Dr. Nickolas Hein

Nickolas Hein grew up in Wichita and came to BC in 2016. He earned a Ph.D. in mathematics at Texas A&M after receiving B.A. and M.A. degrees in mathematics at KU in Lawrence. His research includes algebraic geometry, combinatorics, and numerical analysis. In particular, his interests include Schubert calculus, Catalan objects, certification of numerical solutions, reality in algebraic geometry, and experimental mathematics. Dr. Hein is an active reviewer for the American Mathematical Society Mathematical Reviews. He began a one-year term as chair for the Kansas section of the Mathematical Association of America in Spring 2022.

Nickolas is grateful to have been called to Atchison and enjoys raising his four children, with his wife, surrounded by the stable presence of the local Benedictine community. Dr. Hein is the faculty advisor to the Mathematics and Computer Science club at BC. He also helps run the St. Benedict Elementary School chess club.

EDUCATION

Ph.D., Texas A&M University, 2013

Dissertation: Reality and Computation in Schubert Calculus

M.A., The University of Kansas, 2006

Thesis: The Riemann-Roch Theorem for Complex Curves

B.A., The University of Kansas, 2003

Honors Thesis: Constructible Numbers and the Insolvability of the Quintic

CLASSES

• Intermediate College Algebra, College Algebra
• Pre-Calculus
• Applied Differential Calculus, Applied Integral Calculus
• Calculus 1,2
• Multivariable Calculus
• Discrete Mathematical Structures
• Euclidean Geometry
• Linear Algebra
• Modern Algebra I, II
• Complex Analysis
• Directed Research
• Senior Comprehensive Exam
• Numerical Solution of Polynomial Systems Arising in Engineering and Science
• Introduction to Graph Theory
• Groebner Bases and Convex Polytopes
• Modern Algebra with Geometry (graduate)
• Algebraic Geometry (graduate)

RESEARCH INTERESTS

Reformulating classical problems to optimize solution by modern techniques, computational solution of problems with structure, experiments involving reality in algebraic geometry, variants of the Catalan numbers.

For some computational archives, click here.

PUBLICATIONS

Nickolas Hein and Jia Huang. Variations of the Catalan numbers from some nonassociative binary operations, Discrete Mathematics, Volume 345, Issue 3, 2022, ISSN 0012-365X, 21 pp., https://doi.org/10.1016/j.disc.2021.112711 .

Timothy Duff, Nickolas Hein, and Frank Sottile. Certification for Polynomial Systems via Square Subsystems, Journal of Symbolic Computation, July 2020, 21 pp., https://doi.org/10.1016/j.jsc.2020.07.010 .

Nickolas Hein, Frank Sottile, Igor Zelenko. A congruence modulo four for real Schubert Calculus with isotropic flags, Canadian Mathematical Bulletin, Published electronically ahead of print, 2017, 9 pp., http://dx.doi.org/10.4153/CMB-2016-087-2 .

Nickolas Hein and Frank Sottile. A lifted square formulation for certifiable Schubert calculus, Journal of Symbolic Computation (2017), pp. 594-608, http://dx.doi.org/10.1016/j.jsc.2016.07.021 .

Nickolas Hein, Jia Huang. Modular Catalan Numbers, European Journal of Combinatorics, Vol. 61 Iss. C, Mar. 2017, pp. 197-218, http://dx.doi.org/10.1016/j.ejc.2016.11.004 .

Jonathan Hauenstein, Nickolas Hein, and Frank Sottile. A primal-dual formulation for certifiable computations in Schubert calculus, Foundations of Computational Mathematics, 16 (2016), pp. 941-963, http://dx.doi.org/10.1007/s10208-015-9270-z .

Nickolas Hein, Frank Sottile, and Igor Zelenko. A congruence modulo four for real Schubert calculus Journal für die reine und angewandte Mathematik, vol. 2016, iss. 714 (2016), pp. 151-174, http://dx.doi.org/10.1515/crelle-2013-0122 .

Nickolas Hein, Christopher Hillar, Abraham Martín del Campo-Sanchez, Frank Sottile, and Zach Teitler. The monotone secant conjecture in the real Schubert calculus, Experimental Mathematics, 24 (2015), pp. 261-269, http://dx.doi.org/10.1080/10586458.2014.980044 .

Nickolas Hein, Christopher Hillar, and Frank Sottile. Lower bounds in real Schubert calculus, São Paulo Journal of Mathematics, 7, 1 (2013), pp. 33-58. DOI: 10.11606/issn.2316-9028.v7i1p33-58

Luis García-Puente, Nickolas Hein, Christopher Hillar, Abraham Martín del Campo-Sanchez, James Ruffo, Frank Sottile, and Zach Teitler. The secant conjecture in the real Schubert calculus, Experimental Mathematics, 21, 3 (2012), pp. 252-265.

REFEREED EXTENDED ABSTRACTS

N. Hein and J. Huang. Variations of the Catalan Numbers From Some Nonassociative Binary Operations, extended abstract, presented at Séminaire Lotharingien de Combinatoire - FPSAC 2018, published in Issue 80B,  Proceedings of the 30th International Conference on "Formal Power Series and Algebraic Combinatorics", July 2018, Extended Abstract, 12 pp.

J.D. Hauenstein, N. Hein, and F. Sottile, Certifiable Numerical Computations in Schubert Calculus, extended abstract, presented at MEGA 2013, Frankfurt, Germany.

J.D. Hauenstein, N. Hein, C.J. Hillar, A. Martín del Campo-Sanchez, F. Sottile, and Z. Teitler. The monotone secant conjecture in the real Schubert calculus, extended abstract, presented at MEGA 2011, Stockholm, Sweden.

REFEREED EXTENDED ABSTRACTS

Hein and J. Huang. Variations of the Catalan Numbers From Some Nonassociative Binary Operations, extended abstract, presented at Séminaire Lotharingien de Combinatoire - FPSAC 2018, published in Issue 80B,  Proceedings of the 30th International Conference on "Formal Power Series and Algebraic Combinatorics", July 2018, Extended Abstract, 12 pp.

J.D. Hauenstein, N. Hein, and F. Sottile, Certifiable Numerical Computations in Schubert Calculus, extended abstract, presented at MEGA 2013, Frankfurt, Germany.

J.D. Hauenstein, N. Hein, C.J. Hillar, A. Martín del Campo-Sanchez, F. Sottile, and Z. Teitler. The monotone secant conjecture in the real Schubert calculus, extended abstract, presented at MEGA 2011, Stockholm, Sweden.