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 Hu 在这个项目中，研究人员打算考虑代数拓扑中的一系列思想及其与其他数学领域的相互作用。 这些思想的切入点之一是面向实数的同伦理论，它利用复配边谱上的复共轭作用来逼近球面的同伦群。这里使用的方法是下降的情况，这也出现在 Voevodsky 的代数簇同伦理论中。在这个方向上，研究者对代数共边（复共边的代数几何类似物）感兴趣。另一个密切相关的想法是代数几何和等变同伦理论中的维尔迪埃和格洛腾迪克对偶性。另一种相关但不同的二元性是科祖尔二元性。利用这种对偶性，研究人员与合作者共同证明了 Kontsevich 关于 k 代数 Hochschild 上同调的猜想的一个版本。这又与数学物理中的形变量子化问题有关。 Koszul 对偶性进入的另一个领域是 Chas 和 Sullivan 的弦拓扑，它给出了代数和同伦理论之间的另一种联系。 研究者还参与了另一个与物理相关的项目，即构建椭圆上同调的几何模型。拓扑学是对可以连续变形的空间的研究。在某种程度上，该提案旨在通过“稳定”方法更好地理解此类对象之间的映射，即。 e.同时考虑一系列所有更高维度的对象，并对它们使用某些概括。 类似的方法也可以用于研究更刚性的几何对象，例如在代数几何中代数方程组的解集。这又导致了对抽象层次上的代数结构的研究。最后，弦拓扑的相关思想不仅考虑空间本身，还考虑由其中的环组成的空间上的结构。这对于物理学中的弦理论尤其重要，弦理论认为宇宙不是由点状粒子组成，而是由环状“弦”组成。