Danica Kosanović and Vera Traub receive Hausdorff Memorial Prize

Danica Kosanović spectacularly proves in her PhD thesis a 30 year old conjecture about invariants of classical knots, that is the embedding of a circle in the three dimensional euclidean space. The classification of knots has long been a topic in topology and is addressed from various points of views and mathematical fields. Two knots can look completely different and still be the „same“ topologically, in the sense that they are identical up to a homeomorphism. It is hence difficult to prove that two knots are not the same. One therefore takes an indirect way such as searching for knot invariants. Danica Kosanović shows in her work that the classical knot invariants introduced by Vassiliev and Kontsevich are universal in important cases. The PhD thesis excels through an impressive range of methods, technical virtuosity and brilliant presentation. Her work was supervised by Peter Teichner from the Max Planck Institute for Mathematics in Bonn.

Vera Traub has made essential breakthroughs regarding open questions of the so-called Traveling Salesman Problem (TSP), the most popular problem of combinatorial optimization. The problem is famous all over the world and was first formulated in 1930: Given a starting, an endpoint and further points which have to be visited the aim is to find the shortest tour passing through all the points by optimizing the visiting order of the points. Vera Traub has found better approximation algorithms, hence efficient algorithms that guarantee a good solution. In the asymmetric case, that is with "one way streets", she showed that the integrality gap is constant; this measures how good a classical fast-computable estimate is. In the symmetric case she even undermatched the integrality gap. She moreover showed how the problem can be reduced to the case where the starting and endpoint accord. Vera Traub has introduced various new techniques and methods which can be applied not only to the TSP, but to various parts of combinatorial optimization. Her PhD thesis’ significance hence far outreaches the TSP. The thesis was supervised by Jens Vygen from the Research Institute for Discrete Mathematics in Bonn.