The AB Program in Geometric Analysis - Sharp Sobolev Inequalities and Related Problems
Function theory and Sobolev inequalities have been the target of investigation for decades. Sharp constants in these inequalities constitute a critical tool in geometric analysis. The $AB$ program is concerned with sharp Sobolev inequalities on compact Riemannian manifolds. Important and significant progress has been made during recent years. We summarize the present state and describe new results.