AMR

Ordnung muss sein! (Order is inevitable.) This saying , attributed to Theodore Motzkin, aptly summarizes Ramsey theory , which aims to solve problems of the form: “From what size on can a structure no longer avoid having a certain property ?"

AUTHOR: V. Ovsienko. The goal of this short review is to explain the main ideas of the emerging new theory. of ``quantum rational',' based on modular, or PSL(2,]-invariance, and that of ``quantum irrationals''.

AUTHOR: A. Felikson (Updated 4/4/23): In recent decades, identities similar to the one in Ptolemy’s theorem started to pop up in many fields in connection to the notion of cluster algebras introduced and studied since 2000 by Fomin and Zelevinsky. In this brief note we will try to describe several animals from this big and rich zoo.

AUTHOR : R. Kirby :EDITORS : J. Hass and R. Ghrist ART : J. Hass INTRODUCTION In his celebrated paper [8], Mike Freedman showed that analogues of higher dimensional smooth manifold theorems also worked in dimension four in the topological category if the fundamental group was good, i.e., did not grow too fast. In particular he proved […]

AUTHORS : B. Durand and A. Shen : EDITOR/ART : R. Ghrist : This is a review of E. Jeandel and M. Rao, “An aperiodic set of 11 Wang tiles”, Advances in Combinatorics, 2021, https://doi.org/10.19086/aic.18614; see also the preprint https://arxiv.org/abs/1506.06492 from 2015. HISTORIC ACCOUNT. Assume that a finite set of geometric shapes, called tiles, is given. We […]

AUTHOR : F. Morgan : EDITORS : J. Hass, R. Ghrist : ART : R. Ghrist In 1884 Hermann Schwarz proved that a single round soap bubble is the least-perimeter way to enclose a given volume in $R^3$. In 2000, Michael Hutchings, Frank Morgan, Manual Ritoré, and Antonio Ros proved the Double Bubble Conjecture — […]

AUTHOR : C. Adams : EDITOR/ART : R. Ghrist : To move mathematics forward, researchers are constantly seeking new relationships between various areas of mathematics. The farther apart two areas are, the more dramatic the implications when relationships are discovered. But how to find these relationships? Certainly, studying the fields and developing intuition plays a critical role. […]

AUTHOR/EDITOR/ART : R. Ghrist : This is a review of recent work by R. Cardona, E. Miranda, D. Peralta-Salas, and F. Presas, as per the following works: R. Cardona, E. Miranda, D. Peralta-Salas, Looking at Euler flows through a contact mirror: Universality and undecidability (2021) https://arxiv.org/abs/2107.09471 Now published: R. Cardona, E. Miranda, D. Peralta-Salas.  Looking […]

AUTHOR : S. Tabachnikov : EDITOR/ART : R. Ghrist : This review is for the preprint: F. Wegner, From elastica to floating bodies of equilibrium, https://arxiv.org/abs/1909.12596. Problem No 19 in [14] reads: Is a solid of uniform density which will float in water in every position a sphere? This question by S. Ulam can be […]

AUTHOR : S. Tabachnikov : EDITOR/ART : R. Ghrist : This review concerns the following preprint: M. Bialy and A. Mironov, The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables, https://arxiv.org/abs/2008.03566 Now published:  M. Bialy, A. Mironov.  The Birkhoff-Poritsky conjecture for centrally-symmetric billiard tables.   Ann. of Math. (2) 196 (2022), no. 1, 389–413.  The dynam…