Unsolved nonstandard problems

In the fall of 1990 a small colloquium on nonstandard analysis was arranged at the request of a group of graduate and postgraduate students of Novosibirsk State University. At the meetings many unsolved problems were formulated stemming from various branches of analysis and seemingly deserving attention of the novices of nonstandard analysis. In 1994 some discussion took place on combining nonstandard methods at the international conference «Interaction Between Functional Analysis, Harmonic Analysis and Probability» (Missouri University, Columbia USA). The same topics were submitted to the international conference «Analysis and Logic» held in Belgium in 1997. In 1998 an INTAS research project was submitted. The problems raised in the framework of these projects are the core of this article. The list of the problems contains not only simple questions for drill but also topics for serious research intended mostly at the graduate and post graduate level. Some problems need creative thought to clarify and specify them.