XXIII National Mathematics Symposium, Kayseri Turkey, August 2010.
Lecture 1, Lecture2
Lecture notes
Lectures in the CIMPA/TUBITAK/GSU summer school, June 2014 (Notes)
Lecture in the Gokova Geometry and Topology Conference, May 2014. Notes together with Jack Huizenga (Notes)
Lectures in Warsaw Poland on homogeneous varieties, December 2013
(Notes)
Lectures at the Utah School on birational geometry and moduli spaces, June 2010 (Notes)
Books:
Surveys on recent developments in algebraic geometry (edited with Tommaso de Fernex and Angela Gibney) (Available here)
Papers and preprints:
Comments, corrections and
suggestions are always welcome.
Stability of Tschirnhausen bundles (joint with Eric Larson and Isabel Vogt) (pdf) Preliminary version. Comments, corrections and suggestions are welcome.
Stability and cohomology of kernel bundles on projective space (joint with Jack Huizenga and Geoffrey Smith) (pdf) Preliminary version. Comments, corrections and suggestions are welcome.
The normal bundle of a general canonical curve of genus at least 7 is semistable (joint with Eric Larson and Isabel Vogt) (pdf) Preliminary version. Comments, corrections and suggestions are welcome.
The cohomology of the general stable sheaf on a K3 surface (joint with Howard Nuer and Kota Yoshioka) (pdf) The data for the paper (data) Preliminary version. Comments, corrections and suggestions are welcome.
Very free rational curves in Fano varieties (joint with Geoffrey Smith) (pdf) Journal of Algebra Volume 611 (2022), 246--264.
Disconnected moduli spaces of stable bundles on surfaces (joint with Jack Huizenga and John Kopper) (pdf) Bulletin of the London Mathematical Society Volume 54, no. 2 (2022), 812--824.
Clustered families and applications to Lang-type conjectures (joint with Eric Riedl) (pdf) To appear in the Proceedings of the London Mathematical Society.
The cohomology of general tensor products of vector bundles on the projective plane (joint with Jack Huizenga and John Kopper) (pdf) Selecta Mathematica vol. 27 no. 5 Article no: 94 (2021), 46 pages.
Stability of normal bundles of space curves (joint with Eric Larson and Isabel Vogt) (pdf) Algebra and Number Theory 16 no.4 (2022), 919--953.
Algebraic hyperbolicity of very general surfaces (joint with Eric Riedl) (pdf) To appear in Israel Journal of Mathematics.
Existence of semistable sheaves on Hirzebruch surfaces (joint with Jack Huizenga) (pdf) Advances in Mathematics 381 (2021), 96 pages.
Brill-Noether Theorems, Ulrich bundles and the cohomology of moduli spaces of sheaves (joint with Jack Huizenga) (pdf) Proceedings of the ICM Satellite Conference on Moduli Spaces in Algebraic Geometry and Applications, Matematica Contemporanea 47 (2020), 21--72.
The stable cohomology of moduli spaces of sheaves on surfaces (joint with Matthew Woolf) (pdf) Journal of Differential Geometry 121 no.2 (2022), 291--340.
Effective bounds on ampleness of cotangent bundles (joint with Eric Riedl) (pdf) Bulletin of the London Mathematical Society 52 (2020), 237--243.
Algebraic hyperbolicity of the very general quintic surface (joint with Eric Riedl) (pdf) Advances in Mathematics 350 (2019), 1314--1323.
The ample cone of moduli spaces of sheaves on surfaces and the Brill-Noether Problem (joint with Jack Huizenga) for the Proceedings of the Abel Conference 2017 (pdf) in Geometry of Moduli, Abel Symposia 14 (2018), 75--106.
Normal bundles of rational curves on complete intersections (joint with Eric Riedl) (pdf) Macaulay2 calculations (txt) Communications in Contemporary Mathematics 21 no. 2 (2019), 23 pages.
Brill-Noether Theorems and globally generated vector bundles on Hirzebruch surfaces (joint with Jack Huizenga) (pdf) Nagoya Mathematical Journal 238 (2020), 1--36.
Weak Brill-Noether for rational surfaces (joint with Jack Huizenga) (pdf) in Local and Global Methods in Algebraic Geometry, Contemporary Mathematics 712 (2018), 81--104.
