Numerical Schubert Calculus in Macaulay2

The Macaulay2 package NumericalSchubertCalculus provides methods for the numerical computation of Schubert problems on Grassmannians. It implements both the Pieri homotopy algorithm and the Littlewood-Richardson homotopy algorithm. Each algorithm has two independent implementations in this package. One is in the scripting language of Macaulay2 using the package NumericalAlgebraicGeometry, and the other is in the compiled code of PHCpack.

Decomposable sparse polynomial systems

The Macaulay2 package DecomposableSparseSystems implements methods for studying and numerically solving decomposable sparse polynomial systems. We describe the structure of decomposable sparse systems and explain how the methods in this package may be used to exploit this structure, with examples.

Solving Decomposable Sparse Systems

Amendola et al. proposed a method for solving systems of polynomial equations lying in a family which exploits a recursive decomposition into smaller systems. A family of systems admits such a decomposition if and only if the corresponding Galois group is imprimitive. When the Galois group is imprimitive we consider the problem of computing an explicit decomposition. A consequence of Esterovs classification of sparse polynomial systems with imprimitive Galois groups is that this decomposition is obtained by inspection. This leads to a recursive algorithm to solve decomposable sparse systems, which we present and give evidence for its efficiency.

Generic properties of dispersion relations for discrete periodic operators

An old problem in mathematical physics deals with the structure of the dispersion relation of the Schrodinger operator $-Delta+V(x)$ in $R^n$ with periodic potential near the edges of the spectrum. A well known conjecture says that generically (with respect to perturbations of the periodic potential) the extrema are attained by a single branch of the dispersion relation, are isolated, and have non-degenerate Hessian (i.e., dispersion relations are graphs of Morse functions). The important notion of effective masses in solid state physics, as well as Liouville property, Greens function asymptotics, etc. hinge upon this property. The progress in proving this conjecture has been slow. It is natural to try to look at discrete problems, where the dispersion relation is (in appropriate coordinates) an algebraic, rather than analytic, variety. Such models are often used for computation in solid state physics (the tight binding model). Alas, counterexamples exist in some discrete situations. We start establishing the following dichotomy: the non-degeneracy of extrema either fails or holds in the complement of a proper algebraic subset of the parameters. The known counterexample has only two free parameters. This might be too tight for genericity to hold. We consider the maximal $Z^2$-periodic two-atomic nearest-cell interaction graph, with nine edges per unit cell and the discrete Laplace-Beltrami operator on it. We then use methods from computational and combinatorial algebraic geometry to prove the genericity conjecture for this graph. We show three different approaches to the genericity, which might be suitable in various situations. It is also proven in this case that adding more parameters does not destroy the genericity result. We list all bad periodic subgraphs of the one we consider and discover that in all these cases genericity fails for trivial reasons only.

Classification of Schubert Galois groups in Gr(4,9)

We classify Schubert problems in the Grassmannian of 4-planes in 9-dimensional space by their Galois groups. Of the 31,806 essential Schubert problems in this Grassmannian, only 149 have Galois group that does not contain the alternating group. We identify the Galois groups of these 149---each is an imprimitive permutation group. These 149 fall into two families according to their geometry. This study suggests a possible classification of Schubert problems whose Galois group is not the full symmetric group, and it begins to establish the inverse Galois problem for Schubert calculus.

Numerical Schubert Calculus via the Littlewood-Richardson Homotopy Algorithm

We develop the Littlewood-Richardson homotopy algorithm, which uses numerical continuation to compute solutions to Schubert problems on Grassmannians and is based on the geometric Littlewood-Richardson rule. One key ingredient of this algorithm is our new optimal formulation of Schubert problems in local Stiefel coordinates as systems of equations. Our implementation can solve problem instances with tens of thousands of solutions.

Convex Hull of Two Circles in R^3

We describe convex hulls of the simplest compact space curves, reducible quartics consisting of two circles. When the circles do not meet in complex projective space, their algebraic boundary contains an irrational ruled surface of degree eight whose ruling forms a genus one curve. We classify which curves arise, classify the face lattices of the convex hulls, and determine which are spectrahedra. We also discuss an approach to these convex hulls using projective duality.

Injectivity of generalized Wronski maps

We study linear projections on Pluecker space whose restriction to the Grassmannian is a non-trivial branched cover. When an automorphism of the Grassmannian preserves the fibers, we show that the Grassmannian is necessarily of m-dimensional linear subspaces in a symplectic vector space of dimension 2m, and the linear map is the Lagrangian involution. The Wronski map for a self-adjoint linear differential operator and pole placement map for symmetric linear systems are natural examples.

Trace test

The trace test in numerical algebraic geometry verifies the completeness of a witness set of an irreducible variety in affine or projective space. We give a brief derivation of the trace test and then consider it for subvarieties of products of projective spaces using multihomogeneous witness sets. We show how a dimension reduction leads to a practical trace test in this case involving a curve in a low-dimensional affine space.

Real Algebraic Geometry for Geometric Constraints

Real algebraic geometry adapts the methods and ideas from (complex) algebraic geometry to study the real solutions to systems of polynomial equations and polynomial inequalities. As it is the real solutions to such systems modeling geometric constraints that are physically meaningful, real algebraic geometry is a core mathematical input for geometric constraint systems.