Abstract:
Algebraic statistics is concerned with the study of probabilistic models and techniques for statistical inference using methods from algebra and geometry. This article presents a list of open mathematical problems in this emerging field, with main emphasis on graphical models with hidden variables, maximum likelihood estimation, and multivariate Gaussian distributions. This article is based on a lecture presented at the IMA in Minneapolis during the 2006/07 program on Applications of Algebraic Geometry.

Abstract:
Integer programming is concerned with solving linear systems of equations over the non-negative integers. The basic question is to find a solution which minimizes a given linear objective function for a fixed right hand side. Here we also consider parametric versions where the objective function and the right hand side are allowed to vary. The main emphasis is on Gr\"obner bases, rational generating functions, and how to use existing software packages. Concrete applications to problems in statistical modeling will be presented. These are the notes for a lecture to be given during the AMS Short Course, ``Trends in Optimization'', prior to the Joint Mathematics Meetings in Phoenix on January 5-6, 2004.

Abstract:
The Hurwitz form of a variety is the discriminant that characterizes linear spaces of complementary dimension which intersect the variety in fewer than degree many points. We study computational aspects of the Hurwitz form, relate this to the dual variety and Chow form, and show why reduced degenerations are special on the Hurwitz polytope.

Abstract:
We study algebras k[x_1,...,x_n]/I which admit a grading by a subsemigroup of N^d such that every graded component is a one-dimensional k-vector space. V.I.~Arnold and coworkers proved that for d = 1 and n <= 3 there are only finitely many isomorphism types of such A-graded algebras, and in these cases I is an initial ideal (in the sense of Groebner bases) of a toric ideal. In this paper it is shown that Arnold's finiteness theorem does not extend to n = 4. Geometric conditions are given for I to be an initial ideal of a toric ideal. The varieties defined by A-graded algebras are characterized in terms of polyhedral subdivisions, and the distinct A-graded algebras are parametrized by a certain binomial scheme.

Abstract:
This article will appear in the proceedings of the AMS Summer Institute in Algebraic Geometry at Santa Cruz, July 1995. The topic is toric ideals, by which I mean the defining ideals of subvarieties of affine or projective space which are parametrized by monomials. Numerous open problems are given.

Abstract:
We determine an explicit Gr\"obner basis, consisting of linear forms and determinantal quadrics, for the prime ideal of Raftery's mixture transition distribution model for Markov chains. When the states are binary, the corresponding projective variety is a linear space, the model itself consists of two simplices in a cross-polytope, and the likelihood function typically has two local maxima. In the general non-binary case, the model corresponds to a cone over a Segre variety.

Abstract:
One of the major successes in computational biology has been the unification, using the graphical model formalism, of a multitude of algorithms for annotating and comparing biological sequences. Graphical models that have been applied towards these problems include hidden Markov models for annotation, tree models for phylogenetics, and pair hidden Markov models for alignment. A single algorithm, the sum-product algorithm, solves many of the inference problems associated with different statistical models. This paper introduces the \emph{polytope propagation algorithm} for computing the Newton polytope of an observation from a graphical model. This algorithm is a geometric version of the sum-product algorithm and is used to analyze the parametric behavior of maximum a posteriori inference calculations for graphical models.

Abstract:
In tropical algebraic geometry, the solution sets of polynomial equations are piecewise-linear. We introduce the tropical variety of a polynomial ideal, and we identify it with a polyhedral subcomplex of the Grobner fan. The tropical Grassmannian arises in this manner from the ideal of quadratic Plucker relations. It is shown to parametrize all tropical linear spaces. Lines in tropical projective space are trees, and their tropical Grassmannian G_{2,n} equals the space of phylogenetic trees studied by Billera, Holmes and Vogtmann. Higher Grassmannians offer a natural generalization of the space of trees. Their facets correspond to binomial initial ideals of the Plucker ideal. The tropical Grassmannian G_{3,6} is a simplicial complex glued from 1035 tetrahedra.

Abstract:
Statistical models of evolution are algebraic varieties in the space of joint probability distributions on the leaf colorations of a phylogenetic tree. The phylogenetic invariants of a model are the polynomials which vanish on the variety. Several widely used models for biological sequences have transition matrices that can be diagonalized by means of the Fourier transform of an abelian group. Their phylogenetic invariants form a toric ideal in the Fourier coordinates. We determine generators and Gr\"obner bases for these toric ideals. For the Jukes-Cantor and Kimura models on a binary tree, our Gr\"obner bases consist of certain explicitly constructed polynomials of degree at most four.

Abstract:
We determine the maximal gap between the optimal values of an integer program and its linear programming relaxation, where the matrix and cost function are fixed but the right hand side is unspecified. Our formula involves irreducible decomposition of monomial ideals. The gap can be computed in polynomial time when the dimension is fixed.