Past PHD Projects


Description of Project:  The main focus of this project is to solve polynomial systems which have roots with multiplicities or roots forming small clusters. The motivation for this investigation comes from the fact that such systems are considered ill-conditioned and traditional numerical methods are generally unable to handle them. In this project we gave algorithms to compute the radical or approximate radical of a given zero-dimensional ideal based on linear algebra only on a so called matrix of traces.Then we analyzed how these algorithms behave under perturbation of the given ideal or its underlying roots. Finally we gave simple and efficient methods to compute matrices of traces. The results of this project already appeared in papers including: Approximate radical for clusters: a global approach using Gaussian elimination or SVD., Mathematics in Computer Science, Vol. 1, no. 2, pp 393–425, (2007) and Moment matrices, trace matrices and the radical of ideals, Proceedings of the ISSAC 2008, 125--132, ACM, New York, (2008)



Description of Project:  The usual Galois theory of algebraic equations associates a finite group to each polynomial.  Properties of this polynomial (e.g., solvability in terms of radicals) are reflected by properties of the groups (e.g., solvability of the group).  A similar Galois theory was developed for linear differential equations where the associated groups are linear algebraic groups, that is, groups of matrices whose entries satisfy polynomial equations. Again, properties of the linear differential equation are reflected in properties of the group.  Using this Galois theory, algorithms have been developed to determine if a given linear differential equation can be solved in terms of iterations of algebraic functions, exponentials and integrals.  The key to many of these algorithms has been an understanding of the representation theory of linear algebraic groups, that is, understanding all the different ways in which a group can be represented as a group of matrices. In the last century, PIerre Deligne gave a category theory based approach to this Galois theory and the representations of these groups. This new way of looking at this subject has been crucial to the algorithms.

In the last 8 years, a Galois theory has been developed for parameterized differential and difference equations. The Galois groups are linear differential algebraic groups, that is, groups of matrices whose entries are functions of the parameters that satisfy certain differential equations. This dissertaion gives a category theoretical approach to deriving this Galois theory and gives a categorical description of the representation theory of these groups with the goal of deriving algorithms in this theory.  The results of the thesis have already appeared in print as: Tannakian Categories, Linear Differential Algebraic Groups, and Parametrized Linear Differential Equations, Transformation Groups 14 (1), 2009, 195--223 and Differential Tannakian categories, Journal of Algebra 321 (10), 2009, 3043--3062



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