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.