Normal bundles of rational curves in projective space (joint with Eric Riedl) (pdf) Mathematische Zeitschrift vol 288 (2018), 803--827.
Restriction varieties and the rigidity problem (Notes for lectures in Poland) (pdf) in Schubert varieties, equivariant cohomology and characteristic classes Impanga 15 (2018), 49--96.
Effective cones of cycles on blow-ups of projective space (joint with John Lesieutre and John Ottem) (pdf) Algebra and Number Theory vol. 10 no. 9 (2016), 1983--2014.
The nef cone of the moduli spaces of sheaves and strong Bogomolov inequalities (joint with Jack Huizenga) (pdf) Israel Journal of Mathematics. 226 no. 1 (2018), 205--236.
Fano manifolds of index n-2 and the cone conjecture (joint with Artie Prendergast-Smith) (pdf) Sage code (txt) Mathematical Proceedings of the Cambridge Philosophical Society 166 no. 1 (2019), 1--31.
Ulrich Schur bundles on flag varieties (joint with Laura Costa, Jack Huizenga, Rosa Maria Miro-Roig and Matthew Woolf) (pdf) Journal of Algebra vol 474 (2017), 49--96.
A follow-up paper with my undergraduate student Luke Jaskowiak: Ulrich partitions for two-step flag varieties (pdf) Involve vol 3 no.3 (2017), 531--539.
Castelnuovo-Mumford regularity and Bridgeland stability of points in the projective plane (with Donghoon David Hyeon and Jonyoung Park) (pdf) Proceedings of the American Mathematical Society vol 145 (2017), 4573--4583.
The ample cone of the moduli spaces of sheaves on the plane (joint with Jack Huizenga) (pdf) Algebraic Geometry vol 3 no. 1 (2016), 106--136.
Extremal higher codimension cycles on moduli spaces of curves (joint with Dawei Chen) (pdf) Proceedings of the London Mathematical Society vol 111 no. 1 (2015), 181--204.
The birational geometry of moduli spaces of sheaves on the plane (lecture notes for Gokova conference, joint with Jack Huizenga) (pdf) Proceedings of the Gokova Geometry-Topology Conference 2014, (2015), 114--155.
Dense PGL-orbits in products of Grassmannians (joint with Majid Hadian and Dmitry Zakharov) (pdf) Journal of Algebra 429 (2015), 75--102.
The effective cone of the moduli space of sheaves on the plane (joint with Jack Huizenga and Matthew Woolf) (pdf) Journal of the European Mathematical Society vol. 19 no. 5 (2017), 1421--1467.
Oberwolfach abstract on the subject (pdf)
Eckardt loci on hypersurfaces (joint with Artie Prendergast-Smith) (pdf) Communications in Algebra 43 no. 8 (2015), 3083--3101.
Interpolation, Bridgeland stability and monomial schemes in the plane (joint with Jack Huizenga) (pdf) Journal de Mathematiques pures et appliquees 102 (2014), 930--971.
Extremal effective divisors on the moduli space of n-pointed genus one curves (joint with Dawei Chen) (pdf) Mathematische Annalen, 359 no. 3 (2014), 891--908.
Flexibility of Schubert classes (joint with Colleen Robles) (pdf) Differential Geometry and its applications, 31 no. 6 (2013), 759--774.
Fano manifolds of index n-1 and the cone conjecture (joint with Artie Prendegast-Smith) (pdf) International Mathematics Research Notices, Volume 2014, no. 9 (2014), 2401--2439.
Symplectic retriction varieties and geometric branching rules II. Revised version: (pdf) old file: (pdf) Journal of Combinatorial Theory Series A, volume 125 (2014), 47--97.
The birational geometry of the Hilbert scheme of points on surfaces. (pdf) (joint with Aaron Bertram). Birational geometry, rational curves and arithmetic, Simons Symposia, Springer 2013, 15--55.
The minimal model program for the Hilbert scheme of points on the plane and Bridgeland stability. (pdf) (joint with Daniele Arcara, Aaron Bertram and Jack Huizenga) Advances in Mathematics, volume 235 (2013), 580--626.
