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: $ 7.29万
• 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: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=642740
• 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.

分类是数学中的一种哲学，表明迄今为止，在各个领域发展的许多理论只是更深层次，更复杂的阴影。在过去几年的研究中，我一直在遵循这种思维方式，研究了几个密切相关的领域：代数几何（这是对多项式的研究），代表理论（大致，矩阵的研究）和拓扑（对表面，结，结和其他更为复杂的形状的研究）。 通过思考一个层次，这种理念在上述领域中发现了令人惊讶的应用。阴影只会给您一个粗略的（可能是错误的）描述事物的真实本质。发现实际对象可以揭示出惊人和意外的细节。同样，分类揭示了数学中的新，杰出的深层结构。