Mathematical Sciences: Theory and Application of Linear Ordinary Differential Equations

约翰逊教授将研究具有有界但非衰减系数(可能是几乎周期的或遍历过程的路径)的线性微分算子。利用动力系统理论中的技术，他将研究这类算子的谱性质和广义弗洛奎指数。他将在下列领域应用他的结果：非线性动力系统中不变环面的分叉；一维概周期薛定谔算子的谱性质；线性最优控制理论；非周期常微分方程和偏微分方程拓扑指数理论及其与某些C*代数的上同调理论的关系；初值有界的Korteweg-DeVries方程约翰逊对微分方程式的研究基础非常广泛，涉及从纯数学到物理和工程的各种领域。

Professor Johnson will study linear differential operators with bounded but non-decaying coefficients (which might be almost periodic or paths of an ergodic process). Using techniques from dynamical systems theory, he will study spectral properties and generalized Floquet exponents for such operators. He will apply his results in the following areas: bifurcation from invariant tori in non-linear dynamical systems; spectral properties of the one-dimensional almost periodic Schroedinger operator; linear optimal control theory; the topological index theory of non- periodic ordinary and partial differential equations, and its relation with cohomology theory of certain C* algebras; the Korteweg-deVries equation with bounded initial data. Johnson's research in differential equations is very broadly based and has implications for a variety of fields ranging from pure mathematics to physics and engineering.