Midwest Algebra, Geometry and their Interactions Conference MAGIC05 University of Notre Dame, Notre Dame October 7-11, 2005 Tight closure and positivity by Karen E. SMITH, University of Michigan Abstract: I'll survey the development of three different aspects of tight closure theory in recent years, all of which touch on some way with issues of positivity in algebraic geometry. These are first, globally F-regular varieties, which are in some sense 'positively curved;' second, singularities and test ideals, which correspond to multiplier ideals; and third, the question of when an element is in tight closure of an ideal, which has to do with positivity of certain vector bundles.