A new preprint 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 “round” constant curvature metric. It was proved by Hatcher in … Continue reading What’s new on the ArXiv: 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 metrics: they have to be atoroidal and have infinite fundamental … Continue reading What’s new on the ArXiv: A locally hyperbolic 3-manifold that is not hyperbolic

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 Christopher Hacon and James McKernan for their contributions to the … Continue reading Breakthrough Prize for higher-dimensional 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 acts geometrically (i.e., properly and cocompactly) on a space $X$, then it is quasi-isometric … Continue reading What’s new on the ArXiv: Quasi-isometric groups with no common model geometry

A week ago, the long-awaited preprint Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet by Arkani-Hamed, Bai, He, and Yan, appeared on the ArXiv. Michael Rios and David Chester in two videos try to explain the essence of the new work and, for example, the compatibility with Garrett Lise’s E8 theory. The video is remarkable not only for the content but … Continue reading S-matrices and the big unification

The four color theorem from graph theory is certainly the most famous problem for which so far only a brute force computational proof exists. A new preprint of Kronheimer-Mrowka supports an approach towards this theorem via homology theories. Kronheimer. Mrowka: A deformation of instanton homology for webs, https://arxiv.org/pdf/1710.05002.pdf The four color theorem says that every planar map can be colored by four colors such that … Continue reading What‘s new on the ArXiv: A deformation of instanton homology for webs

The German Mathematical Society (DMV) offers twice a year the „Gauß Lecture“, an overview lecture with a well-known mathematician. The lecture is intended to show current developments in mathematics and addresses the interested public. At this link is the chronicle of previous lectures. The last of this years lectures has been given by Cédric Villani at October 23. It was a big success with more than … Continue reading Gauss in Regensburg

Always wondered what the work of the 2016 Nobel prize laureates might have to do with topology? This recent video from Fan Zhang may give a first idea:   Continue reading What in the world is topological quantum matter?

Today, there was the Séminaire Bourbaki, which takes place every four months in Paris. The list of the talks can be found here. Interesting from the point of view „Geometry at Infinity“ was especially the talk of Olivier Guichard regarding work of Kapovich-Leeb-Porti and Labourie on convex-cocompact groups in higher rank symmetric spaces: Another talk was by Serge Cantat about the recent progress towards Zimmer‘s … Continue reading Séminare Bourbaki: Convex-cocompactness in higher rank