西的影子。在我过去几年的研究中，我一直遵循这一思路，研究几个密切相关的领域：代数几何（研究多项式）、表示论（粗略地说，研究矩阵）和拓扑学（研究曲面、结点和其他更复杂的形状）。通过思考一个更高的层次，这种哲学在上述领域中发现了令人惊讶的应用。阴影只给你一个粗略的（可能是错误的）描绘事物的真实的本质。发现实际的物体可以揭示惊人的和意想不到的细节。同样，范畴化揭示了数学中新的、显著的、深刻的结构。