Frank Sottile: Publications and Preprints

1	Real Solutions to Equations From Geometry, University Lecture Series volume 57, AMS, 2011. 200+viii pages. Older version: math.AG/0609829.       Errata.

2	Phase limit sets of linear spaces and discriminants, with Mounir Nisse, November 2024.
3	A quantum Murnaghan--Nakayama rule for the flag manifold. with Carolina Benedetti, Nantel Bergeron, Laura Colmenarejo, and Franco Saliola. 34 pages.
4	Numerical Nonlinear Algebra, with Daniel Bates, Paul Breiding, Tianran Chen, Jonathan Hauenstein, and Anton Leykin. arXiv.org/2302.08585. In honor of Bernd Sturmfels on his 60th birthday.
5	Numerical Schubert Calculus in Macaulay2, with Anton Leykin, Abraham Martín del Campo-Sanchez, Ravi Vakil and Jan Verschelde. 10 May 2021. ArXiv.org/2105.04494.
6	Galois Groups in Enumerative Geometry and Applications, with Thomas Yahl. Some computations about the splitting types of cubic surfaces. arXiv.org/2108.07905.

7	Some problems at the interface of approximation theory and algebraic geometry, From ``Approximation Theory and Numerical Analysis meet Algebra, Geometry, and Topology'', Palazzone Cortona 5–9 September 2022. arXiv.org/2308.03211.

8	Real solutions to systems of polynomial equations in Macaulay2, with Jordy Lopez. Kelly Maluccio. Thomas Yahl. Journal of Software for Algebra and Geometry, 14-1 (2024), 87–95. DOI 10.2140/jsag.2024.14.87.
9	Critical points of Discrete Periodic Operators, with Matt Faust. Journal of Spectral Theory, 14 (2024), no. 1, pp. 1–35 arXiv.org/2206.13649
10	Classification of Schubert Galois groups in Gr(4,9), with Abraham Martín del Campo-Sanchez and Robert Williams. 41 pages. Arnold Mathematical Journal, 9 (2023) no. 3, 393–433. https://doi.org/10.1007/s40598-022-00221-2. Read-only, online version.
11	Numerical homotopies from Khovanskii Bases, with Michael Burr and Elise Walker. Mathematics of Computation, 92 (2023), No. 343, 2333–2353.
12	On the fundamental group of open Richardson varieties, with Changzheng Li and Chi Zhang. Communications in Number Theory and Physics, Volume 17 (2023), Number 1, 77–101.
13	Euclidean Distance Degree via Mixed Volume, with Paul Breiding and James Woodcock. Foundations of Computational Mathematics. 22 (2022), 1743–1765. DOI: 10.1007/s10208-021-09534-8.
14	Describing Amoebas, with Mounir Nisse, Pacific Journal of Mathematics, 317 No. 1, (2022) 187–205.
15	Galois Groups of Composed Schubert Problems, with Robert Williams. and Li Ying. Facets of algebraic geometry. Vol. II, London Math. Soc. Lecture Note Ser. 473, Cambridge Univ. Press, Cambridge, pp. 336–366, 2022.
16	Irrational toric varieties and secondary polytopes with Ata Firat Pir. 24 pages, 2020. Discrete and Computational Geometry 67 (2022) No, 4, 1053–1079. On-line version. DOI:10.1007/s00454-021-00334-z.
17	Certification for Polynomial Systems via Square Subsystems, with Timothy Duff and Nickolas Hein. Journal of Symbolic Computation, 109 (2022), 367 383. DOI:10.1016/j.jsc.2020.07.010.
18	Nontrivial linear projections of the Grassmannian Gr_3(C^6) with Yanhe Huang, George Petroulakis, and Igor Zelenko. Linear Algebra and its Applications, 610, (2021), 488–505.
19	Solving Decomposable Sparse Systems, with Taylor Brysiewicz, Jose Israel Rodriguez, and Thomas Yahl. Numerical Algorithms, 88, (2021), 453–474.
20	Decomposable sparse polynomial systems, with Taylor Brysiewicz, Jose Israel Rodriguez, and Thomas Yahl. Journal of Software for Algebra and Geometry, Vol. 11 (2021), 53–59. DOI: 10.2140/jsag.2021.11.53.
21	General witness sets for numerical algebraic geometry, ISSAC '20: Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation, ACM, July 2020, pp.418–425. DOI:10.1145/3373207.3403995.
