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Algebraic and geometric combinatorics
I am interested in the interplay between algebra, geometry and combinatorics. The overarching theme of my current research is to develop a new perspective on classical multivariate polynomials by recasting them in terms of polytopes. In recent work with collaborators we show a close connection of flow polytopes, generalized permutahedra and Schubert polynomials, thereby bringing new ideas to study these and related polynomials.
- MATH 4410 : Introduction to Combinatorics I
- MATH 4900 : Supervised Research
- MATH 4901 : Supervised Reading
- MATH 7900 : Supervised Reading and Research
- Schubert polynomials as integer point transforms of generalized permutahedra (with Alex Fink and Avery St. Dizier) Advances in Mathematics, accepted. http://arxiv.org/abs/1706.04935
- From generalized permutahedra to Grothendieck polynomials via flow polytopes (with Avery St. Dizier) preprint. http://arxiv.org/abs/1705.02418
- The polytope of Tesler matrices (with A. H. Morales and B. Rhoades), Selecta Mathematica, to appear. http://arxiv.org/abs/1409.8566
- Toric matrix Schubert varieties and their polytopes (with L. Escobar), Proceedings of the American Mathematical Society, to appear. http://arxiv.org/abs/1508.03445
- Subword complexes via triangulations of root polytopes (with L. Escobar), preprint. http://arxiv.org/abs/1502.03997
- Product formulas for volumes of flow polytopes, Proceedings of the American Mathematical Society 143 no. 3, (2015), 937-954.
- Flow polytopes of signed graphs and the Kostant partition function (with A. H. Morales), International Mathematical Research Notices no. 3, (2015), 830-871.
- Root polytopes, triangulations, and the subdivision algebra, II, Transactions of the American Mathematical Society 363 no. 11, (2011), 6111-6141.
- Root polytopes, triangulations, and the subdivision algebra, I, Transactions of the American Mathematical Society 363 no 8, (2011), 4359-4382.