Geometric Aspects of Algebraic Topology

代数拓扑的几何方面

• 批准号: 0204080
• 负责人: Po Hu
• 金额: $ 8.92万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2002-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204080&HistoricalAwards=false
• 关键词: Geometric; Aspects; Algebraic; Topology

DMS-0204080Po HuIn this project, the investigator intends to consider a circle of ideas in algebraic topology and its interactions with other areas of mathematics. One point of entry to these ideas is Real-oriented homotopy theory, which uses the complex conjugation action on the complex cobordism spectrum to approach homotopy groups of spheres. The method used here is a case of descent, which appears also in Voevodsky's homotopy theory of algebraic varieties. In this direction, the investigator is interested in algebraic cobordism, the algebro-geometric analogue of complex cobordism. Another closely related idea is that of Verdier and Grothendieck dualities, both in algebraic geometry and in equivariant homotopy theory. Yet another related but different kind of duality is Koszul duality. Using this duality, the investigator, jointly with collaborators, proved a version of Kontsevich's conjecture on Hochschild cohomology of k-algebras. This in turn is related to the question of deformation quantazation in mathematcial physics. Another area into which Koszul duality enters is the string topology of Chas and Sullivan, which gives another connection between algebra and homotopy theory. The investigator is also involved with another project related to physics, namely constructing geometric models of elliptic cohomology.Topology is the study of spaces that can be deformedcontinuously. In part, this proposal seeks to better understand maps between such objects by ``stable'' methods, i. e. considering a sequence of objects of all higher dimensions at once, and by using certain summetries upon them. Similar methods can also be used to study more rigid geometric objects, for example in algebraic geometry the solution sets of systems of algebraic equations. This leads in turn to the study of algebraic structures on an abstract level.Finally, the related idea of string topology considers not just a space itself, but structures on the space consisting of loops in it. In particular, this is important to string theory in physics, which sees the universe as composed not of point-like particles but of loop-like ``strings''.

DMS-0204080PO HUIN这个项目，研究人员打算考虑代数拓扑中的思想循环及其与其他数学领域的相互作用。 这些思想的一个进入点是实现的同质理论，该理论使用复杂的共轭作用在复杂的恢复频谱上来接近球形的同置群体。这里使用的方法是下降的情况，它也出现在Voevodsky的代数品种的同质性理论中。在这个方向上，研究者对代数的恢复感兴趣，即复杂恢复的代数几何类似物。另一个密切相关的思想是verdier和Grothendieck二元性，无论是代数几何学和均等同型理论。 Koszul二元性是另一个相关但不同的二元性。使用这种二元性，研究人员与合作者共同证明了Kontsevich关于K-Algebras Hochschild共同体的猜想的版本。反过来，这与数学物理学中变形定量的问题有关。 Koszul二元性进入的另一个领域是Chas和Sullivan的弦拓扑，它在代数和同型理论之间提供了另一种联系。 研究者还参与了与物理学有关的另一个项目，即构建椭圆形的几何模型。topogology是对可以变形的空间的研究。在某种程度上，该提案试图通过``稳定''方法更好地理解此类对象之间的地图。 e。同时考虑所有更高维度的对象，并通过对它们的某些避孕序进行使用。 类似的方法也可以用于研究更多刚性几何对象，例如在代数几何形状中，代数方程系统的解决方案集。这又导致了在抽象层面上研究代数结构的研究。在本文中，弦拓扑的相关思想不仅考虑了一个空间本身，还考虑了其​​中包含循环的空间上的结构。特别是，这对于物理学中的字符串理论很重要，该理论认为宇宙不是像点状的粒子，而是类似循环的``字符串''组成。