22	Numerical Schubert Calculus via the Littlewood-Richardson Homotopy Algorithm, with Anton Leykin, Abraham Martín del Campo-Sanchez, Ravi Vakil and Jan Verschelde. Mathematics of Computation, 90 (2021), 1407–1433.
23	A numerical toolkit for multiprojective varieties, with Jonathan D. Hauenstein, Anton Leykin and Jose Israel Rodriguez. Mathematics of Computation, 90 (2021) No. 327, 413–430.
24	Generic properties of dispersion relations for discrete periodic operators, with Peter Kuchment and Ngoc T. Do. Journal of Mathematical Physics, 61, No. 10, 2020. 19 pp. DOI: 10.1063/5.0018562.
25	Bivariate Semialgebraic Splines, with Michael Di Pasquale. Journal of Approximation Theory, 254 (2020) 105392 19 pages.
26	A geometric proof of an equivariant Pieri rule for flag manifolds with Changzheng Li, Vijay Ravikumar, and Mingzhi Yang. 6 pages 2018. Forum Mathematicum, 31 (2019), 779–783. arXiv.org/1809.04242. DOI 10.1515/forum-2018-0213
27	Convex hull of two circles in R3 with Evan D. Nash, Ata Firat Pir, and Li Ying. Combinatorial Algebraic Geometry, Fields Institute Communications 80, G. Smith and B. Sturmfels, Eds., 2017, 297–319.
28	Real Algebraic Geometry for Geometric Constraints, CRC Handbook on Geometric Constraint Systems Principles, M. Sitharam, A. St. John, and J. Sidman, eds. CRC Press, 2019. 273–285.
29	Trace test, with Anton Leykin and Jose Israel Rodriguez. Arnold Mathematical Journal, 4 (2018), 113–125.
3 30	Two Murnaghan-Nakayama rules in Schubert Calculus with Andrew Morrison. Annals of Combinatorics, 22(2) (2018), 363–375. DOI: 10.1007/s00026-018-0387-z On-line version from Springer.
31	Injectivity of generalized Wronski maps with Yanhe Huang and Igor Zelenko. Canadian Mathematical Bulletin, 60 (2017), No. 4, 747–761. DOI: 10.4153/CMB-2017-001-0.
32	Semialgebraic Splines with Michael Di Pasquale and Lanyin Sun. Computer Aided Geometric Design, 55, (2017), 29–47.
33	Numerical computation of Galois groups, with Jonathan Hauenstein and Jose Israel Rodriguez. Foundations of Computational Mathematics, 18 (2018) 867–890 DOI 10.1007/s10208-017-9356-x On-line Version
34	Equivariant cohomology theories and the pattern map, with H. Praise Adeyemo. Houston Journal of Mathematics, 43 (2017), 375–393.
35	Experimentation in the Schubert Calculus with Abraham Martín del Campo. "Schubert Calculus — Osaka 2012", ed. by H.Naruse, T.Ikeda, M.Masuda, and T.Tanisaki, Advanced Studies in Pure Mathematics 71, Mathematical Society of Japan, 295–336, 2016.
36	A congruence modulo four for real Schubert calculus with isotropic flags with Nickolas Hein and Igor Zelenko. Canadian Mathematical Bulletin 60 (2017), No. 2, 309–318. DOI: 10.4153/CMB-2016-087-2
37	A lifted square formulation for ceritifiable Schubert calculus with Nickolas Hein. Journal of Symbolic Computation, 79 (2017), Part 3, 594–608. DOI: dx.doi.org/10.1016/j.jsc.2016.07.021. Description and archive of computations.
38	A primal-dual formulation for certifiable computations in Schubert calculus with Jonathan Hauenstein and Nickolas Hein. Foundations of Computational Mathematics, 16, (2016), 941–963. DOI:10.1007/s10208-015-9270-z. Description and archive of computations.
39	A congruence modulo four in real Schubert calculus with Nickolas Hein and Igor Zelenko. Journal für die reine und angewandte Mathematik, Volume 2016, Issue 714 (May 2016), 151–174, DOI: 10.1515/crelle-2013-0122
40	Software for the Gale transform of fewnomial systems and a Descartes rule for fewnomials, with Daniel J. Bates, Jonathan D. Hauenstein, and Matthew E. Niemerg. Numerical Algorithms, article, DOI: 10.1007/s11075-015-0095-2. (2016). View the published version.
41	Double transitivity of Galois Groups in Schubert Calculus of Grassmannians with Jacob White. Algebraic Geometry, 2, Issue 4 (September 2015), 422—445.
