@article{2006,
  abstract     = {The monotone secant conjecture posits a rich class of polynomial systems, all of whose solutions are real. These systems come from the Schubert calculus on flag manifolds, and the monotone secant conjecture is a compelling generalization of the Shapiro conjecture for Grassmannians (Theorem of Mukhin, Tarasov, and Varchenko). We present some theoretical evidence for this conjecture, as well as computational evidence obtained by 1.9 teraHertz-years of computing, and we discuss some of the phenomena we observed in our data. },
  author       = {Hein, Nicolas and Hillar, Christopher and Martin Del Campo Sanchez, Abraham and Sottile, Frank and Teitler, Zach},
  journal      = {Experimental Mathematics},
  number       = {3},
  pages        = {261 -- 269},
  publisher    = {Taylor & Francis},
  title        = {{The monotone secant conjecture in the real Schubert calculus}},
  doi          = {10.1080/10586458.2014.980044},
  volume       = {24},
  year         = {2015},
}

@article{1579,
  abstract     = {We show that the Galois group of any Schubert problem involving lines in projective space contains the alternating group. This constitutes the largest family of enumerative problems whose Galois groups have been largely determined. Using a criterion of Vakil and a special position argument due to Schubert, our result follows from a particular inequality among Kostka numbers of two-rowed tableaux. In most cases, a combinatorial injection proves the inequality. For the remaining cases, we use the Weyl integral formulas to obtain an integral formula for these Kostka numbers. This rewrites the inequality as an integral, which we estimate to establish the inequality.},
  author       = {Brooks, Christopher and Martin Del Campo Sanchez, Abraham and Sottile, Frank},
  journal      = {Transactions of the American Mathematical Society},
  number       = {6},
  pages        = {4183 -- 4206},
  publisher    = {American Mathematical Society},
  title        = {{Galois groups of Schubert problems of lines are at least alternating}},
  doi          = {10.1090/S0002-9947-2014-06192-8},
  volume       = {367},
  year         = {2015},
}

@article{2178,
  abstract     = {We consider the three-state toric homogeneous Markov chain model (THMC) without loops and initial parameters. At time T, the size of the design matrix is 6 × 3 · 2T-1 and the convex hull of its columns is the model polytope. We study the behavior of this polytope for T ≥ 3 and we show that it is defined by 24 facets for all T ≥ 5. Moreover, we give a complete description of these facets. From this, we deduce that the toric ideal associated with the design matrix is generated by binomials of degree at most 6. Our proof is based on a result due to Sturmfels, who gave a bound on the degree of the generators of a toric ideal, provided the normality of the corresponding toric variety. In our setting, we established the normality of the toric variety associated to the THMC model by studying the geometric properties of the model polytope.},
  author       = {Haws, David and Martin Del Campo Sanchez, Abraham and Takemura, Akimichi and Yoshida, Ruriko},
  journal      = {Beitrage zur Algebra und Geometrie},
  number       = {1},
  pages        = {161 -- 188},
  publisher    = {Springer},
  title        = {{Markov degree of the three-state toric homogeneous Markov chain model}},
  doi          = {10.1007/s13366-013-0178-y},
  volume       = {55},
  year         = {2014},
}

@article{5920,
  abstract     = {We study chains of lattice ideals that are invariant under a symmetric group action. In our setting, the ambient rings for these ideals are polynomial rings which are increasing in (Krull) dimension. Thus, these chains will fail to stabilize in the traditional commutative algebra sense. However, we prove a theorem which says that “up to the action of the group”, these chains locally stabilize. We also give an algorithm, which we have implemented in software, for explicitly constructing these stabilization generators for a family of Laurent toric ideals involved in applications to algebraic statistics. We close with several open problems and conjectures arising from our theoretical and computational investigations.},
  author       = {Hillar, Christopher J. and Martin del Campo Sanchez, Abraham},
  issn         = {0747-7171},
  journal      = {Journal of Symbolic Computation},
  pages        = {314--334},
  publisher    = {Elsevier},
  title        = {{Finiteness theorems and algorithms for permutation invariant chains of Laurent lattice ideals}},
  doi          = {10.1016/j.jsc.2012.06.006},
  volume       = {50},
  year         = {2013},
}