The problem From the usual Euclidean plane we form the following graph: the points of the plane are the vertices of our graph, and two vertices are connected by an edge if they are exactly unit distance apart. The so-called Hadwiger-Nelson problem is to compute the chromatic number of this graph, i.e., the least amount … Continue reading "Chromatic Number of the Plane is at least 5"
This is the first post of a series of posts in which we will eventually venture deep into the realm of coarse geometry. But we will always be motivated by questions which are related to the one that we will discuss here. But our first steps into coarse geometry will be very gently: we will … Continue reading "Norms of infinite matrices"
John Roe, the founder of coarse index theory, passed away last month after a long fight against cancer. His web page is still online ( http://sites.psu.edu/johnroe/ ) and can be visited to get a glimpse not only of his mathematical work, but also of his personal life and all the things in the world that … Continue reading "John Roe 1959-2018"
The Frederic Esser Nemmers Prize in Mathematics goes this year to Assaf Naor “for his profound work on the geometry of metric spaces, which has led to breakthroughs in the theory of algorithms.”
The countless shapes of snowflakes have long raised the curiosity of many scientists, among others the famous Kepler. They have by now been classified by empirical observation into 80 different shapes, but a mathematical explanation for this classification seems to be missing. A striking point about them is that, even though two snowflakes are almost … Continue reading "Snowflakes at infinity"
A new paper proves the contractibility of the space of constant curvature metrics on all 3-manifolds except possibly real projective space. Bamler, Kleiner: Ricci flow and diffeomorphism groups of 3-manifolds, https://arxiv.org/pdf/1712.06197.pdf The Smale conjecture in its original form asserted that the diffeomorphism group of the 3-sphere deformation retracts onto O(3), the isometry group of its … Continue reading "Ricci flow and diffeomorphism groups of 3-manifolds"
A preprint with a new example shows that the understanding of infinitely generated Kleinian groups will be more complicated than for the finitely generated ones. Cremaschi: A locally hyperbolic 3-manifold that is not hyperbolic, https://arxiv.org/pdf/1711.11568 By the proofs of hyperbolization and tameness, one knows precisely which irreducible 3-manifolds with finitely generated fundamental groups admit hyperbolic … Continue reading "A locally hyperbolic 3-manifold that is not hyperbolic"
The two ways Feynman diagrams appear in mathematics
These days, there was the 2nd Bavarian Geometry/Topology Meeting, organized by Fabian Hebestreit and Markus Land, and hopefully becoming a tradition as the NRW topology meeting which by now had its 28th recurrent. Main event of the meeting were the lectures of Oscar Randal-Williams from Oxford, who discussed work on the cohomology of the mapping … Continue reading "Bavarian Geometry/Topology Meeting"
The award ceremony is certainly not what mathematicians are used to, and there are certainly many things that one can say for and against such monstrous awards and the ambience around. In any case, if you‘d like to see the ceremony, the math part starts at 1:22:30. The breakthrough prize for 2018 was given to … Continue reading "Breakthrough Prize for higher-dimensional geometry"