Algebraic Statistics and Applied Algebraic Geometry

Neighbor-Joining Cones: These are supporting files, computations, and other material for the paper (w/ R. Davidson) “Combinatorial and Computational investigations of Neighbot-Joining bias.”   [ arXiv ]

1. •[ RegularNJ.m ] Mathematica code to simulate the spherical fraction of the polyhedral subdivision of the Neighbor-Joining algorithm (without any bias correction in the last step).
   

2. •[ UniformNJ.m ] Simulates the spherical fraction of the polyhedral subdivision of the NJ algorithm with the Uniform bias correction on the last step.
   

3. •[ BaggageNJ.m ] Simulates the spherical fraction of the polyhedral subdivision of the NJ algorithm with the Baggage bias correction on the last step.
   

4. •[ Simulation-Data ] The data of uniformly taking 1,000,000 points from the unit sphere and input into the three algorithms above for 4,5,6,7,and 8 taxa.
   

Critical points via monodromy and local methods: These are some files with examples for computing critical points using monodromy and numerical algebraic geometry, described in the paper (w/ J. Rodriguez) “Critical points via monodromy and local methods.”   [ arXiv ]

                    The official page with the supporting material for this project is hosted by Jose [ Page ]

1. •[ Bertini.m2 ] Interface between Macaulay2 and Bertini that includes the monodromy functions.
   

2. •[GDHomotopy.m2] Some auxiliary functions for gradient descent homotopies.
   

3. •[ Ellipse.m2 ] M2 code that was shown in the paper to illustrate our method. It computes the critical points for finding the distance between a point and an ellipse.
   

Goodness-of-fit testing in Ising models: These are supporting files, computations, and other material for the paper (w/ S. Cepeda and C. Uhler) “Goodness-of-fit testing in Ising models.”   [ arXiv ]

1. •[ M2 Code ] Computes the Markov basis of the Ising model using 4ti2 (as in Section 2).
   

2. •[ R code ] Creates an MCMC sample from the Ising model by simple swaps (as described in Section 3).
   

3. •[ Simulation-Data ] The data of the 2 experiments described in Section 5, to compare the effectiveness of different test statistics.
   

Finiteness for Toric Ideals

Finiteness of Laurent lattice ideals: The following Macaulay2 code is an implementation of the algorithm described in the paper (w/ C. Hillar ) “Finiteness theorems and algorithms for permutation invariant chains of Laurent lattice ideals.”   [ arXiv ] [ M2 Code ]

Toric homogeneous Markov chain models: The following Macaulay2 code computes the Markov basis for the toric homogeneous Markov chain model in S states at time T. It was used for the paper:

(w/ D. Haws and R. Yoshida) “Degree bounds for a minimal Markov basis for the three-state toric homogenous Markov chain model.”  [ arXiv ]  [ M2 Code ]

Schubert Calculus

[ NumericalSchubertCalculus package ]: This Macaulay2 package is an implementation of the Littlewood-Richardson homotopy algorithm that uses numerical continuation to solve Schubert problems. The algorithm is based on the one described in the extended abstract  (F. Sottile, R. Vakil, and J. Verschelde) “Solving Schubert problems with Littlewood-Richardson homotopies” [ arXiv ]. This package also includes the Pieri homotopies algorithm described in (A. Leykin and F. Sottile) “Galois groups of Schubert problems”  [ arXiv ].  This package is still in development

[ M2 Code ] you can find the code in the M2 Github.  How to use: ...

Caveat:  This package depends on the NAG4M2 package, which is the numerical algebraic geometry package in Macaulay2. Also, this package allows you to compute the Galois group of a simple Schubert problem by interacting with GAP, so the user needs to have GAP installed in order to compute the Galois group.

Reality in Schubert Calculus:  I have helped in the programming for the large scale experiments to explore the number of real solutions to Schubert problems. You can find more information here: FRSC  LowerBounds.

I have also created the PHP pages that displays the data from our Database. Here is an example Sample 

Software

In my work I use the following mathematical software:

1. •Maple
   

2. •Macaulay2
   

3. •Singular
   

4. •4ti2
   

5. •R
   

I also have some experience (relatively small) programming in the following languages:

1. •Python
   

2. •PHP
   

3. •MySQL
   

4. •Perl
   

5. •Bash / Shell script