Mathematical Sciences: Theory and Application of Linear Ordinary Differential Equations

Johnson教授将研究系数有界但不衰减的线性微分算子（可能几乎是周期的或遍历过程的路径）。利用动力系统理论的技术，他将研究这些算子的谱性质和广义Floquet指数。他将把他的成果应用于以下领域：非线性动力系统的不变环面分岔；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的拓扑指标理论及其与某些C*代数上同调理论的关系初始数据有界的Korteweg-deVries方程。约翰逊在微分方程方面的研究基础非常广泛，对从纯数学到物理和工程的各个领域都有影响。

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.