Snowflakes at infinity

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"

What’s new on the ArXiv: Ricci flow and diffeomorphism groups of 3-manifolds

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 "What’s new on the ArXiv: Ricci flow and diffeomorphism groups of 3-manifolds"

What’s new on the ArXiv: A locally hyperbolic 3-manifold that is not hyperbolic

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 "What’s new on the ArXiv: A locally hyperbolic 3-manifold that is not hyperbolic"

Bavarian Geometry/Topology Meeting

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"

Breakthrough Prize for higher-dimensional geometry

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"

What’s new on the ArXiv: Quasi-isometric groups with no common model geometry

Do quasi-isometries between groups always arise from actions on a common model space? Previous counterexamples invoked central extensions of lattices, e.g., of surface groups. A new construction of infinitely many classes is now using amalgams of surface groups. Stark, Woodhouse: Quasi-isometric groups with no common model geometry, https://arxiv.org/pdf/1711.05026.pdf If a group \(\Gamma\) acts geometrically (i.e., … Continue reading "What’s new on the ArXiv: Quasi-isometric groups with no common model geometry"