42	Cohomological consequences of the pattern map, with H. Praise Adeyemo. Illinois Journal of Mathematics, 58 (2014), No 4, 997—1008.
43	Higher convexity of coamoeba complements, with Mounir Nisse, Bulletin of the London Mathematical Society, 47, No. 5 (2015), 853—865.
44	Algebraic Geometry, Princeton Companion to Applied Mathematics, ed. N.J. Higham, 2015, 570—579.
45	Degenerations of Irrational Toric Varieties with Elisa Postinghel and Nelly Villamizar Journal of the London Mathematical Society, 92 (2015), No. 2, 223—241, DOI: 10.1112/jlms/jdv024.
46	Higher convexity for complements of tropical varieties, with Mounir Nisse, Mathematische Annalen, 365 (2016), No 1, pp. 1–12.
47	The monotone secant conjecture in the real Schubert calculus with Nickolas Hein, Chris Hillar, Abraham Martín del Campo-Sanchez, Zach Teitler. Experimental Mathematics, 24 (2015), 261–269. DOI:10.1080/10586458.2014.980044
48	Galois groups of Schubert problems of lines are at least alternating with Christopher J. Brooks Abraham Martín del Campo-Sanchez Transactions of the AMS, 367 No. 6 (2015), 4183–4206.
49	Newton Polytopes and Witness Sets with Jonathan Hauenstein. Mathematics in Computer Science, 8 (2014), 235–251.
50	Non-archimedean coamoebae, with Mounir Nisse, in Tropical and Non-Archimedean Geometry, Contemporary Mathematics, 605 (2013). 73–91.
51	Discriminant coamoebas through homology with Mikael Passare†, J. Commutative Algebra, 5, Number 3 (2013), 413–440.
52	Lower Bounds in Real Schubert Calculus with Nickolas Hein and Chris Hillar. São Paulo Journal of Mathematics, 7, No. 1, (2013) 33–58.
53	Static skew-symmetric output feedback control with Chris Hillar SIAM J Optimization and Control, 51, 4 (2013). DOI: 10.1137/110855363
54	Arithmetic Toric Varieties with E. Javier Elizondo, Paulo Lima-Filho, and Zach Teitler. Mathematische Nachrichten, Volume 287, Issue 2-3, pages 216–241, DOI 10.1002/mana.201200305/
55	The phase limit set of a variety, with Mounir Nisse, Algebra and Number Theory, 7-2 (2013), 339–352. DOI 10.2140/ant.2013.7.339
56	Orientability of real toric varieties, with Jenya Soprunova, Discrete and Computational Geometry, 50, No. 2 (2013), 509–519. DOI 10.1007/s00454-013-9498-9.
57	Cofree compositions of Coalgebras with Stefan Forcey and Aaron Lauve. Annals of Combinatorics, Volume 17 (2013), Issue 1, 105–130. DOI: 10.1007/s00026-012-0170-5.
58	The Secant Conjecture in the real Schubert calculus with Luis García-Puente, Nickolas Hein, Chris Hillar, Abraham Martín del Campo-Sanchez, James Ruffo, and Zach Teitler. Experimental Mathematics, 21, Issue 3 (2012), pp. 252–265.
59	Algorithm 921: alphaCertified: certifying solutions to polynomial systems with Jonathan Hauenstein. ACM Transactions on Mathematical Software, 48, No. 4 (2012) Software and guide.
60	Toric degenerations of Bézier patches with Chungang Zhu, and Luis Garcia. ACM Transactions on Graphics, 30 (2011), Issue 5, Article 110. 11 pages.
61	Injectivity of 2D toric Bézier patches with Chungang Zhu, Proceedings of 12th International Conference on Computer-Aided Design and Computer Graphics, Ralph Martin, Hiromasa Suzuki, and Changhe Tu Eds. 15-17 September 2011, Jinan China, IEEE CPS, pp. 235–238.
62	Dense Fewnomials with Korben Rusek and Jeanette Shakalli. Randomization, Relaxation, and Complexity, Leonid Gurvits, Philippe Pébay, J. Maurice Rojas, and David Thompson, eds., Contemporary Mathematics 556, AMS, 2011, pp. 167–186.
63	Khovanskii-Rolle continuation for real solutions, with Daniel J. Bates, Foundations of Computational Mathematics, Volume 11, Number 5 (2011), 563-587. DOI: 10.1007/s10208-011-9097-1. 25 pages.
