PhD defense: Oskar Henriksson
Title of the thesis: On the generic geometry of parametrized polynomial systems in biology and statistics
This thesis is based on seven papers about polynomial systems of equations arising in biology and statistics which are parametrized, in the sense that their coefficients depend on parameters. Using techniques from algebraic geometry, the papers explore various properties of the solution sets of these systems that hold for generic parameter values. Paper A concerns the problem of numerically approximating the solutions of systems with generically finite solution sets. We give an algorithm for constructing homotopies and start systems from tropical intersection data, that allows numerically solving such systems with homotopy continuation methods in a way that leads to a generically optimal number of paths. We also develop techniques for computing the necessary tropical intersection data and generic root counts for several classes of systems, including steady state systems of chemical reaction networks. Papers B, C and D are centered around vertically parametrized systems, which arise in both chemical reaction network theory and optimization. In Paper B, we draw up a general framework for studying generic consistency and nondegeneracy of such systems over both the complex and real numbers. This framework is then applied in Paper C to study reaction-network-theoretic properties such as absolute concentration robustness and nondegenerate multistationarity, and in Paper D to investigate parametric toricity. Papers E and F focus on the method of moments for statistical parameter estimation. In Paper E, we study the determinantal structure and singularities of the moment varieties of the exponential, chi-squared, gamma, and inverse Gaussian distribution. We also determine the number of moments needed to obtain generic unique parameter identifiability for mixtures of the exponential and chi-squared distribution. In Paper F, we build on the results from Paper E, combined with the theory of secant defectivity of surfaces, to determine the number of moments needed to obtain generic finite identifiability for mixtures of the inverse Gaussian and gamma distribution. Finally, in Paper G, we investigate the problem of identifiability in the field of 3D genome reconstruction for diploid organisms. We prove generic finite identifiability from unphased Hi-C data, and also show that the algebraic complexity of the problem significantly decreases in the presence of even a small amount of phased data that distinguishes maternal and paternal genomic loci. Based on this result, we devise a new reconstruction approach, based on homotopy continuation and optimization.
PhD advisor: Professor Elisenda Feliu, Department of Mathematical Sciences, University of Copenhagen
PhD Committe:
Fabien Pazuki, Professor (chair) KU Copenhagen
Paul Breiding, Professor, (University of Osnabrück)
Fatemeh Mohammadi, Professor, (KU Leuven)