Faculty & Staff

Dr. Nickolas Hein

Dr. Nickolas Hein

Assistant Professor
Founders Hall, 2142
(308) 865-8635
heinnj@unk.edu

Dr. Hein earned a Ph.D. in mathematics from Texas A&M University and a M.A. from the University of Kansas.

Syllabi
Math 115-03 (Fall 2014)
Math 310-01 (Fall 2014)

Classes Taught at the University of Nebraska, Kearney
Modern Algebra with Geometry (MATH 871)
Linear Algebra (MATH 440)
College Geometry (MATH 310)
5 hour Calculus 1 (MATH 115)
3 hour Calculus 1 (MATH 123)
College Algebra (MATH 102)

Classes Taught at Texas A&M University
3 hour Calculus 1
5 hour Calculus 1 (TA and lab instructor)
5 hour Calculus 2 (TA and lab instructor)
Functions (TA)

Classes Taught at the University of Kansas
5 hour Calculus 1
5 hour Calculus 2
3 hour Calculus 1
3 hour Calculus 2
College Algebra
Intro to College Algebra

I use Schubert calculus to investigate phenomena in algebraic geometry, particularly in problems involving intersections or incidence conditions. Combinatorial commutative algebra provides results about the structure of intersection problems from Schubert calculus, and this lends Schubert problems to study in computational algebraic geometry. I investigate methods of formulating overdetermined systems as square systems which allows the certification of approximate solutions obtained via modern numerical software tools.

Preprints

arXiv The monotone secant conjecture in the real Schubert calculus. (with Hillar, Martín del Campo, Sottile, and Teitler). [links to full journal version]
arXiv A primal-dual formulation for certifiable computations in Schubert calculus. Preprint. (with Hauenstein and Sottile).
arXiv A congruence modulo four in real Schubert calculus with isotropic flags. Preprint. (with Sottile and Zelenko).
arXiv A congruence modulo four in real Schubert calculus, Crelles Journal, 2014 (with Sottile and Zelenko).
arXiv Lower bounds in real Schubert calculus, Resenhas, 2014 (with Hillar and Sottile).
arXiv The secant conjecture in the real Schubert calculus, Experimental Mathematics, 2012 (with García-Puente, Hillar, Martín del Campo, Ruffo, Sottile, and Teitler).

Data sets

www Beyond the Shapiro Conjecture and Eremenko-Gabrielov lower bounds. Data Set, 2013 (with Sottile).
www Monotone [Secant] Experimental Project. Data Set, 2013 (with Hillar, Martín del Campo, Sottile, and Teitler).
www Beyond the Shapiro Conjecture and Eremenko-Gabrielov lower bounds. Preliminary Data Set, 2010 (with Hauenstein, Martín del Campo, and Sottile).
www Secant Experimental Project. Data Set, 2010 (with Garcia-Puente, Hillar, Martín del Campo, Ruffo, Sottile, and Teitler).

Extended abstracts

arXiv Certifiable numerical computations in Schubert calculus, Extended Abstract presented at MEGA 2013 (with Hauenstein and Sottile).
arXiv The monotone secant conjecture in the real Schubert calculus. Extended Abstract in MEGA 2011 proceedings (with Hauenstein, Hillar, Martín del Campo, Sottile, and Teitler). [links to full journal version]

Dissertation

arXiv Reality and computation in Schubert calculus

Here are examples of primal-dual square formulations for solving (traditionally overdetermined) Schubert problems.  They were solved using regeneration in bertini and certified using rational arithmetic in alphaCertified.