Projects Page - Macaulay2/Workshop-2015-Boise GitHub Wiki

Positive Characteristic

  • Erin Bela
  • David Bruce
  • Daniel Hernandez
  • Zhibek Kadyrsizova
  • Emily Witt

Our main goal was to write an efficient and fast algorithm for approximating F-thresholds. We improved some of the existing codes by making them more efficient and fast. We also implemented codes for the natural generalizations of Frobenius powers of ideals and F-thresholds. In addition, we wrote a code for Fedder's criterion for F-purity of quotients of polynomial rings.

Tensors

  • Hirotachi Abo
  • Roberto Barrera
  • Robert Krone
  • Benjamin Reames
  • Zach Teitler

The goal of this project is to increase the functionality of the package "Tensors," which was created by Andrew Critch and Claudiu Raicu. During the workshop at Boise State University, we added 9 new features to this package including eigenvectors of tensors.

PHCpack

The goal of this project was two-fold: to upgrade PHCpack and to apply PHCpack to investigate the problem of structural identifiability.

Power Series

  • Suprajo Das
  • Nathan Grieve
  • Brent Holmes
  • Svenja Huntemann

We discussed problems related to filtered algebras and their Hilbert Series.
One source of motivation for the problems we discussed was Lemma 3.1 of the paper G. Faltings and G. Wustholz, Diophantine approximations on projective spaces, Inventiones Mathematicae, 116, 109—138 (1994). Our main goal was develop Macaulay2 code to work with classes of filtered algebras and their associated graded objects.

Quantum Cohomology

  • Corey Harris
  • Anna Kazanova
  • David Swinarski
  • Robert Williams

We wrote a package called QuantumCohomology that implements the small quantum cohomology rings of ordinary Grassmannians as top-level rings in Macaulay2. Our algorithm for multiplication in these rings uses the quantum Pieri and quantum Giambelli formulas. We also created an interface to Anders Buch's Maple code.

Splines

We began writing the package AlgebraicSplines for computations involving piecewise polynomials over pure, embedded, polyhedral complexes. More details on the functions we coded can be found by clicking the Splines link above.

Toric Varieties

  • Nathan Bliss
  • Nathan Fieldsteel
  • Jeff Poskin
  • Robert Walker

We implement a new type "ToricMaps", for toric maps between normal toric varieties. We implemented several basic methods to test whether a map is an isomorphism, find inverses of isomorphisms, composition of toric maps, and pullbacks for Cartier divisors.

Versal Deformations

  • Cesar Lozano Huerta
  • Julio Urenda

Visualize

We would like to use various JavaScript libraries to visualize algebraic objects. For example, monomial ideals in two or three variables can be visualized on an integer lattice. With the Visualize.m2 package, a user can give M2 a monomial ideal and an interactive image of the lower boundary of the exponent set will be displayed in the default browser. We have also implemented similar visualizations for graphs, digraphs, posets, and simplicial complexes.