Hodge theory and L2-cohomology, Fall 2014

该提案是为了支持将于2014年11月21日至23日在约翰霍普金斯大学举行的“霍奇理论和l2 -上同源性”会议。这次会议将把两个学科的研究人员聚集在一起，这两个学科在历史上有着密切的关系，但现在往往被认为是不相关的。会议的主题之一将是宣传和建立他们之间的新联系。pi将试图吸引年轻的数学家（最近获得博士学位）和研究生，以及代表性不足的群体的成员。因此，他们会鼓励演讲者在演讲中提供一些说明性的材料。我们提出的问题会议将为这些数学家（和其他人）在该领域的工作提供一个重点。组织者打算出版一份会议论文集，其中包括一些说明性文章。更具体地说，上面提到的两个主题是Hodge理论和l2 -上同论。Hodge定理的第一个“现代”证明，断言紧黎曼流形上的（实）上同调用调和形式表示，使用L2方法（M. Gaffney, 1955）；流形的de Rham上同调是它的l2上同调。复Kaehler流形（特别是光滑投影流形）的Hodge分解如下。随后，研究了非紧流形的l2 -上同调，其中l2 -上同调是度量相关的。一个基本的问题是l2上同调是否具有拓扑意义，就像在紧化情况下一样；一个典型的例子是1980年的扎克猜想（在1987年被证明）。在相反的方向上，我们可以问一个给定的拓扑上同群是否由l2 -上同给出，例如大致同一时间的Cheeger-Goresky-MacPherson猜想。在这两个给定的实例中，拓扑对象是（中）交上同调。另一方面，术语“霍奇理论”现在指的是由霍奇分解（或霍奇结构）产生的任何东西。这涉及了大量的数学领域。这次会议的一个小小的目的是让与会者更好地了解这个问题的根源。然而，它将提供更多的展望未来。例如，复射射角的交上同调带有“几何”Hodge结构（Mo. Saito, 1988; de Cataldo, 2010），但不知道来自l2上同调的交上同调是否与之重合（如果在上述第二个例子中存在交集的话）。需要处理与Dolbeault l2 -上同的联系。此外，地形可以扩展到Lp条件对p不同于2的作用。保守党预计，许多谈判将沿着这条路线进行。会议应该促进来自不同群体的人之间的合作，从而开辟新的研究方向。有关会议的更多细节，请访问http://www.mathematics.jhu.edu/new/hodgetheoryconf/。

The proposal is for support for a conference, "Hodge Theory and L2-Cohomology", to be held at Johns Hopkins University, November 21-23, 2014. The conference will bring together researchers in two subjects that have a close relationship historically, yet which tend to be viewed as unrelated nowadays. One of the themes of the conference will be to advertise and establish new connection between them. The PIs will try to attract younger mathematicians (with a recent Ph.D.) and graduate students, as well as members of under-represented groups. Thus, they will encourage the speakers to give some expository material in their talks. Our proposed problem session(s) will give these mathematicians (and others) a point of focus for working in the field. The organizers intend to produce a conference proceedings that contains some expository articles.To be more specific, the two subjects mentioned above are Hodge theory and L2-cohomology. The first "modern" proof of the Hodge theorem, asserting that on a compact Riemannian manifold the (real) cohomology is represented by harmonic forms, used L2 methods (M. Gaffney, 1955); the de Rham cohomology of the manifold is its L2-cohomology. The Hodge decomposition for complex Kaehler manifolds (in particular, for smooth projective varieties) follows. Subsequently, L2-cohomology was investigated when the manifold is non-compact, where the former is metric dependent. A fundamental question is whether the L2-cohomology has a topological meaning, as it does in the compact case; a typical instance of this was the Zucker conjecture from 1980 (proved in 1987). In the opposite direction, one can ask whether a given topological cohomology group is given by L2-cohomology, e.g., the Cheeger-Goresky-MacPherson conjecture from roughly the same time. In both given instances, the topological object is (middle) intersection cohomology. On the other hand, the term "Hodge theory" now refers to anything emanating from the Hodge decomposition (or Hodge structure). This covers a vast amount of mathematical turf. The conference will serve, in a small part, to have the participants better understand the subject's roots. However, it will offer much more in that which looks forward. For instance, the intersection cohomology of a complex projective variety carries a "geometric" Hodge structure (Mo. Saito, 1988; de Cataldo, 2010), yet it is not known whether the one coming from L2-cohomology coincides with it (if there is one at all in the second instance above). The connection with Dolbeault L2-cohomology needs to be treated. Moreover, the terrain can be expanded to the role of Lp conditions for p different from 2. The PIs expect many of the talks to proceed along these lines. The conference should promote collaboration between people from the different groups, thereby opening new directions in research. More details about the conference can be found at http://www.mathematics.jhu.edu/new/hodgetheoryconf/.