The goal of the activities supported by this grant is to create strong scientific interactions among the centers in Algebraic Geometry and Commutative Algebra that are located at Purdue University, the University of Illinois at Chicago, the University of Illinois at Urbana-Champaign, the University of Kentucky, and the University of Notre Dame. The various research groups at these institutions have compatible interests, but at the same time complementary expertise, and collaborations are already ongoing. Primarily, this goal will be achieved by promoting a series of bi-annual rotating conferences and by supporting the mobility of graduate students, junior faculty members, and visitors. These types of initiatives will help graduate students, postdocs and young faculty broaden their mathematical background and start new collaborations.
Algebraic Geometry and Commutative Algebra have seen a great deal of advances in the past years, most notably through a recent breakthrough in the minimal model program. Particularly fruitful has been the interaction between the two fields, which will also be the focus of the activities supported by this grant. In addition, the advent of high-speed computers and the development of algorithms have led to a line of research in Algebra and Geometry that is close to applications. An area of common interest of the groups at Purdue, UI Chicago, Kentucky, and Notre Dame have been the topics of multiplier ideals, integral closures, and cores. The other areas of common interest among the groups at Purdue, UIC, and UIUC are intersection multiplicities and related problems, and the homological conjectures. Our groups approach these subjects from different points of views, ranging from complex analytic geometry to commutative algebra and combinatorics. This makes an exchange of ideas and the mutual training of graduate students even more crucial. The series of conferences will showcase recent progress in various areas of algebra and geometry, including Hilbert functions and multiplicities, Groebner bases and computational algebra, liaison theory, the homological conjectures, problems in positive and mixed characteristic, tight closure and its interaction with birational geometry, resolution of singularities, Rees algebras, multiplier ideals, cores and Briancon-Skoda type theorems. To this end many of the national and international experts in these areas will be brought together, but there will also be ample opportunities for young researchers to present and discuss their work. It is an implicit expectation that many new fruitful collaborations will emerge.
The goal of this project was to train scientists and to advance research in Commutative Algebra and Algebraic Geometry. These are vibrant mathematical fields with rich and ever evolving connections among them and with other disciplines: Intriguing parallels between algebraic phenomena in prime characteristic and birational geometry are being explored, and maximal Cohen-Macaulay modules have found applications in physics. Algebraic Geometry and Commutative Algebra have seen a great deal of advances in the past years, most notably through a breakthrough in the minimal model program and the solution of the Boji-Soderberg conjecture. Particularly fruitful has been the interaction between the two fields, which was also the focus of the activities supported by this grant. In addition, the advent of high-speed computers and the development of algorithms have led to a line of research in Algebra and Geometry that is close to applications. The Midwest region has a sizable number of established centers in Commutative Algebra and Algebraic Geometry. The activities supported by this grant have created strong scientific interactions among the centers located at Purdue University, the University of Illinois at Chicago, the University of Illinois at Urbana-Champaign, the University of Kentucky, and the University of Notre Dame. Primarily, this goal was achieved by promoting a series of seven major conferences in the region and by supporting the mobility of graduate students and junior scientists through short term exchange visits at neighboring institutions. These types of initiatives have helped graduate students, postdocs, and young faculty broaden their mathematical background, initiate new collaborations, and advance their research. The exposure they received also led to employment opportunities.
