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 "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 … Continue reading "What’s new on the ArXiv: 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"
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"
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. … Continue reading "S-matrices and the big unification"
Concepts in topology successfully transferred to graph theory and combinatorics with non-trivial applications?
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 … Continue reading "What‘s new on the ArXiv: A deformation of instanton homology for webs"