Many processes in life, ranging from credit card transactions to the growth of a sunflower, are modeled by systems of polynomial equations. Algebraic geometry studies solutions of such systems. A major feature of these systems is that they vary in families by varying the coefficients of the polynomials. Some equations in the family are easier to solve, and properties of more complicated systems can be deduced from the solutions of the simpler systems. The investigators study the geometry of certain spaces defined by polynomial equations that are ubiquitous in mathematics and physics, called moduli spaces of vector bundles. They compute geometric invariants of these spaces by relating them to simpler spaces using a recent breakthrough called Bridgeland stability. The investigators are also dedicated to training the next generation of U.S. scientists and researchers. In this project, they will train undergraduate, graduate, and postdoctoral researchers to use the new technique of Bridgeland stability. The Focused Research Group grant will support these young researchers to visit and collaborate with several senior researchers and to attend conferences and workshops on the topic. The investigators will also organize two large conferences and four workshops to help attract young talent to the area.
Moduli spaces of vector bundles are fundamental objects in algebraic geometry, with applications to commutative algebra, representation theory, combinatorics, and mathematical physics. In the last five years, Bridgeland stability conditions have revolutionized the understanding of moduli spaces of vector bundles on surfaces. They have allowed the computation of the ample and effective cones of divisors on these moduli spaces and led to the solution of longstanding problems such as the existence of Lagrangian fibrations on certain hyperkähler manifolds of K3 type and the higher rank interpolation problem for general sheaves on the plane. It is timely to apply these new techniques to central problems in the geometry of moduli spaces of vector bundles on surfaces and threefolds. This Focused Research Group project centers on three lines of inquiry: (1) Prove cohomology vanishing results using Bridgeland stability and consequently construct Ulrich bundles on surfaces and threefolds and effective Brill-Noether divisors on moduli spaces of vector bundles on surfaces. Give applications to Le Potier's Strange Duality Conjecture. (2) Determine when special bundles, such as Lazarsfeld-Mukai bundles or null-correlation bundles on surfaces and threefolds, are Bridgeland stable. Apply the stability to classical problems on syzygies and Koszul cohomology. (3) Study the birational geometry of moduli spaces of Bridgeland stable objects via wall-crossing. The investigators plan to train ten undergraduates, ten graduate students, and seven postdoctoral associates through research involvement in the project.
