Geometric Aspects of Algebraic Topology

在这个项目中，研究者打算考虑代数拓扑中的一圈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实数取向同伦理论，它利用复配谱上的复共轭作用来逼近球的同伦群。这里使用的方法是一个下降的例子，它也出现在Voevodsky的代数变异同伦理论中。在这个方向上，研究者感兴趣的是代数共体，复共体的代数-几何模拟。另一个密切相关的思想是在代数几何和等变同伦理论中的Verdier和Grothendieck对偶。另一种相关但不同的对偶是科祖尔对偶。利用这种对偶性，研究者和合作者证明了Kontsevich关于k-代数的Hochschild上同调猜想的一个版本。这又与数学物理中的变形量子化问题有关。Koszul对偶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个联系。研究者还参与了另一个与物理相关的项目，即建立椭圆上同调的几何模型。拓扑学研究的是可以连续变形的空间。在某种程度上，这个提议试图通过“稳定”的方法来更好地理解这些对象之间的映射，即同时考虑所有更高维度的对象序列，并通过对它们使用某些求和。类似的方法也可用于研究更刚性的几何对象，例如在代数几何中研究代数方程组的解集。这反过来又导致了在抽象层次上对代数结构的研究。最后，弦拓扑的相关思想不仅考虑空间本身，而且考虑空间上由环路组成的结构。特别是，这对物理学中的弦理论很重要，弦理论认为宇宙不是由点状粒子组成的，而是由环状的“弦”组成的。

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''.