Research in Geometric Analysis

摘要：Krantz提出利用实复分析、微分几何、李理论和偏微分方程等方法研究复空间中域的全纯自映射群（自同构群）。目标是识别不变量，这将有助于分类域到生物全纯等价。在另一个方向上，Krantz将根据之前与Chang和Stein的合作研究域上的实变量Hardy空间。最后，Krantz将与Marco Peloso合作，继续研究Sobolev拓扑中的非齐次Cauchy-Riemann方程。映射理论的应用和蠕虫领域的研究正在发展。Krantz教授将研究与对称性相关的复杂分析。对称的概念是通过域的映射来测量的。空间中一个区域到其自身的这种映射的集合具有称为“群”的代数结构。这个群的性质揭示了所研究区域的几何结构，反之亦然。这种将代数不变量附加到几何对象上的想法是20世纪数学的重大突破之一，并揭示了这一学科的新深度和结构。Krantz对这些“自同构群”的研究补充了他对这些空间区域上某些偏微分方程的持续研究。偏微分方程可以用来构造映射，并计算它们的正则性。作为这两个项目的辅助，Krantz将研究一些函数和映射的空间，这些空间被称为Hardy空间，它们的存在受到映射问题的启发。复变函数理论的几何研究在宇宙学（多亏了彭罗斯的工作）和地球物理学中都有应用。

ABSTRACT:Krantz proposes to study the groups of holomorphic self-mappings(automorphism groups) of domains in complex space using techniques of realand complex analysis, differential geometry, Lie theory, and partialdifferential equations. The goal is to identify invariants that will helpto classify domains up to biholomorphic equivalence. In anotherdirection, Krantz will study real-variable Hardy spaces on domains,pursuant to earlier collaborative work with Chang and Stein. Finally,Krantz will collaborate with Marco Peloso in his continuing study of theinhomogeneous Cauchy-Riemann equations in the Sobolev topology.Applications to mapping theory and the study of the worm domain are beingdeveloped.Professor Krantz will study aspects of complex analysis related tosymmetry. The notion of symmetry is measured by means of mappings ofdomains. The set of such mappings of a region in space to itself hasan algebraic structure which is called a ``group''. The properties ofthis group reveal geometric structure of the region under study andvice versa. This idea of attaching an algebraic invariant to ageometric object is one of the big thrusts in twentieth-centurymathematics, and uncovers new depth and texture in the subject.Krantz's study of these ``automorphism groups'' complements hiscontinuing study of certain partial differential equations on thesespatial regions. The partial differential equations can be used toconstruct mappings, and to calculate their regularity properties. Asan ancillary to these two projects, Krantz will study certain spacesof functions and mappings which are called Hardy spaces, and whoseexistence was inspired by the mapping questions. The geometric studyof complex function theory in several variables has applications incosmology (thanks to work of Penrose) and in geophysics.