Society for Industrial and Applied Mathematics: SIAM Review: Table of Contents

Survey and Review

Marlis Hochbruck — 2026-05-13T07:00:00Z

SIAM Review, Volume 68, Issue 2, Page 215-215, June 2026.

Localized Patterns

Jason J. Bramburger — 2026-05-13T07:00:00Z

SIAM Review, Volume 68, Issue 2, Page 217-292, June 2026.
Abstract.Localized patterns are coherent structures embedded in a quiescent state and occur in both discrete and continuous media across a wide range of applications. While it is well-understood how domain-covering patterns (for example, stripes and hexagons) emerge from a pattern-forming/Turing instability, analyzing the emergence of their localized counterparts remains a significant challenge. There has been considerable progress in studying localized patterns over the past few decades, often by employing innovative mathematical tools and techniques. In particular, the study of localized pattern formation has benefited greatly from numerical techniques; the continuing advancement in computational power has helped to both identify new types of pattern and further our understanding of their behavior. We review recent advances regarding the complex behavior of localized patterns and the mathematical tools that have been developed to understand them, covering various topics from spatial dynamics, exponential asymptotics, and numerical methods. We observe that the mathematical understanding of localized patterns decreases as the spatial dimension increases, thus providing significant open problems that will form the basis for future investigations.

Gradient Flow Equations for Deep Linear Neural Networks: A Survey from a Network Perspective

Joel Wendin — 2026-05-13T07:00:00Z

SIAM Review, Volume 68, Issue 2, Page 293-345, June 2026.
Abstract.This paper surveys recent progress in understanding the dynamics and loss landscape of the gradient flow equations associated with deep linear neural networks, i.e., the gradient descent training dynamics (in the limit when the step size goes to 0) of deep neural networks missing the activation functions and subject to quadratic loss functions. When formulated in terms of the adjacency matrix of the neural network, as is done in this paper, these gradient flow equations form a class of converging matrix ODEs which is nilpotent, polynomial, isospectral, and with conservation laws. A detailed description of the loss landscape shows that it is described in detail and is characterized by infinitely-many global minima and saddle points, both strict and nonstrict, but that it lacks local minima and maxima. The loss function itself is a positive semidefinite Lyapunov function for the gradient flow, and its level sets are unbounded invariant sets of critical points with critical values that correspond to the amount of singular values of the input-output data learnt by the gradient along a certain trajectory. The adjacency matrix representation we use in the paper allows us to highlight the existence of a quotient space structure in which each critical value of the loss function is represented only once, while all other critical points with the same critical value belong to the fiber associated to the quotient space. It also allows us to easily determine stable and unstable submanifolds at the saddle points, even when the Hessian fails to obtain them.

Research Spotlights

Stefan M. Wild — 2026-05-13T07:00:00Z

SIAM Review, Volume 68, Issue 2, Page 347-347, June 2026.

Spanning Trees and Redistricting: New Methods for Sampling and Validation

Sarah Cannon — 2026-05-13T07:00:00Z

SIAM Review, Volume 68, Issue 2, Page 349-381, June 2026.
Abstract.Deciding whether a political districting plan was distorted by a hidden agenda, or whether it dilutes the voting power of some group, requires a neutral baseline for comparison. Remarkably, all nine U.S. Supreme Court justices have now signed on to decisions finding that computational methods can provide key evidence. Today, the leading approaches to the benchmarking of districting plans are based on the use of spanning trees for sampling graph partitions. We present a new reversible recombination algorithm and rigorously prove its fundamental properties. Furthermore, we argue for a canonical sampling distribution called the spanning tree distribution that is well adapted to redistricting and provides a principled foundation for comparing and validating methods. Together with a highly efficient (and open-source) implementation that can generate and handle large datasets, this work provides the most powerful null model to date for the gerrymandering problem, meeting an urgent democratic challenge with sound scientific methodology.

SIGEST

The Editors — 2026-05-13T07:00:00Z

SIAM Review, Volume 68, Issue 2, Page 383-383, June 2026.

Bounds on Small Ramsey Numbers by Semidefinite Programming

Bernard Lidický — 2026-05-13T07:00:00Z

SIAM Review, Volume 68, Issue 2, Page 385-403, June 2026.
Abstract.Finding exact Ramsey numbers is a problem typically restricted to relatively small graphs. The flag algebra method introduced by Razborov in 2007 was developed to find asymptotic results for very large graphs, so it seems that the method should not be suitable for finding small Ramsey numbers. But this intuition is wrong, and we will develop a technique to do just that in this paper. We find new upper bounds for many small graph and hypergraph Ramsey numbers. As a result, we prove several exact values. The main power of the method relies on utilizing semidefinite programming to find certificates that are sums of squares. We hope that this technique will be adapted to address other questions for smaller graphs with the flag algebra method.

Education

Hélène Frankowska — 2026-05-13T07:00:00Z

SIAM Review, Volume 68, Issue 2, Page 405-405, June 2026.

Tauberian Identities and Suspended Falls: A Wile E. Coyote Time Lag in the Fall of Stretched Elastic Bodies

Roberto Camassa — 2026-05-13T07:00:00Z

SIAM Review, Volume 68, Issue 2, Page 407-422, June 2026.
Abstract.The application of the motion of a vertically suspended mass-spring system released under tension is studied, focusing upon the delay timescale for the bottom mass fall as a function of the spring constants and masses. This “hang-time,” reminiscent of the Coyote and Road Runner cartoons, is quantified using the far field asymptotic expansion of the bottom mass’s Laplace transform. These asymptotics are connected to the short time mass dynamics through Tauberian identities and explicit residue calculations. It is shown, perhaps paradoxically, that this delay timescale is maximized in the large mass limit of the top “boulder.” Experiments are presented and compared with the theoretical predictions. This system is an exciting example for the teaching of mass-spring dynamics in classes on ordinary differential equations and does not require any normal mode calculations for these predictions.

