Geometric hypoelliptic Laplacian and orbital integrals (after Bismut, Lebeau, and Shen)

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2017-04-26 DOI:10.24033/ast.1068

X. Ma

引用次数: 7

Abstract

About 15 years ago, Bismut gave a natural construction of a Hodge theory for a hypoelliptic Laplacian acting on the total space of the cotangent bundle of a Riemannian manifold. This operator interpolates between the classical elliptic Laplacian on the base and the generator of the geodesic flow. We will describe recent developments of the theory of hypoelliptic Laplacians, in particular the explicit formula obtained by Bismut for orbital integrals and the recent solution by Shen of Fried's conjecture (dating back to 1986) for locally symmetric spaces. The conjecture predicts the equality of the analytic torsion and the value at 0 of the dynamic zeta function.

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几何准椭圆拉普拉斯积分和轨道积分(继Bismut、Lebeau和Shen之后)

大约在15年前，Bismut给出了作用于黎曼流形的协切束的总空间的准椭圆拉普拉斯算子的Hodge理论的自然构造。该算子在基上的经典椭圆拉普拉斯算子和测地线流的生成算子之间进行插值。我们将描述准椭圆拉普拉斯算子理论的最新发展，特别是Bismut关于轨道积分的显式公式和Shen关于局部对称空间的Fried猜想(追溯到1986年)的最新解。该猜想预测了解析扭转的相等性和动态zeta函数在0处的值。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.