My research in pure mathematics is centered on algebraic geometry, which studies the interplay between systems of polynomial equations and the geometry of their solution sets. 

After my early work, which combined birational geometry and the geometry of curves, my focus moved towards vector bundles on higher-dimensional varieties.  While a good share of my work on this topic involves its connections to (non)commutative algebra, its scope has recently expanded to include positivity, i.e. the algebraic study of when a vector bundle can have positive curvature.

Rational Curves on Moduli Spaces of Vector Bundles (with Montserrat Teixidor i Bigas), submitted.

Effective global generation on varieties with numerically trivial canonical class (with Alex Kuronya), submitted.

Castelnuovo-Mumford Regularity and GV-sheaves on Irregular Varieties, submitted.

Continuous CM-Regularity of Semihomogeneous Vector Bundles (with Alex Kuronya), Advances in Geometry 20 (2020), no. 3, p. 401-412.
 

Vector bundles whose restriction to a linear section is Ulrich (with Rajesh Kulkarni and Ian Shipman), Mathematische Zeitschrift 287 (2017), no. 3-4, p. 1307-1326.

A few questions about curves on surfaces (with Ciro Ciliberto, Andreas L. Knutsen, John Lesieutre, Victor Lozovanu, Rick Miranda, and Damiano Testa), Rend. Circ. Mat. Palermo, II. 66 (2017), no. 2, p. 195-204.

The characteristic polynomial of an algebra and representations (with Rajesh Kulkarni and Ian Shipman), Linear Algebra and its Applications 530 (2017), p. 47-56.

Ulrich Sheaves and Higher-Rank Brill-Noether Theory (with Rajesh Kulkarni and Ian Shipman), Journal of Algebra 474 (2017), p. 166-179.

 

Stability of Syzygy Bundles on an Algebraic Surface (with Lawrence Ein and Robert Lazarsfeld), Mathematical Research Letters 20 (2013), no. 1, 87-94.

The Geometry of Ulrich Bundles on del Pezzo Surfaces (with Emre Coskun and Rajesh Kulkarni), Journal of Algebra 375 (2013), p. 280–301.

 

Pfaffian Quartic Surfaces and Representations of Clifford Algebras (with Emre Coskun and Rajesh Kulkarni), Documenta Mathematica 17 (2012), 1003-1028.

 

On representations of Clifford algebras of ternary cubic forms (with Emre Coskun and Rajesh Kulkarni), in New trends in noncommutative algebra, 91–99, Contemporary Mathematics, 562, Amer. Math. Soc., Providence, RI, 2012.

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Kernel bundles, syzygies of points, and the effective cone of C_{g−2}, International Mathematics Research Notices 2011, no. 6, 1417–1437.

 

Residuation of linear series and the effective cone of C_d, American Journal of Mathematics 133 (2011), no. 2, 393–416.