2022 Lecturer: Marianna Csörnyei
The Kakeya needle problem for rectifiable sets
A planar set admits the “Kakeya property” if it can be moved continuously to any other position covering arbitrary small area during the movement. It was known for more than 100 years that line segments have this property, but until recently there were only very few other known examples.
In the talk we will study two variants of this problem, the geometric and the analytic version. In the classical, geometric version, we find all connected closed sets with the Kakeya property. In the analytic version, where we are allowed to delete a null set at each time moment, we will show that every rectifiable set admits the Kakeya property, moreover, they can be moved to any other position covering not only arbitrary small but zero area.
Marianna Csörnyei is a Professor of Mathematics at The University of Chicago. She received her Ph.D. from the Loránd Eötvös University in Budapest in 1999. Before joining the faculty at the University of Chicago, she was a Research Fellow at the University College London during 1999 – 2003, a member of the Institute for Advanced Study in Princeton in 2003/2004, and a Professor of Mathematics at the University College London during 2004 – 2011.
Csörnyei has made significant contributions to several areas of Mathematical Analysis, including Geometric Measure Theory, Functional Analysis and Real Analysis. While she was still an undergraduate, she established a reputation as a brilliant problem solver. (Even before that, in her high school years, she won a gold medal at the International Mathematical Olympiad.) Later she worked on deep innovative long-term projects. She is known for example for her results concerning various versions of the Kakeya needle problem and for her work on the structure of Lebesgue null sets in Euclidean spaces. The latter work is connected to problems in partial differential equations and the calculus of variations, questions concerning the possibility/impossibility of strengthening the Rademacher Theorem about the almost everywhere differentiability of Lipschitz functions, as well as to some combinatorial problems.
Csörnyei was an invited speaker at the 2010 International Congress of Mathematicians and has given lectures at distinguished institutions around the world. She won the Whitehead Prize from the London Mathematical Society in 2002 and the Philip Leverhulme Prize in 2008. In 2019 she was elected an External Member of the Hungarian Academy of Sciences.