Hodge theory and L2-cohomology, Fall 2014

Hodge 理论和 L2-上同调，2014 年秋季

• 批准号: 1449104
• 负责人: Steven Zucker
• 金额: $ 2.48万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-11-01 至 2015-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1449104&HistoricalAwards=false
• 关键词: Hodge; theory; L2; cohomology; Fall

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/.

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-上同调"会议。这次会议将汇集两个主题的研究人员，这两个主题在历史上有着密切的关系，但现在往往被视为无关。 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他们的演讲中提供一些临时材料。我们提出的问题会议将为这些数学家（和其他人）提供一个在该领域工作的焦点。 组织者打算制作一个会议记录，其中包含一些简要的文章，更具体地说，上面提到的两个主题是霍奇理论和L2-上同调。霍奇定理的第一个"现代"证明，断言在紧致黎曼流形上的（真实的）上同调由调和形式表示，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（特别是光滑射影流形）的Hodge分解如下。 随后，研究了非紧流形的L2-上同调，其中前者是度量依赖的。 一个基本的问题是L2-上同调是否具有拓扑意义，就像在紧情形下一样;一个典型的例子是1980年的Zucker猜想（1987年证明）。 在相反的方向，人们可以问一个给定的拓扑上同调群是否由L2-上同调给出，例如，奇格-戈雷斯基-麦克弗森猜想的诞生 在这两个例子中，拓扑对象是（中）交上同调。另一方面，术语“霍奇理论”现在指的是霍奇分解（或霍奇结构）产生的任何东西。这涵盖了大量的数学领域。 会议将在很小程度上帮助与会者更好地了解这一主题的根源。 然而，它将提供更多的期待。 例如，复射影簇的交上同调带有一个“几何”霍奇结构（Mo。Saito，1988年; de Cataldo，2010），但不知道来自L2-上同调的一个是否与它重合（如果在上面的第二个例子中有一个）。与Dolbeault L2-上同调的联系需要处理。此外，地形的作用可以扩展到Lp条件，p不同于2。 PI预计许多谈判将沿着这些路线进行。 会议应促进来自不同群体的人们之间的合作，从而开辟新的研究方向。有关会议的更多细节可以在www.example.com上找到。