A Mathematical Model for Nordic Skiing

Jane Shaw MacDonald — 2026-05-13T07:00:00Z

SIAM Review, Volume 68, Issue 2, Page 423-455, June 2026.
Abstract.Nordic skiing provides fascinating opportunities for mathematical modeling studies that exploit methods and insights from physics, applied mathematics, data analysis, scientific computing, and sports science. A typical ski course winds over varied terrain with frequent changes in elevation and direction, and so its geometry is naturally described by a three-dimensional space curve. The skier travels along a course under the influence of various forces, and their dynamics can be described using a nonlinear system of ordinary differential equations (ODEs) that are derived from Newton’s laws of motion. We develop an algorithm for solving the governing equations that combines Hermite spline interpolation, numerical quadrature, and a high-order ODE solver. Numerical simulations are compared with measurements of skiers on actual courses to demonstrate the effectiveness of the model. Throughout, we aim to illustrate how elementary concepts from undergraduate courses in calculus and scientific computing can be applied to the study of real problems in sport, which we hope will provide stimulating examples for both instructors and students. At the same time, we demonstrate how these concepts are capable of providing novel insights into skiing that should also be of interest to sport scientists.

Book Reviews

Anita T. Layton — 2026-05-13T07:00:00Z

SIAM Review, Volume 68, Issue 2, Page 457-458, June 2026.

Featured Review:; Probability Theory: The Logic of Science

Krešimir Josić — 2026-05-13T07:00:00Z

SIAM Review, Volume 68, Issue 2, Page 459-462, June 2026.
Probability Theory: The Logic of Science by E. T. Jaynes is a highly ambitious, unusual, and personal book. Its goal is no less than to lay the foundation for how to reason under uncertainty, not only in science, but generally. Inspired by Polya’s Mathematics and Plausible Reasoning, throughout his career Jaynes pursued a program of understanding the rules of plausible reasoning that would be broadly applicable across domains. Starting not from axioms, but from the “desiderata,” or desirable properties, that one would like to have in a quantitative system of reasoning, Jaynes arrives at the traditional rules of probability theory, namely, the sum and product rules. In so doing, he argues that probability theory is not constrained to any discipline, but constitutes a widely applicable extension of traditional logical systems, wherein statements have truth values that have a quantifiable uncertainty. Jaynes viewed probability theory as a kind of “theory of everything” for reasoning. The book contains an extensive and impressive collection of examples and discussions aimed at underlining this idea and showing the ways in which it aligns or not with traditional views.

Book Review:; Applied Statistical Learning: With Case Studies in Stata

Hyukjun (Jay) Gweon — 2026-05-13T07:00:00Z

SIAM Review, Volume 68, Issue 2, Page 462-463, June 2026.
Statistical learning has become an essential component across a wide range of research fields. Among the many textbooks that lay the foundations of theory and practice, Matthias Schonlau’s Applied Statistical Learning: With Case Studies in Stata is a valuable addition—not only for undergraduate and graduate students and course instructors, but also for applied researchers, particularly those who use Stata for data analysis.

Book Review:; Non-Local Cell Adhesion Models: Symmetries and Bifurcations in 1-D

Nadia Loy — 2026-05-13T07:00:00Z

SIAM Review, Volume 68, Issue 2, Page 463-464, June 2026.
If you’ve ever wondered how math and biology can play complex, yet harmonious jazz, this book is your backstage pass. Non-Local Cell Adhesion Models: Symmetries and Bifurcations in 1-D takes you on an intellectually thrilling journey starting from a rather (apparently) humble but powerfully insightful heuristic model introduced by Armstrong, Painter, and Sherratt. Despite its seemingly simple beginnings, this model, a nonlocal nonlinear PDE, is a heavyweight champion in applied mathematics capable of mimicking certain cell sorting experiments in the fascinating and fundamental realm of cell adhesion.

Book Review:; Biological Rhythms

Anita T. Layton — 2026-05-13T07:00:00Z

SIAM Review, Volume 68, Issue 2, Page 464-466, June 2026.
Daniel B. Forger’s Biological Rhythms is an engaging, compact, and intellectually generous exploration of the clocks that govern life. Written for the MIT Press “Essential Knowledge” series, the book aims to offer a rigorous but accessible introduction to the science of circadian and other biological rhythms. Forger succeeds admirably, distilling a wide swath of chronobiology—spanning physiology, neuroscience, endocrinology, and applied mathematics—into a volume that is approachable, fascinating, and often surprisingly personal.

Book Review:; A Brief History of Mathematics for Curious Minds

J. M. Almira — 2026-05-13T07:00:00Z

SIAM Review, Volume 68, Issue 2, Page 466-467, June 2026.
Reading a book about the history of mathematics is never a waste of time, especially if you are a working mathematician, whether applied or pure. Obviously, there are many excellent monographs devoted to specific periods such as nineteenth-century mathematics, or to specific subjects such as mathematical analysis, and these serve specialists very well. The book under review is more ambitious and attempts to cover the whole history of mathematics under the additional assumption that the intended reader is a layperson, not necessarily a professional mathematician. Remarkably, this ambitious task is carried out in just over 200 pages organized into ten chapters, complemented by a bibliography, a name index, and 32 short appendices containing mathematical gems.