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Yusuf Mustopa
Norbert Wiener Assistant Professor

Yusuf Mustopa
Contact Info:
Tufts University
Department of Mathematics
503 Boston Avenue
Room 106
Medford, MA 02155

Email @tufts.edu:
Phone: 617-627-2357

Algebraic geometry.

My current research program explores the interplay between vector bundles and the algebraic structure of projective varieties (i.e. solution sets of systems of homogeneous polynomials).

One research direction is centered around Ulrich bundles; these are a natural generalization of spinor bundles on quadric hypersurfaces, and they form a fertile meeting ground for various topics in algebra and algebraic geometry, including Brauer groups, moduli of vector bundles on curves, determinantal hypersurfaces, and matrix factorizations of polynomials. At the moment, Ulrich bundles are known to exist on relatively few projective varieties, and the importance of when they exist has recently increased due to a link with Boij-Soderberg theory; informally, if a projective variety of dimension m admits an Ulrich bundle, then the cohomology of its coherent sheaves is exactly as complicated as that of projective space. My recent joint work with R. Kulkarni and I. Shipman shows that the existence of an Ulrich sheaf on an arbitrary projective variety is equivalent to the solubility of a Brill-Noether problem for vector bundles of high rank on a curve which is special in moduli. In addition to being a gauge for the difficulty of proving every variety admits an Ulrich sheaf, this result opens a geometric path to constructing new examples.

Another direction is centered around syzygy bundles, whose exterior powers help encode minimal graded free resolutions of projective varieties. These have played a big role in my research almost from the start of my career, when I used syzygy bundles on projective curves to produce divisors on their symmetric products. My previous joint work with L. Ein and R. Lazarsfeld shows that syzygy bundles on projective surfaces of sufficiently high degree are slope-stable. In ongoing work with S. Kitchen and A. Kuronya, I am studying how the Bridgeland stability of syzygy bundles (viewed as objects in the derived category) can be used to estimate the shape of minimal graded free resolutions.