Summer 2004 VIGRE Mini-courses




High School: Math Circle State Math Contest Summer High School Program Undergraduate: GRE Prep Course Undergraduate Colloquium Research Experience for Undergraduates Graduate: Mini-Courses Qualifying Exam Problem Sessions Graduate Fellowships Committees: Steering Internal Advisory External Advisory Outreach Advisory Other Information: People Master Calendar VIGRE Award Publications Undergraduates Graduates Postdocs 04-05 AR		Summer 2004 VIGRE Mini-courses
		The Synthetic Geometry of the Weil-Petersson Metric Jeffrey F. Brock (Brown University) May 10 - 21, 2004 The last few years have seen many new results and broadening interest in the study of the Weil-Petersson metric on Teichmuller space and its metric completion. Improved estimates on the behavior of the metric near infinity have allowed for a deeper understanding of the behavior of geodesics than was previously available. In this mini-course, Dr. Brock gave an overview of the central background results and estimates needed, described recent developments in the large-scale and synthetic geometry of the Weil-Petersson metric, and described an emerging conjectural picture of the behavior of geodesics. Topics included: Wolpert's non-completeness and convexity theorems and Masur's description of the Weil-Petersson completion. The large-scale geometry of the Weil-Petersson metric and the combinatorial geometry of simple closed curves on surfaces. The CAT(0) geometry of the Weil-Petersson completion and the non-refraction of geodesics at the frontier. The Masur-Wolf theorem that the isometry group of the Weil-Petersson metric is the mapping class group. The Weil-Petersson visual sphere: density of cusps, non-continuity of Mod(S), and an ending lamination conjecture for Weil-Petersson geodesics. Geometric limits of Weil-Petersson geodesics on Moduli space and the Weil-Petersson geodesic flow. The idea of the course was to give an introduction containing enough background to understand current open problems in the field, and to provide a starting point for attacking these problems. While background in hyperbolic geometry and Teichmuller theory was very helpful, the discussion were self-contained as possible. Mini-Course Schedule Classical Problems in Commutative Algebra June 7 - 18, 2004 The Department of Mathematics at the University of Utah hosted a two week mini-course on homological conjectures in commutative algebra. During the past 30 years, the homological conjectures and related questions have had a significant impact on the development of commutative algebra. These problems originated in the work of Serre, Auslander, Peskine, Szpiro, and others. In 1974, Mel Hochster gave an overview of these problems in a series of lectures providing answers to some of the questions and indicating further directions of research. Since then, important contributions were made by various experts and some of these conjectures have been solved. However, some of them still await answers. This area of research remains a rich one and is as influential today in the development of commutative algebra as it was decades ago. The aim of the mini-course was to introduce graduate students with some background in commutative algebra and young researchers to this area of research. The lectures provided an introduction to the subject, an overview of the main contributions of the past decades, as well as a discussion of the remaining open questions. The first week developed the fundamental concepts needed in the second week when the major directions of research were discussed. First Week Speakers: Florian Enescu (University of Utah) Dr. Enescu's lecture notes taken by graduate student Bahman Engheta Claudia Miller (Syracuse University) Dr. Miller's lecture notes taken by graduate student H. Ananth Sean Sather-Wagstaff (University of Illinois at Urbana-Champaign) Dr. Sather-Wagstaff's lecture notes taken by graduate student Erin Chamberlain Sandra Spiroff (University of Utah) Dr. Spiroff's lecture notes: Set 1, Set 2, and Set 3 Second Week Speakers: Sankar Dutta (University of Illinois at Urbana-Champaign) Dr. Dutta's lecture notes taken by graduate student Bart Snapp Ray Heitmann (University of Texas) Dr. Heitmann's lecture notes taken by graduate student Hailong Dao Appendix to Dr. Heitmann's notes Melvin Hochster (University of Michigan) Dr. Hochster's lecture notes taken by graduate student Neil Epstein Paul Roberts (University of Utah) Dr. Roberts' lecture notes taken by graduate students Frank Moore and Nick Baeth Jan Strooker (University of Utrecht) Dr. Strooker's lecture notes taken by graduate student Neil Epstein The organizers wish to thank the speakers and participants for their contributions to the mini-course. Drs. Hochster and Strooker's notes have not been revised yet. We will post the revised versions when we have them. Schedules: Week 1 and Week 2 Speakers, References, and Lectures: Week 1 and Week 2 Photos Past mini-courses: 2005 2004 2003 2002

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