64	Orbitopes with Raman Sanyal, and Bernd Sturmfels. Mathematika, 57 (2011), issue 02, 275–314.
65	Fewnomial bounds for completely mixed polynomial systems, with Frédéric Bihan, Advances in Geometry, (2011), DOI:10.1515/ADVGEOM.2011.019.
66	Hopf Structures on the Multiplihedra with Stefan Forcey and Aaron Lauve. SIAM Journal on Discrete Mathematics, 24, No. 4, (2010), 1250-1271. Published online, DOI: 10.1137/090776834.
67	Solving Schubert Problems with Littlewood-Richardson Homotopies, with Ravi Vakil and Jan Verschelde. Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, edited by Stephen M. Watt, pages 179-186, ACM 2010.
68	Experimentation at the Frontiers of Reality in Schubert Calculus with Chris Hillar, Luis García-Puente, Abraham Martín del Campo-Sanchez, James Ruffo, Zach Teitler, and Stephen L. Johnson. Contemporary Mathmematics, 317, Amer. Math. Soc., Providence, RI, 2010, pp. 365-380.
69	A Skew Littlewood-Richardson rule from Hopf algebras with Thomas Lam and Aaron Lauve. IMRN, doi:10.1093/imrn/rnq104, Published June 2, 2010.
70	Some geometrical aspects of control points for toric patches with Gheorghe Craciun and Luis Garcia, in Mathematical Methods for Curves and Surfaces, Lecture Notes in Computer Science 5862, Springer 2010, 111–135.
71	General isotropic flags are general (for Grassmannian Schubert calculus), Journal of Algebraic Geometry, 19 (2010), 367-370.
72	Frontiers of Reality in Schubert Calculus Bulletin of the AMS, 47, No 1. (2010), 31-71.
73	Betti number bounds for fewnomial hypersurfaces via stratified Morse theory, with Frédéric Bihan, Proc. Amer. Math. Soc., 137, No. 9 (2009), 2825-2833.
74	Linear precision for parametric patches with Luis Garcia Advances in Computational Mathematics. 33 (2010), Page 191--214. DOI: 10.1007/s10444-009-9126-7
75	Linear precision for toric surface patches, with Kristian Ranestad and Hans-Christian Graf von Bothmer. 25 pages. ArXiv.org/0806.3230. Foundations of Computational Mathematics, 10, Issue 1 (2010), 37--66, DOI: 10.1007/s10208-009-9052-6.
76	Convex Hulls of Orbits and Orientations of a Moving Protein Domain, with Marco Longinetti and Luca Sgheri, Discrete and Computational Geometry, 43 No. 1 (2010), 54–77.
33 77	Galois groups of Schubert problems via homotopy computation, with Anton Leykin, Mathematics of Computation, 78 (2009) 1749--1765.
78	A Littlewood-Richardson rule for Grassmannian permutations with Kevin Purbhoo. Proc. Amer. Math. Soc., 137 (2009) 1875--1882.
79	(Theorems of Brion, Lawrence, and Varchenko on rational generating functions for cones), with Matthias Beck and Christian Haase. Mathematical Intelligencer, 31 (2009), No. 1, 9--17.
80	The equivariant cohomology rings of quot schemes, with Tom Braden, and Linda Chen. Pacific Journal of Mathematics, 238, No. 2 (2008) 201--232.
81	Gale duality for complete intersections, with Frédéric Bihan, Annales de l'Institut Fourier, Tome 58 (2008) fasicule 3, 877--891. A Singular script which computes examples of the Kouchnirenko Theorem for systems of Master functions.
82	The recursive nature of cominuscule Schubert calculus with Kevin Purbhoo Advances in Mathematics, 217 (2008), pp. 1962--2004.
83	Line problems in nonlinear computational geometry, with Thorsten Theobald. Surveys on Discrete and Computational Geometry - Twenty Years Later, Contemporary Mathematics, 453 AMS, 2008, 411--432.
84	Real Hessian Curves with Adriana Ortiz-Rodríguez. Boletín de la Sociedad Matemática Mexicana, Volume 13 Number 3 (2007), 157--166. Companion web page.
85	Bounds on the number of real solutions to polynomial equations, with Daniel J. Bates, Frédéric Bihan, IMRN, 2007, 2007:rnm114-7.
