Deepening the understanding of the interplay between optimal mass transportation and metric geometry (Schultz)

The history of optimal mass transportation theory goes way back to the 18th century. In the past few decades, a broad spectrum of applications in mathematics have been found for the theory of optimal mass transportation. In his dissertation being audited at the University of Jyväskylä, MSc Timo Schultz studied curvature bound conditions in metric spaces, which is one target of application of optimal mass transportation.
Published
15.6.2020

One of the fundamental questions in optimal mass transportation theory is the question of the existence of so-called optimal transport maps. In his thesis, Timo Schultz gives an affirmative answer to this question in the framework of metric measure spaces admitting suitable curvature lower bounds. As an application in metric geometry, Schultz proves that in the presence of optimal transport maps, one can deduce a global one-dimensionality for a metric measure space in the case where it contains a one-dimensional part.

Different notions of curvature play a crucial role in geometry. While a surface of a ball is positively curved, the surface of a horse saddle is negatively curved. In his dissertation, Schultz studies metric measure spaces having suitable synthetic lower bounds on the Ricci curvature.

One of the purposes of the thesis is to fill in the gap between a weak, but stable curvature bound condition and a stronger, but unstable one. Towards this goal, Schultz introduces a new curvature bound condition which lies in between the aformentioned ones. For metric measure spaces satisfying the new condition, he generalises results that are known for the stronger curvature bound condition. Such results are for example the existence of optimal transport maps, and the convexity of suitable entropy functionals.

The dissertation is published in JYU Dissertations series, number 232, 2020, Jyväskylä. ISBN 978-951-39-8183-9 URN:ISBN:978-951-39-8183-9, ISSN 2489-9003

Link to publications: http://urn.fi/URN:ISBN:978-951-39-8183-9

Timo Schultz earned his high school diploma in 2009 from Turun normaalikoulun lukio, and his master degree in mathematics in 2015 from University of Jyväskylä.

M.Sc. Timo Schultz defends his doctoral dissertation in Mathematics "Existence of optimal transport maps with applications in metric geometry" on Monday June 15th 2020 at 12 noon. Opponent Professor is Nicola Gigli (Scuola Internazionale Superiore di Studi Avanzati, Italy) and Custos is Docent Tapio Rajala (University of Jyväskylä). The doctoral dissertation is held in English.

The dissertation is held online. Link to the Zoom Webinar (Zoom application or Google Chrome web browser recommended): https://jyufi.zoom.us/j/64480582758
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