Disputation: Martin Helsø

Doctoral candidate Martin Helsø at the Department of Mathematics, Faculty of Mathematics and Natural Sciences, is  defending the thesis Investigations into real determinantal quartic hypersurfaces for the degree of Philosophiae Doctor.

Picture of the candidate.

Doctoral candidate Martin Helsø

The University of Oslo is closed. The PhD defence and trial lecture will therefore be fully digital and streamed directly using Zoom. The host of the session will moderate the technicalities while the chair of the defence will moderate the disputation.

Ex auditorio questions: the chair of the defence will invite the audience to ask questions ex auditorio at the end of the defence. If you would like to ask a question, click 'Raise hand' and wait to be unmuted.

Trial lecture

20th November, 10:15, Zoom

"Spectrahedra and Their Shadows: Results and Open Questions"

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Abstract

Mathematical optimisation is finding the input that yields the smallest or largest output value of a given function. This has countless applications in fields such as engineering, economics and machine learning. When solving an optimisation problem, it is essential to understand the geometry imposed by its constraints. For a long time, people worked mainly with the simplest optimisation problems, whose constraints result in geometric objects with only flat sides. In order to solve more problems, researchers are constantly considering other types of constraints that produce more complicated shapes. For an important class of optimisation problems, the region satisfying the constraints is called a spectrahedron.

The thesis uses tools from algebraic geometry to study spectrahedra. The boundary of a spectrahedron is part of a larger geometric object known as a determinantal hypersurface. The hypersurface may have special points where it is spiked or intersects itself. These points are called singularities. The tip of a cone is a singularity. If an optimisation problem has an optimal solution, it is attained at the boundary of the spectrahedron, often in a singularity. The thesis answers the following questions for certain types of determinantal hypersurfaces: What kinds of singularities do they have, how many singularities are there and which of the singularities lie on the spectrahedron?

A determinantal hypersurface with one singularity. The singularity is disjoint from the spectrahedron, which is the grey blob.
A determinantal hypersurface with one singularity. The singularity is disjoint from the spectrahedron, which is the grey blob.

 

Published Oct. 30, 2020 9:15 AM - Last modified Apr. 21, 2021 2:29 PM