Professor Beer has been involved in the study of topologies on the closed convex subsets of a normed linear space. Such topologies give rise to topologies on convex functions, with a function identified with its epigraph, the set of points on or above its graph. In the last two years, Professor Beer introduced a topology compatible with Mosco convergence of sequences of convex sets, showing that it is stable with respect to duality if and only if the underlying space is reflexive. Several characterizations of the topology were obtained, and continuity properties of a variety of operators and functionals on convex sets were studied. In response to the unsuitability of this Mosco topology in the nonreflexive case, and in view of some prior promising results regarding the norm convergence of linear functionals and their level sets, Professor Beer studied the topology of uniform convergence of distance functionals on bounded sets. Not only is this topology stable with respect to duality without restriction, but it is also the right convergence notion in terms of the stability of solutions of convex optimization problems. Professor Beer will continue his studies on convergence of convex sets, with particular attention to connections with Banach space geometry, approximation of multifunctions, and specific operations and functionals. He also hopes to become involved in applications of set convergence in evolving research in the foundations of nonsmooth analysis, e.g., to certain optimization problems.
