Entropic regularization for linear programming leads to intersecting a toric variety with the feasible polytope. In semidefinite programming, the toric variety is replaced by a new geometric object, called Gibbs manifold, and the feasible polytope becomes a spectrahedron. I will explain these concepts and present the example of (quantum) optimal transport. This is based on joint work with Dmitrii Pavlov, Bernd Sturmfels, François-Xavier Vialard and Max von Renesse.
Chemical master equations (CMEs) are of great interest in systems biology where they are used to model transcriptional dynamics. I will explain how CMEs arise naturally in attempts to interpret and analyze single-cell genomics data, and then survey some of interesting mathematics and statistics questions that emerge as a result.
The classical iterative proportional scaling algorithm, or IPS, numerically computes the maximum likelihood estimate of a given vector of counts for a log-linear partition model. We investigate the conditions under which IPS produces the exact maximum likelihood estimate, or MLE, in finitely many steps. Since IPS produces a rational function at each step, a necessary condition is that the model must have rational maximum likelihood estimator. However, the convergence is highly parametrization-dependent; indeed, one parametrization of a model may exhibit exact convergence in finitely many steps while another does not. We introduce the generalized running intersection property, which guarantees exact convergence of IPS. As the name suggests, this strictly generalizes the well-known running intersection property for hierarchical models. This generalized running intersection property can be understood in terms of the toric geometry of the log-linear model, and models that satisfy this property can be obtained by performing repeated toric fiber products of linear ideals. We also draw connections between models that satisfy the generalized running intersection property and balanced, stratified staged trees.
In the course of investigation evolutionary relationships found in the genomes a set of species, several binary relations appear. For example “best matches” refer pairs of genes x and y so that y is one of the closest relatives of x in the species that harbors y. Orthology designates pairs of genes whose last common ancestor is a speciation event. Horizontal gene transfer is related to the lower diverence time relation, satisfied by a pair of genes that is younger than the divergence of the species in which they reside. I will sketch the connections between the relations and the the information that they convey about the gene trees, the phylogeny of the underlying species, and the reconciliation of gene and species trees.
