Mathematical Sciences: Problems in Modern Analysis

Enflo和Lomonosov教授将探索一些分析领域，包括：(1)涉及数论和的界的数论问题，(2)Banach空间中的非线性问题，包括Banach空间的同态和同构之间的关系，以及(3)不变子空间问题。恩弗洛和洛蒙诺索夫教授将研究的问题非常不同。也许最重要的是他们在不变子空间问题上的联合项目。在有限维空间中，每个线性算子(矩阵)都有一个特征向量，因此这个向量的乘积形成一个不变子空间。这在无限维度中不再适用，但在重要的特殊情况下仍然适用。主要研究者将研究在特殊但重要的无限维空间中，某些算子类在多大程度上具有非平凡不变子空间。

Professors Enflo and Lomonosov will explore a number of areas of analysis including: (1) problems in number theory involving bounds on number theoretic sums, (2) non-linear problems in Banach spaces including the relationship between homeomorphisms and isomorphisms of Banach spaces, and (3) the invariant subspace problem. The problems which Professors Enflo and Lomonosov will study are extremely diverse. Perhaps the most important is their joint project on the invariant subspace problem. In a finite dimensional space, every linear operator (matrix) has an eigenvector and thus multiples of this vector form an invariant subspace. This is no longer true in infinite dimensions but remains true in important special cases. The principal investigators will study to what extent certain classes of operators have nontrivial invariant subspaces in special but important infinite dimensional spaces.