In statistics, the philosophy of maximum likelihood estimation is of central importance. Algebraic techniques are especially adequate to study the maximum likelihood geometry for discrete exponential statistical models. We will introduce an associated algebraic invariant known as the ML degree and explore its computational aspects, illustrated with several examples and applications.
The sparsity structure of a polynomial system is often described by a graph that captures the interactions among the variables. By carefully exploiting the properties of this graph (in particular, its chordal completions), more efficient algorithms can be developed.
The M2 package "Chordal" provides several specialized routines for computing properties of sparse polynomial ideals, such as: dimension, cardinality, membership, elimination, components. At the heart of our methods is a new representation of polynomial ideals that we call Chordal Networks. Remarkably, several interesting families of polynomial ideals admit compact chordal network representations (of size linear in the number of variables), even though the number of components is exponentially large.
In the presentation, we will survey two packages for biological applications: PhylogeneticTrees.m2 and ReactionNetworks.m2. The first package constructs rings and ideals associated to models of evolution, while the second constructs algebraic objects arising in systems biology. In this talk, we will describe the role of computational algebraic geometry in evolutionary and systems biology and describe the main functionality of the two packages.
We survey conditional independence ideals and graphical models with a focus on the Gaussian case. Here primary decomposition of certain determinantal ideals can be used to infer properties of conditional independence statements, such as implications between sets of them. We demonstrate how to use the GraphicalModels package to save a lot of keystrokes.
(by Madeline Brandt, David J. Bruce, Taylor Brysiewicz, Robert Krone, Elina Robeva)
The set of rotation matrices, SO(n), forms an irreducible real algebraic variety and we derive a formula in n for its degree from the work of Kazarnovskii. To confirm the formula we computed as many values we could using numerical algebraic geometry. The degree of a variety can be defined as the number of points in the intersection of the variety with a generic linear space of the appropriate dimension, and standard homotopy continuation algorithms can count these points. For SO(n), the number of paths to track grows so quickly that we could only feasibly compute the degree this way for n at most 5. An alternative homotopy continuation algorithm starts with a small number of solutions, and populates the rest of the set by moving the linear space through monodromy loops. With this algorithm, we confirmed that the degrees of SO(n) for n equal to 6 and 7 are 4768 and 111616 respectively.
This tutorial will address basic questions that arise when one tries to use "approximate" and "algebraic geometry" in one sentence.
In this talk, we will explore how using and creating Macaulay2 packages can be a powerful research tool. To illustrate our points, we look at a handful of specific packages.
We introduce Macaulay2 by showing its use on some examples, including a graph coloring game. Along the way, we show some basic usage of Macaulay2 for mathematical exploration, and how best to use Macaulay2 with the emacs interface, as well as the web interface.