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Algebraic geometry and algebraic combinatorics
In recent years I have been studying algebraic varieties carrying large groups of symmetries, such as toric varieties, flag manifolds, Schubert varieties, and quiver cycles. While individually complicated, these can be broken up (really, degenerated) into many simple pieces, at which point their study becomes a matter of combinatorics rather than geometry.
- MATH 4340 : Honors Introduction to Algebra
- MATH 4900 : Supervised Research
- MATH 4901 : Supervised Reading
- Complete moduli spaces of branchvarieties (with Valery Alexeev), Crelle's Journal (to appear).
- Four positive formulae for type A quiver polynomials (with Ezra Miller and Mark Shimozono), Inventiones Mathematicae 166 (2006), no. 2.
- Gröbner geometry of Schubert polynomials (with Ezra Miller), Annals of Mathematics 161 (2005), 1245–1318.
- Puzzles and (equivariant) cohomology of Grassmannians (with Terence Tao), Duke Math. J. 119 (2003), no. 2, 221–260.
- The honeycomb model of GL(n) tensor products I: proof of the saturation conjecture (with Terence Tao), Journal of the AMS 12(1999), no. 4, 1055–1090.