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. 

Recent Reviews


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.

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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