What results from the past have proved unusually influential in current directions of mathematical research? These posts have short reviews of classic mathematical research papers. Most will be about results that appeared in the last half-century.
What are the latest breakthroughs? These posts summarize current mathematical research. They cover significant and influential results announced in talks, preprints, and publications. These results are often preliminary, and their correctness may be as yet uncertain.
These posts examine central open problems, their history, and why their solution would have an impact on progress in mathematics.
Modular invariant q-deformed numbers: first steps
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”.
Ptolemy Relations and Friends
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.
Freedman’s Hard Decomposition Theorem
AUTHOR : R. Kirby :EDITORS : J. Hass and R. Ghrist ART : J. Hass INTRODUCTION In his celebrated paper , Mike Freedman showed that analogues of
An Aperiodic Set of Eleven Wang Tiles
AUTHORS : B. Durand and A. Shen : EDITOR/ART : R. Ghrist : This is a review of E. Jeandel and M. Rao, “An aperiodic set of
Milman and Neeman Post Proof of Triple and Quadruple Bubble Conjectures in $R^n$ and $S^n$
AUTHOR : F. Morgan : EDITORS : J. Hass, R. Ghrist : ART : R. Ghrist In 1884 Hermann Schwarz proved that a single round
Artificial Intelligence in Knot Theory
AUTHOR : C. Adams : EDITOR/ART : R. Ghrist : To move mathematics forward, researchers are constantly seeking new relationships between various areas of mathematics. The farther
AMR Reviews Scientific Committee
Gil BOR, CIMAT, Mexico ; Anna FELKISON, Durham University, UK; Sergei GELFAND, AMS ; Peter KUCHMENT, Texas A&M University; Vladimir RETAKH, Rutgers University; Sergei TABACHNIKOV, Penn State University