Beyond polyhedral homotopies

We forgot to post this here in the beginning of the summer… The paper outlines a very general framework for (an) algorithm(s) waiting to be implemented.

trop-two-circles

Beyond polyhedral homotopies
(Anton Leykin and  Josephine Yu)

Abstract: We present a new algorithmic framework which utilizes tropical geometry and homotopy continuation for solving systems of polynomial equations where some of the polynomials are generic elements in linear subspaces of the polynomial ring. This approach generalizes the polyhedral homotopies by Huber and Sturmfels. 

 

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The short story of MAGA

The Meeting on Algebraic Geometry and its Applications was intended as a one-day meeting that attracts people in the South-East of the US. The first two incarnations — at Georgia Tech in 2015 and at Clemson in 2016 — gathered 25-30 participants each.

MAGA had a year off in 2017 due to a large international meeting scheduled in the same year at GT, but the meeting will run in 2018. There is also a plan to continue this as a regular annual event held in April in Atlanta.

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However, the world has changed a bit in the last couple of years and a lot of people associate the acronym MAGA… with something completely different. Therefore a question: what could be a new name/abbreviation?

ICERM Semester Program on “Nonlinear Algebra”


icerm-nonlinear-algebra

The Fall of 2018 is the semester of Nonlinear Algebra at the Institute for Computational and Experimental Research in Mathematics (ICERM) which is in  Brown University.

 

“… This semester will work towards a time when ideas of nonlinear algebra, its theory, methods, and software are as ubiquitous as those of linear algebra.”

The application deadline for students and postdocs is January 1.

 

SIAM AG 2017 in pictures

The SIAM AG meeting gathered close to 450 registered participants! The live stream of the plenary talks brought about as many virtual ones (according to the viewing statistics).

As a group photo was deemed not feasible, here are several pictures from the reception and poster session that will help us to remember that cool summer week. (Yes, the weather did cooperate as well!)

SIAM AG reception

 

 

MonodromySolver

Last month the latest stable version (1.10) of Macaulay2 was released. This includes the first stable version of the package MonodromySolver, developed by some of us at Georgia Tech (Anton, Cvetelina, Kisun, Tim), as well as our collaborators Jeff Sommars and Anders Jensen. The corresponding paper has been on the arXiv for about a year.

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 (F_p )_{p \in B} in n variables over the complex numbers, where the parameter space B is a finite-dimensional affine space, such that, for a Zariski open subset of z \in \mathbb{C}^n, there exists p \in B such that F_p(z) =0. 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!

New paper: Toric h-vectors and Chow Betti Numbers of Dual Hypersimplices

Toric h-vectors and Chow Betti Numbers of Dual Hypersimplices
by Charles Wang and Josephine Yu

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.

http://front.math.ucdavis.edu/1707.04581