 |
Yusuf Mustopa
RTG Assistant Professor, Department of Mathematics |
About Me:
I was born and raised in New York City, where I attended F.H. LaGuardia High School for the Arts with the intention of becoming a visual artist. During my senior year, the allure of precalculus was too strong to ignore; upon arriving at SUNY New Paltz, I took calculus for fun while continuing to major in art. By the time I completed my B.A., I had completely devoted myself to mathematics. Shortly thereafter, I earned an M.A. at the City College of New York and went on to pursue a doctorate at SUNY Stony Brook, where I wrote my thesis in algebraic geometry under Jason Starr. I will be a Visiting Assistant Professor in the Department of Mathematics at Boston College during the 2012-2013 academic year.
Teaching:
I am teaching MATH 217 (Linear Algebra) in Spring 2012.
Research:
I work in algebraic geometry. My current projects concern the connections between ACM bundles, noncommutative algebra, and generalized theta-divisors (joint with Emre Coskun and Rajesh Kulkarni) and the asymptotic stability of syzygy bundles (joint with Lawrence Ein and Rob Lazarsfeld). This work is partially supported by an NSF grant (RTG DMS-0502170).
Upcoming and Recent Invited Talks:
Colloquium, Washington University in St. Louis (August 31, 2011)
Algebra, Geometry and Physics Seminar, Stony Brook University (September 7, 2011)
Algebraic Geometry Seminar, University of Utah (October 18, 2011)
Algebraic Geometry Seminar, University of Georgia (November 2, 2011)
Colloquium, Oakland University (January 17, 2012)
AMS Spring Southeastern Sectional Meeting, Special Session on Computational Algebraic Geometry and Applications, University of South Florida (March 10-11, 2012)Upcoming and Recent Conferences/Workshops:
A Celebration of Algebraic Geometry: A Conference in Honor of Joe Harris' 60th Birthday, Harvard University, Boston, MA (August 25, 2011-August 28, 2011)
Western Algebraic Geometry Symposium, Colorado State University, Fort Collins, CO (October 1, 2011-October 2, 2011)
Michigan Computational Algebraic Geometry Conference, Oakland University, Rochester, MI (May 13, 2012)
Michigan Derived Algebraic Geometry Learning Workshop, University of Michigan, Ann Arbor, MI (May 17, 2012-May 19, 2012)
Syzygies in Algebraic Geometry, with an exploration of a connection with String Theory, Banff International Research Station (August 12, 2012-August 17, 2012)
Papers:
Residuation of Linear Series And The Effective Cone Of C_d, American Journal of Mathematics, vol. 133 (2011), no. 2, p. 393-416:
This is the paper version of my thesis, with some extra results. It contains new bounds for the effective and moving cones of the d-th symmetric power C_d of a curve C of genus g, as well as a precise computation of the effective cone of C_{g-1} and an example of a non-equidimensional stable base locus whose connectedness measures the inflectionary behavior of the canonical image of C.
Kernel Bundles, Syzygies Of Points, And The Effective Cone Of C_{g-2}, International Mathematics Research Notices, no. 6 (2011), p. 1417-1437
This is my second paper on the birational geometry of symmetric products of curves. In addition to containing a calculation of the effective cone of C_{g-2} for a general curve C of genus g at least 4, it introduces secant loci on C_d associated to higher-rank vector bundles on C, which are used to study the question of whether the effective cone of C_{g-2} is determined by the Clifford index of C.
On Representations of Clifford Algebras of Ternary Cubic Forms (with E. Coskun and R. Kulkarni), refereed, to appear in New Trends in Noncommutative Algebra: A Conference in Honor of Ken Goodearl's 65th Birthday
We show that under van den Bergh's correspondence between representations of Clifford algebras and Ulrich bundles on cyclic covers, the irreducible representations correspond precisely to stable Ulrich bundles. We then combine this with work of Casanellas-Hartshorne to construct irreducible representations of the Clifford algebra of a ternary cubic form which have arbitrarily high dimension.
Pfaffian Quartic Surfaces and Representations of Clifford Algebras (with E. Coskun and R. Kulkarni), submitted
We use recent work of Aprodu-Farkas on the Green Conjecture to construct a family of rank-2 orientable Ulrich bundles on every smooth quartic surface in P^3; this implies that every smooth quartic surface is the zero locus of the Pfaffian of an 8 by 8 skew-symmetric matrix of linear forms, strengthening a result of Beauville-Schreyer. These bundles are then applied to produce 8-dimensional irreducible representations of the Clifford algebra of a ternary quartic form.
The Geometry of Ulrich Bundles on Del Pezzo Surfaces (with E. Coskun and R. Kulkarni), submitted
We show that the existence of Ulrich bundles on a del Pezzo surface with fixed Chern class is equivalent to the existence of a theta-divisor for a certain kernel bundle, and relate this to the Minimal Resolution Conjecture via work of Farkas-Mustata-Popa. Moreover, we show that del Pezzo surfaces are the only arithmetically Gorenstein surfaces for which such a characterization can hold.