86	Sharpness of fewnomial bound and the number of components of a fewnomial hypersurface, with Frédéric Bihan and J. Maurice Rojas. IMA Volume 146: Algorithms in Algebraic Geometry edited by Alicia Dickenstein, Frank-Olaf Schreyer, and Andrew J. Sommese. 15--20, Springer, New York, 2007.
87	New fewnomial upper bounds from Gale dual polynomial systems, with Frédéric Bihan, Moscow Mathematics Journal, 7 (2007), Number 3, 387--407.
88	Lines tangent to four triangles in three-dimensional space with Hervé Brönnimann, Olivier Devillers, and Sylvain Lazard. Discrete and Computational Geometry, 37, No. 3, (2007), 369--380. Companion web page.
89	A Pieri-type formula for the K-theory of a flag manifold, with Cristian Lenart, Trans. Amer. Math. Soc., Trans. Amer. Math. Soc. 359 (2007), 2317--2342.
90	Irrational proofs of three theorems of Stanley, with Matthias Beck, European Journal of Combinatorics, 28, Issue 1, January 2007, 403–409.
91	Grothendieck Polynomials via Permutation Patterns and chains in the Bruhat Order, with Cristian Lenart and Shawn Robinson. American Journal of Mathematics, 128, No. 4, (2006), 805--848.
92	Experimentation and conjectures in the real Schubert calculus for flag manifolds, with James Ruffo, Yuval Sivan, Evgenia Soprunova, Experimental Mathematics, 15, No. 2 (2006), 199--221.
93	Cremona Convexity, Frame Convexity, and a Theorem of Santaló, with Jacob E. Goodman, Andreas Holmsen, Richard Pollack, and Kristian Ranestad. Advances in Geometry, 6, No. 3, (2006), 301–322.
94	Polynomial systems with few real zeroes, with Benoît Bertrand and Frédéric Bihan, Mathematisches Zeitschrift, 253 (2006), no. 2, 361--385.
95	Lower Bounds for Real Solutions to Sparse Polynomial Systems, with Evgenia Soprunova, Advances in Mathematics, Volume 204, Issue 1, 1 August 2006, 116--151.
96	Cocommutative Hopf algebras of permutations and trees, with Marcelo Aguiar. Journal of Algebraic Combinatorics, Volume 22 (2005), Number 4, pp. 451--470.
97	Combinatorial Hopf Algebras and generalized Dehn-Sommerville relations, with Marcelo Aguiar and Nantel Bergeron, Compositio Mathematica, 142, January 2006, 1--30.
98	Structure of the Loday-Ronco Hopf algebra of trees, with Marcelo Aguiar, Journal of Algebra, Volume 295, Issue 2 , 15 January 2006, 473--511.
99	Transversals to line segments in R3, with Hervé Brönnimann, Hazel Everett, Sylvain Lazard, and Sue Whitesides, Discrete and Computational Geometry, Volume 34, Number 3, (2005), 381--390. Companion web page.
100	Quiver Coefficients are Schubert Structure Constants, with Anders Buch and Alex Yong, Mathematics Research Letters, Volume 12, Issue 4, (2005) 567–574.
101	Real k-flats tangent to quadrics in Rn, with Thorsten Theobald. Proc. Amer. Math. Soc., 133 (2005), 2835--2844.
102	The envelope of lines meeting a fixed line and tangent to two spheres, with Gábor Megyesi, Discrete and Computational Geometry, 33, Number 4, (2005) 617–644.     Companion web page.
103	Structure of the Malvenuto-Reutenauer Hopf algebra of permutations, with Marcelo Aguiar, Advances in Mathematics 191, (2005), 225--275.
104	A New Approach to Hilbert's Theorem on Ternary Quartics, with Victoria Powers, Bruce Reznick, and Claus Scheiderer, Comptes Rendus Mathematique (Paris), 339, Issue 9, (2004), 617--620.
105	Maximally inflected real rational curves, with Viatcheslav Kharlamov. Moscow Mathematics Journal 3 (2003), no. 3, 947--987, 1199--1200.     Companion Web Page.
106	Toric ideals, real toric varieties, and the algebraic moment map, in Topics in Algebraic Geometry and Geometric Modeling, ed. by R. Goldman and R. Krasuaskas, Contemp. Math. 334, 2003. pp. 225–240. (Proceedings of AGGM, Vilnius, Lithuania.)
107	Elementary transversality in the Schubert calculus in any characteristic. Michigan Mathematical Journal, 51 (2003), 651–666.
