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AMS Notices: October 2017
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Feature Column: Untangling Your Square Dance by David Austin
Explore Origami works by Robert J. Lang on Mathematical Imagery
AMS Bulletin: July 2017 issue
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Markov Chains and Mixing TimesMarkov Chains and Mixing Times
David A. Levin
Yuval Peres
Featuring three new chapters (on monotone chains, the exclusion process, and stationary times), the second edition of this introduction to the modern theory of Markov chains includes smaller additions and corrections throughout, with updated notes at the end of each chapter informing the reader of recent research developments.

A First Course in Sobolev Spaces: Second EditionA First Course in Sobolev Spaces: Second Edition
Giovanni Leoni
The second edition of this book about differentiation of functions focuses more on higher order derivatives and has the potential to be used as two separate textbooks, one for an advanced undergraduate course in functions of one variable and one for a graduate course on Sobolev functions.

Nonlinear PDEs: A Dynamical Systems ApproachNonlinear PDEs: A Dynamical Systems Approach
Guido Schneider
Hannes Uecker
This introductory textbook on nonlinear dynamics of PDEs uses an example-oriented presentation and develops new mathematical tools, giving insight into some important classes of nonlinear PDEs and nonlinear dynamics phenomena which may occur in PDEs.

Bulletin of the AMS

Bulletin of the AMS Laplacian growth, sandpiles, and scaling limits
( view abstract )
Laplacian growth, sandpiles, and scaling limits
Laplacian growth is the study of interfaces that move in proportion to harmonic measure. Physically, it arises in fluid flow and electrical problems involving a moving boundary. We survey progress over the last decade on discrete models of (internal) Laplacian growth, including the abelian sandpile, internal DLA, rotor aggregation, and the scaling limits of these models on the lattice $ \epsilon {\mathbb{Z}}^d$ as the mesh size $ \epsilon $ goes to zero. These models provide a window into the tools of discrete potential theory, including harmonic functions, martingales, obstacle problems, quadrature domains, Green functions, smoothing. We also present one new result: rotor aggregation in $ {\mathbb{Z}}^d$ has $ O(\log r)$ fluctuations around a Euclidean ball, improving a previous power-law bound. We highlight several open questions, including whether these fluctuations are $ O(1)$.

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