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# The public defense of Asaph Keikara Muhumuzas doctoral thesis in Mathematics/Applied Mathematics

The public defense of Asaph Keikara Muhumuza´s doctoral thesis in Mathematics/Applied Mathematics will take place at 15.15 on December 14, 2020.

**Title:**”Extreme points of the Vandermonde determinant in Numerical approximation, Random Matrix Theory and Financial Mathematics”

The faculty examiner is Associate Professor Olga Liivapu, Estonian University of Life Sciences. The examining committee consists of Professor Guglielmo De Amico, University G. d’Annunzio of Chieti-Pescara, Associate Professor Andriy Andreev, Stockholm University and Professor Christos Skiadas, Technical University of Crete.

Reserve: Professor Kimmo Eriksson, Mälardalen University and Associate Professor Oleksandr Borisenko, Taras Shevchenko National University of Kyiv.

The doctoral thesis has serial number 327.

**Abstract**

This thesis discusses the extreme points of the Vandermonde determinant on various surfaces, their applications in numerical approximation, random matrix theory and financial mathematics. Some mathematical models that employ these extreme points such as curve fitting, data smoothing, experimental design, electrostatics, risk control in finance and method for finding the extreme points on certain surfaces are demonstrated.

The first chapter introduces the theoretical background necessary for later chapters. We review the historical background of the Vandermonde matrix and its determinant, some of its properties that make it more applicable to symmetric polynomials, classical orthogonal polynomials and random matrices.

The second chapter discusses the construction of the generalized Vandermonde interpolation polynomial based on divided differences. We explore further, the concept of weighted Fekete points and their connection to zeros of the classical orthogonal polynomials as stable interpolation points.

The third chapter discusses some extended results on optimizing the Vandermonde determinant on a few different surfaces defined by univariate polynomials. The coordinates of the extreme points are shown to be given as roots of univariate polynomials.

The fourth chapter describes the symmetric group properties of the extreme points of Vandermonde and Schur polynomials as well as application of these extreme points in curve fitting.

The fifth chapter discusses the extreme points of Vandermonde determinant to number of mathematical models in random matrix theory where the joint eigenvalue probability density distribution of a Wishart matrix when optimized over surfaces implicitly defined by univariate polynomials.

The sixth chapter examines some properties of the extreme points of the joint eigenvalue probability density distribution of the Wishart matrix and application of such in computation of the condition numbers of the Vandermonde and Wishart matrices.

The seventh chapter establishes a connection between the extreme points of Vandermonde determinants and minimizing risk measures in financial mathematics. We illustrate this with an application to optimal portfolio selection.

The eighth chapter discusses the extension of the Wishart probability distributions in higher dimension based on the symmetric cones in Jordan algebras. The symmetric cones form a basis for the construction of the degenerate and non-degenerate Wishart distributions.

The ninth chapter demonstrates the connection between the extreme points of the Vandermonde determinant and Wishart joint eigenvalue probability distributions in higher dimension based on the boundary points of the symmetric cones in Jordan algebras that occur in both the discrete and continuous part of the Gindikin set.

**Register to take part**

To be able to participate in the PhD defense, you need to pre-apply to administrative research support staff Marja Mutikainen marja.mutikainen@mdh.se by informing your name and e-mail. You also need to inform if you wish to take part in Zoom or in Kappa. Last day to register is November 19, 2020.

## Contact Information

Asaph Keikara Muhumuza

asaph.keikara.muhumuza@mdh.se