108	Common transversals and tangents to two lines and two quadrics in P3, with Gábor Megyesi and Thorsten Theobald. Discrete and Computational Geometry, 30, (2003), 543–571.     Companion Web Page.
109	Enumerative Real Algebraic Geometry, in Algorithmic and Quantitative Aspects of Real Algbraic Geometry, S. Basu and L. Gonzalez-Vega, eds., DIMACS series 60, AMS, 2003. pp. 139--180.       Expanded on-line version.
110	Skew Schubert polynomials, with Cristian Lenart, Proc. Amer. Math. Soc., 131 (2003), 3319-3328.
111	Lines tangent to 2n-2 spheres in Rn, with Thorsten Theobald, Trans. Amer. Math. Soc., 354 (2002), 4815--4829.   Some pictures and computer code.
112	A Pieri-type formula for isotropic flag manifolds, with Nantel Bergeron, Trans. Amer. Math. Soc., 354 No. 7, (2002), 2659--2705.
113	Shifted quasi-symmetric functions and the Hopf algebra of peak functions with Nantel Bergeron, S. Mykytiuk, and S. van Willigenburg, Discrete Math., 256 (2002), 57--66.
114	Skew Schubert functions and the Pieri formula for flag manifolds, with Nantel Bergeron. Trans. Amer. Math. Soc., 354 No. 2, (2002), 651–673.
115	From enumerative geometry to solving systems of polynomial equations with Macaulay 2, in Computations in Algebraic Geometry with Macaulay 2, edited by D. Eisenbud, D. Grayson, M. Stillman, and B. Sturmfels. Algorithms and Computation in Mathematics 8, Springer-Verlag, 2001. pp. 101-129. Macaulay 2 source code for computations in the paper. (This requires the Macaulay 2 package realroots.m2.)
116	Complementary Algorithms for Tableaux, with Tom Roby, Jeff Stroomer, and Julian West. J. Combin. Th. Ser. A, 96, No. 1, October 2001, 127-161.
117	Rational curves on Grassmannians: systems theory, reality, and transversality, In "Advances in Algebraic Geometry Motivated by Physics", ed. by Emma Previato, Contemporary Mathematics, 276, 2001, pp. 9--42.
118	A sagbi basis for the quantum Grassmannian, with Bernd Sturmfels. J. Pure and Appl. Algebra, 158, 24 April 2001 pp. 347-366.
119	Some real and unreal enumerative geometry for flag manifolds, Michigan Math Journal, 48 (2000) pp. 573-592.
120	Non-commutative Pieri operators on posets, Nantel Bergeron, S. Mykytiuk, and S. van Willigenburg, J. Combin. Th. Ser. A, Vol. 91, No. 1/2, Aug 2000, pp. 84-110.
121	Real Schubert Calculus: Polynomial systems and a conjecture of Shapiro and Shapiro, Experimental Mathematics, 9, Number 2, (2000), pp. 161-182. An archive of the computations and Maple scripts for some proofs.
122	Real rational curves in Grassmannians, J. Amer. Math. Soc. 13 (2000), 333-341.
123	Pieri-type formulas for maximal isotropic Grassmannians via triple intersections, Colloquium Mathematicum, 82 (1999), pp. 49--63.
124	Hopf algebras and edge-labeled posets, with Nantel Bergeron, Journal of Algebra, 216 (1999), pp. 641-651.
125	A monoid for the Grassmannian Bruhat order, with Nantel Bergeron, European Journal of Combinatorics, 20, (1999), pp. 197-211.
126	The special Schubert calculus is real, Electronic Research Announcements of the AMS, 5, 1999, pp. 35-39.
127	Numerical Schubert calculus, with Birkett Huber and Bernd Sturmfels, Journal of Symbolic Computation, 26, (1998) pp. 767-788.
128	Schubert polynomials, the Bruhat order, and the geometry of flag manifolds, with Nantel Bergeron, Duke Math. J., 95, (1998), pp. 373-423. A 16 page supplement. A description of how Table 1 was generated.
129	Some remarks on real and complex output feedback, with Joachim Rosenthal, Sys. & Control Lett., 33 (1998), pp. 73-80. A non-trivial real system that cannot be controlled by real output feedback.
130	Pieri's formula via explicit rational equivalence, Canad. J. Math., 46 (1997), pp. 1281-1298. A 5 page supplement.
131	Enumerative geometry for real varieties, in Algebraic Geometry, Santa Cruz 1995, ed. by János Kollár, Proceedings and Symposia in Pure Mathematics, 61, No. 1, AMS 1997. pp. 435-447. An appendix contains further discussion of work of Ronga, Tognoli, and Vust.
