This week’s Macaulay2 event featured presentations, Q&A, and ample practice problems on a wide variety of topics. Many thanks to the speakers and organizers for making this event so successful. Stay tuned for next week’s conference!
People in algebraic geometry have used monodromy groups for a long time—in particular, use of monodromy group to solve polynomial systems has appeared elsewhere recently (see here, and here) and is certainly implicit in earlier algorithms. In fact, the possible variations on this idea seem limitless. Our implementation allows one to compare various approaches and unifies them in a general framework.
The package currently works best for the following problem: we are given a family of square polynomial systems in variables over the complex numbers, where the parameter space is a finite-dimensional affine space, such that, for a Zariski open subset of there exists such that However, the various methods are modular enough to incorporate them into solving more general systems—see here for a great example. Understanding what users want in more general settings will help us improve the package in the future, so try it out and let us know what you think!
Abstract: The toric h-numbers of a dual hypersimplex and the Chow Betti numbers of the normal fan of a hypersimplex are the ranks of intersection cohomology and Chow cohomology respectively of the torus orbit closure of a generic point in the Grassmannian. We give explicit formulas for these numbers. We also show that similar formulas hold for the coordinator numbers of type A^* lattices.
“These tutorials are intended to appeal to participants with any level of prior M2 experience. The topics will range from the basic functionality of M2 to modeling problems in the M2 language to more specialized tutorials on algebraic statistics and numerical algebraic geometry. We will also reserve ample time for practice and Q&A sessions.”