Rigidity of Schubert classes in orthogonal Grassmannians. (pdf) Israel Journal of Mathematics, vol. 200 (2014), 85--126.
Surfaces of low degree containing a canonical curve. (pdf) Computational Algebraic and Analytic Geometry, Contemporary Mathematics, 572, 2012.
Parameter spaces of Schubert varieties in hyperplane sections of Grassmannians (with my student Richard Abdelkerim, this is part of his thesis) (pdf) Journal of Pure and Applied Algebra
vol. 216 (2012), 800--810.
Singularities of generalized Richardson varieties (joint with Sara Billey) (pdf ). Communications in Algebra. 40:4, 1466--1495.
Rigid and non-smoothable Schubert classes (pdf ). Journal of Differential Geometry vol. 87, no. 3 (2011), 493--514.
Towards Mori's program for the moduli space of stable maps (joint with Dawei Chen, with an appendix by Charley Crissman) (pdf). American Journal of Mathematics vol. 133 no.5 (2011), 1389--1419.
Stable base locus decompositions of Kontsevich moduli spaces (joint with Dawei Chen) (pdf). Michigan Mathematical Journal, vol. 59, no. 2 (2010), 435--466.
Hilbert scheme of a pair of codimension two linear subspaces (joint with Dawei Chen and Scott Nollet) (pdf). Communications in Algebra vol. 39 no.8 (2011), 3021--3043.
Rational curves on smooth cubic hypersurfaces (joint with Jason Starr) (pdf). International Mathematics Research Notices Article RNP102 (2009), 16 pages, doi:10.1093/imrn/rnp102.
Restriction varieties and geometric branching rules. (pdf). Advances in Mathematics. vol. 228 no. 4 (2011), 2441--2502.
The quantum cohomology of flag varieties and the periodicity of the Schubert structure constants. (pdf) Mathematische Annalen vol. 346 no. 2 (2010), 419--447.
A Littlewood-Richardson rule for partial flag varieties. (pdf). Currently under revision. This is a preliminary version of a Littlewood-Richardson rule for arbitrary partial flag varieties. Any comments, corrections and suggestions are welcome. GALLERY (html)
A Littlewood-Richardson rule for two-step flag varieties (pdf). For examples see Gallery (html). Old version is available here: (pdf).
Inventiones Mathematicae volume 176, no 2 (2009) p. 325--395 DOI: 10.1007/s00222-008-0165-3
Geometric positivity in the cohomology of homogeneous
spaces and generalized Schubert calculus (joint with Ravi Vakil) (pdf). Algebraic Geometry, Seattle 2005 volume 1, p. 77--124.
Divisors on the space of maps to Grassmannians (joint with
Jason Starr). (pdf). International Mathematics Research Notices, vol. 2006, Article ID 35273, 25 pages, 2006.
The effective cone of the Kontsevich moduli space (joint
with Joe Harris and Jason Starr). (pdf). Canadian Mathematical Bulletin. vol. 51 no. 4 (2008), pages 519-534.
The ample cone of the Kontsevich moduli space (joint with Joe
Harris and Jason Starr) (pdf).
Canadian Journal of Mathematics. vol. 61 no. 1 (2009), pages 109-123.
Gromov-Witten invariants of jumping curves (pdf). Transactions of the American Mathematical Society. 360 (2008), p. 989-1004.
Degenerations of surface scrolls and Gromov-Witten invariants of
Grassmannians (pdf).
Journal of Algebraic Geometry. 15 (2006), p. 223-284.
The enumerative geometry of Del Pezzo surfaces via degenerations
(pdf). American
Journal of Mathematics. vol. 128 no. 3 (2006), p. 751-786.
The arithmetic and the geometry of Kobayashi hyperbolicity (pdf). Snowbird lectures in algebraic geometry. Contemporary Mathematics, vol 388 (2005) p. 77-88.
A first glimpse at the minimal model program (pdf). (with 8 coauthors) Snowbird lectures in algebraic geometry. Contemporary
Mathematics, vol 388 (2005) p. 17-42.
Degenerations of scrolls and Del Pezzo surfaces and applications
to enumerative geometry, Thesis, Harvard University, 2004.