132	Real enumerative geometry and effective algebraic equivalence, J. Pure and Appl. Algebra, 117 & 118 (1997), 601-615.
133	Enumerative geometry for the real Grassmannian of lines in projective space, Duke Math. J., 87, (1997), 59-85. Published version of Ph.D. Thesis. The 8 page supplement.
134	Tableau switching: algorithms and applications, with Georgia Benkart and Jeff Stroomer, J. Combin. Th. Ser. A, 76, (1996), pp. 11-43.
135	Pieri's formula for flag manifolds and Schubert polynomials, Annales de l'Institut Fourier, 46 (1996), pp. 89--110. Lemma 15 in the published version has a typo, ej should be ew(j).
136	Intersection theory on spherical varieties, with Wm. Fulton, R. MacPherson, and Bernd Sturmfels, J. Alg. Geom., 4, (1995), pp. 181-193.
137	Real enumerative geometry for the Grassmannian of lines in projective space, Ph.D. Thesis, University of Chicago, August 1994. Published as Enumerative geometry for the real Grassmannian of lines in projective space

1	Certification for Polynomial Systems via Square Subsystems with Timothy Duff. 11 pages 2018. Extended abstract for MEGA 2019, Madrid. arXiv.org/1812.02851.
2	Certifiable Numerical Computations in Schubert Calculus, With Jonathan D. Hauenstein and Nickolas Hein. Extended abstract.
3	An inequality of Kostka numbers and Galois groups of Schubert problems, With Abraham Martín del Campo-Sanchez, and Christopher J. Brooks. Extended abstract for FPSAC12, July 30 – August 3, 2012 / Nagoya, Japan.
4	Injectivity of 2D Toric Bézier Patches, With Chungang Zhu, Extended abstract for CAD/Graphics, Jinan, China 2011.
5	Cofree compositions of coalgebras, With Stefan Forcey and Aaron Lauve. Extended abstract for FPSAC11, June 13--17, Reykjavik.
6	The Monotone Secant Conjecture in the Real Schubert Calculus, With Jonathan D. Hauenstein, Nickolas Hein, Christopher J. Hillar, Abraham Martín Del Campo, and Zach Teitler Extended abstract for MEGA 2011, Stockholm.
7	Toric Degenerations of Bézier Patches, With Luis García-Puente and Chungang Zhu. Extended abstract for MEGA 2011, Stockholm.
8	Solving Schubert Problems with Littlewood-Richardson Homotopies, With Ravi Vakil and Jan Verschelde. Extended abstract for ISSAC 10, Munich.
9	New Hopf Structures on Binary Trees, With Aaron Lauve and Stefan Forcey. Extended abstract for FPSAC09, July 2009, RISC, Hagenberg, Austria.
10	Gale duality for complete intersections, With Frédéric Bihan. Abstract for Oberwolfach report on the workshop on tropical geometry, 10--14 December 2007.
11	Computing monodromy via parallel homotopy continuation, With Anton Leykin. Extended abstract. For Parallel Symbolic Computation'07, 27 - 28 July 2007 University of Western Ontario May 2007.
12	The Horn recursion for Schur P- and Q- functions, With Kevin Purbhoo. Extended abstract. For the FPSAC`06 conference in San Diego, CA. February 2006.
13	Experimentation and Conjectures in the real Schubert calculus for flag manifolds, Extended abstract. Presented at the MEGA05 Conference. January 2005.
14	On the number of tangents to four triangles in three-dimensional space. With Olivier Devillers, Hervé Brönnimann, and Sylvain Lazard. Presented by Brönnimann at 16th Canadian Conference on Computational Geometry, Concordia University, Montreal. August 9-11, 2004.     Companion web page.
15	Transversals to line segments in R3. With Hervé Brönnimann, Hazel Everett, Sylvain Lazard, and Sue Whitesides. Presented by Brönnimann at 15th Canadian Conference on Computational Geometry, Dalhousie University, Halifax. 11–13 August, 2003.     Companion web page.
16	Structure of the Malvenuto-Reutenauer Hopf algebra of permutations, with Marcelo Aguiar. Presented by Aguiar at Formal Power Series and Algebraic Combinatorics, Melbourne, Australia, 8-13 July 2002.
