SIAM Review, Volume 66, Issue 2, Page 391-399, May 2024.
As I sat down to write this introduction, I became curious how the books chosen for review have changed over the past decades. So I scanned through a few SIREV Book Review section introductions written 10, 20 or more years ago by former section editors. That act of procrastination allows me to put the current collection of reviews in “historical context.” Some of the topics are relatively new, whereas others have remained relevant over decades. Our featured review was written by Zachary Kilpatrick, on the book Neurodynamics: An Applied Mathematics Perspective, authored by Stephen Coombes and Kyle Wedgwood. As Kilpatrick noted, while neuroscience is not a new topic, it has inspired a number of new mathematical methods, like network science and topological data analysis. Authors Coombes and Wedgwood take the reader “on a thrilling and addictive journey developing and analyzing many canonical models in neuroscience.” Sounds exciting! Kilpatrick concluded his review by noting that the book “keeps this tradition of mechanistic model analysis alive, while modernizing with handy methods from stochastic processes and piecewise smooth dynamical systems.” I wrote the next review for a book that likely would not have been on the shelf twenty years ago: Math for Deep Learning: What You Need to Know to Understand Neural Networks, by Ronald T. Kneusel. Unlike most other math book authors, Kneusel is not a university faculty member but an AI developer. His practitioner's background shows in his ability to explain advanced mathematical concepts in an accessible way, accompanied by practical examples. The next review was written by Noah Rosenberg, on the book Tree Balance Indices: A Comprehensive Survey, by Mareike Fischer, Lina Herbst, Sophie Kersting, Luise Kühn, and Kristina Wicke. Rosenberg noted that the literature on tree balance has been fragmented across different fields (e.g., mathematics, theoretical computer science, and evolutionary biology), often featuring slightly different definitions or indexing methods, which can be a barrier for researchers. By helping to unify some of these indices, this is a “wonderfully welcome book,” according to Rosenberg. The next book review by Vittorio Romano is on Advanced Calculus and Its Applications in Variational Quantum Mechanics and Relativity Theory, by Fabio Silva Botelho. The title of the book gives a pretty clear clue about its topics. Romano found the book's objective rather ambitious, which may have made its content challenging for readers who aren't already familiar with the methods. Stephania Bellavia provided a review for An Introduction to Optimization on Smooth Manifolds, by Nicolas Boumal. The book covers numerical methods for unconstrained optimization on smooth manifolds and, contrary to its title, is not an introduction to these topics, as Bellavia noted. Read the review and you will get a pretty good idea of what the book is about. We wrap up with a lively review written by Volker Schulz, on the book A Journey through the History of Numerical Linear Algebra, by Claude Brezinski, Gérard Meurant, and Michela Redivo-Zaglia. The history of numerical linear algebra is rich, so this is bound to be a long journey. Indeed, this is a book of roughly 800 pages that looks at the role numerical linear algebra plays in recent developments in science and technology, particularly the progress in data science and machine learning. As you can tell at the beginning of this introduction, I am often fascinated by the evolution of things (science, law, culture). Schulz's review has inspired me to find time to learn about the historical roots of a field that lies at the heart of current progress in math, science, and technology.

中文翻译：

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 书评


《SIAM 评论》，第 66 卷，第 2 期，第 391-399 页，2024 年 5 月。

当我坐下来写这篇介绍时，我开始好奇被选为评论的书籍在过去几十年里发生了怎样的变化。因此，我浏览了一些 SIREV 书评部分的介绍，这些介绍是由前部分编辑在 10 年、20 年前或更长时间之前撰写的。这种拖延行为让我能够将当前的评论集合放在“历史背景”中。其中一些主题相对较新，而另一些主题几十年来仍然具有相关性。我们的专题评论是由扎卡里·基尔帕特里克 (Zachary Kilpatrick) 撰写的，内容涉及斯蒂芬·库姆斯 (Stephen Coombes) 和凯尔·韦奇伍德 (Kyle Wedgwood) 合着的《神经动力学：应用数学视角》一书。正如基尔帕特里克指出的那样，虽然神经科学并不是一个新主题，但它激发了许多新的数学方法，例如网络科学和拓扑数据分析。作者库姆斯和韦奇伍德带领读者“踏上一段令人兴奋且令人上瘾的旅程，开发和分析神经科学中的许多典型模型”。听起来很令人兴奋！基尔帕特里克在总结他的评论时指出，这本书“保留了机械模型分析的传统，同时利用随机过程和分段平滑动力系统的便捷方法进行现代化。”我为一本二十年前可能不会上架的书写了下一篇评论：《深度学习的数学：理解神经网络需要了解什么》，作者：Ronald T. Kneusel。与大多数其他数学书籍作者不同，克努塞尔不是大学教员，而是人工智能开发人员。他的从业者背景体现在他能够以通俗易懂的方式解释高级数学概念并附有实际例子。下一篇评论由诺亚·罗森伯格 (Noah Rosenberg) 在《树木平衡指数：综合调查》一书中撰写，作者为 Mareike Fischer、Lina Herbst、Sophie Kersting、Luise Kühn 和 Kristina Wicke。 罗森伯格指出，有关树木平衡的文献分散在不同的领域（例如数学、理论计算机科学和进化生物学），通常定义或索引方法略有不同，这可能对研究人员来说是一个障碍。罗森伯格表示，通过帮助统一其中一些指数，这是一本“非常受欢迎的书”。维托里奥·罗马诺 (Vittorio Romano) 的下一篇书评是法比奥·席尔瓦·博特略 (Fabio Silva Botelho) 的《高级微积分及其在变分量子力学和相对论中的应用》。这本书的标题给出了关于其主题的非常清晰的线索。罗马诺发现这本书的目标相当雄心勃勃，这可能使它的内容对于不熟悉这些方法的读者来说具有挑战性。 Stephania Bellavia 对 Nicolas Boumal 所著的《平滑流形优化简介》进行了评论。正如贝拉维亚指出的那样，这本书涵盖了平滑流形上无约束优化的数值方法，与书名相反，它并不是对这些主题的介绍。阅读评论，您将对这本书的内容有一个很好的了解。我们以 Volker Schulz 对 Claude Brezinski、Gérard Meurant 和 Michela Redivo-Zaglia 合着的《数值线性代数历史之旅》一书的生动评论作为结束。数值线性代数的历史是丰富的，因此这注定是一个漫长的旅程。事实上，这是一本大约 800 页的书，探讨了数值线性代数在科学技术的最新发展中所扮演的角色，特别是数据科学和机器学习的进展。正如您在本介绍的开头所看到的，我常常对事物（科学、法律、文化）的演变着迷。 舒尔茨的评论激励我抽出时间来了解这个领域的历史根源，这个领域是当前数学、科学和技术进步的核心。