Norbert Wiener Assistant Professor
Department of Mathematics
503 Boston Avenue
Medford, MA 02155
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.