17	Jeux de Tableaux. With Tom Roby, Jeff Stroomer, and Julian West. Presented by Julian West at FPSAC00. Published in Proceedings, 12th International Conference, FPSAC`00, Moscow, June 2000, D. Krob, A.A. Mikhalev, A.V. Mikhalev, eds. Springer-Verlag, 2000. pp. 332-343.
18	Shifted quasi-symmetric functions and the Hopf algebra of peak functions, with Nantel Bergeron, S. Mykytiuk, and S. van Willigenburg, Presented by Mykytiuk at FPSAC, Barcelona, 7-11 June 1999.
19	Identities of Littlewood-Richardson coefficients for Schubert polynomials and orders on Sn , with Nantel Bergeron, Formal Power Series and Algebraic Combinatorics, 14-18 July, 1997, Universität Wien.
20	A geometric approach to the combinatorics of Schubert polynomials, Formal Power Series and Algebraic Combinatorics, May 29 - June 2, 1995, Marne-la-Valée, Paris.

1. Enumerative Real Algebraic Geometry. A web-based survey of the subject. Web version of paper by the same name. August 2002.
2. Some Maximally inflected curves. Work with Viatcheslav Kharlamov. This page accompanies the paper maximally Inflected Curves. June 2002.
3. Lines transversal to 2 lines and a quadric. Work with Gábor Megyesi and Thorsten Theobald. This page supplements the paper of the same name. April 2002.
4. The conjectures of Shapiro and Shapiro: an archive of computations. This describes the conjectures of Shapiro and Shapiro, providing more context than the paper Real Schubert Calculus: Polynomial systems and a conjecture of Shapiro and Shapiro. Local version. April 2000.
5. Identities of Littlewood-Richardson Coefficients for Schubert polynomials. This describes how Nantel Bergeron and I generated Table 1 of Schubert polynomials, the Bruhat order, and the Geometry of Schubert varieties and discusses the Grassmannian Bruhat order. August 1998.
6. A non-trivial real system that cannot be controlled by real output feedback. An amplification of part of Some remarks on real and complex output feedback. June 1997.

1. Ibadan Lectures on Toric Varieties, expanded version of notes from 2017 CIMPA Research School in Computational and Combinatorial Algebraic Geometry in Ibadan, Nigeria, 12–23 June 2017. .
2. Review of the book Toric Varieties by D. Cox, J. Little, and H. Schenck. For the Bulletin of the AMS, complete August 2013. Published ????. .
3. What we are doing about the high cost of textbooks, AIM Editorial Board, Scripta Manent, Notices of the AMS, August 2013. .
4. Frontiers of Reality in Schubert Calculus. AMS Current Events Bulletin, January 2009.   .
5. Real Solutions to Equations from Geometry. Lecture notes from short course at the Centre Émile Borel, Institut Henri Poincaré, Given in November 2005.   .
6. Tropical Interpolation, Article for Autumn 2004 issue of the Emissary, the newsletter of MSRI. math.AG/0501146,   .
7. Letter to the Editor of the Notices of the AMS concerning the Digital Millenuim Coptright Act, Volume 43, Number 3, (2002), page 310. Text of letter. Also signed by George Avrunin, Anders Buch, and Andrew Hwang.
8. Supplement III, Encyclopaedia of Mathematics. M. Hazewinkel, Managing Editor Schubert Calculus 343-344, Schubert Cell 344-344, Schubert Cycle 344-345, and Schubert Polynomial 345-346. Kluwer Academic Publishers, 2002
9. Proceedings, 10 International Conference on Formal Power Series and Algebraic Combinatorics (10^ième conférence, Les Séries Formelles et la Combinatoire Algébriques) 14-19 June 1998, Fields Institute, Toronto. Ed. by Nantel Bergeron, Ian Goulden, and Frank Sottile. (See conference web pages:FPSAC'98, SFCA'98)
10. Starting Our Careers: A Collection of Essays and Advice on Professional Development from the Young Mathematicians' Network. Edited by Analissa Crandel and Curt Bennett. AMS 1999, 116pp.
    The following articles I wrote appear in that compilation:
    • Beating the Cost of Meetings Originally in CoYM v3.i1
    • Giving Math Talks, with Curt Bennett. Originally in CoYM v3.i4
    • AMS Special Sessions, with Curt Bennett. Originally in CoYM v3.i35
    • Getting Started in Research: a personal perspective Originally in CoYM v5.i6
11. A graduate student's view of the international congress of mathematicians, Mathematical Intelligencer, 9 No. 2 (1987) 33-35.