Research in Geometric Analysis

点击翻译按钮获取中文摘要

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.