Bridgeland Stability, Moduli Spaces, and Applications

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Polynomial systems govern processes in many diverse fields ranging from computer science to physics and economics to biology. The PI specializes in algebraic geometry, the field which studies solutions of polynomial systems. Such systems can often be simplified by varying the coefficients of the polynomials appropriately. As a result, properties of a more complicated system can be deduced from the simpler system. The PI will apply this principle to study the geometry of spaces that are ubiquitous in mathematics and physics, namely moduli spaces of vector bundles. Using a novel technique called Bridgeland stability, the PI will investigate fundamental geometric properties of these spaces. The results will have a wide range of applications in algebraic geometry, commutative algebra, topology, and mathematical physics. The PI is also dedicated to educating the next generation of mathematicians and building a strong workforce in STEM in the US. Towards this goal, the PI actively supervises numerous PhD students and postdocs, as well as high school students and undergraduate students in related research. The grant will provide partial support to the graduate students. The moduli spaces of sheaves on surfaces play a fundamental role in mathematics and physics. They carry essential information about linear series and Chow groups and are key players in Donaldson’s theory of four manifolds, combinatorics, representation theory and mathematical physics. In the last decade, Bridgeland stability has revolutionized our understanding of moduli spaces of sheaves. Using this novel technique, the PI will advance understanding of moduli spaces of sheaves. Specifically, the PI will compute the cohomology of the general stable sheaf on a surface using Bridgeland stability and wall-crossing. In cases where the generic cohomology is already understood, such as minimal rational surfaces and K3 surfaces, the PI will initiate a systematic study of the cohomology jumping loci and compute the cohomology of the tensor product of two general stable sheaves. In addition, the PI aims to prove a conjecture due to the PI and Woolf that states that the Betti numbers of the moduli spaces of sheaves stabilize as the discriminant tends to infinity and the stable Betti numbers are independent of the rank and polarization. Finally, the PI will also study the stability of normal bundles of curves on algebraic varieties with a view towards applications to hyperbolicity, Lang conjectures and separable rational connectedness.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.