Geometric Aspects of Algebraic Topology

DMS-0204080Po Huu在这个项目中，研究人员打算考虑代数拓扑学中的一系列思想及其与其他数学领域的相互作用。这些思想的一个切入点是实定向同伦理论，它利用复余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，这也出现在沃沃茨基的代数簇的同伦理论中。在这个方向上，研究者对代数余边法感兴趣，代数余边法是复余边法的代数几何类比。另一个密切相关的概念是Verdier和Grothendieck对偶，在代数几何和等变同伦理论中都是如此。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，研究者与合作者一起证明了康采维奇关于k-代数的Hochschild上同调的猜想的一个版本。这反过来又与数学物理中的形变量子化问题有关。Koszul对偶进入的另一个领域是查斯和Sullivan的弦拓扑，它给出了代数和同伦理论之间的另一种联系。这位研究人员还参与了另一个与物理有关的项目，即构造椭圆上同调的几何模型。拓扑学是研究可以连续变形的空间。在某种程度上，这项建议试图通过“稳定”的方法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的序列，并对它们使用某些求和。类似的方法也可以用来研究更严格的几何对象，例如在代数几何中，代数方程组的解集。最后，弦拓扑的相关概念不仅考虑了空间本身，还考虑了空间中由环组成的空间上的结构。这对物理学中的弦理论尤其重要，该理论认为宇宙不是由点状粒子组成，而是由环状“弦”组成。

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''.