August 2023
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Ergin Bayram's review of Differential Geometry of Plane Curves, by Hilário Alencar, Walcy Santos, and Gregório Silva Neto
"The authors focus their attention on the differential geometry of planar curves with great depth, although phenomenal books on differential geometry already exist. Many interesting and inspiring geometrical and topological results on planar curves are here presented in an elementary form. The topics are chosen to sharpen the reader’s mathematical intuition for asserted geometric concepts and results. A very good selection of examples guides the reader towards a better understanding of each notion."
Read the full review in zbMathOPEN
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"While this is a book for a first (graduate) course in analytic number theory, it contains recent results. For example, the book includes several theorems by Terrence Tao and his collaborators on primes in arithmetic progressions and on twin primes. In sum, Analytic Number Theory for Beginners provides a brisk introduction to analytic number theory, with more than enough material for a graduate course."
Read the full review in MAA Reviews
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July 2023
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"It aims to ease the transition from primarily calculus-based mathematics courses to more conceptually advanced proof-based courses. As such, it is intended for early undergraduate students who wish to become familiar with the language, fundamental knowledge and methods of abstract mathematics. Although this is a niche market with a lot of competition, the authors – Sebastian M. Cioaba and Werner Linde – have nevertheless come up with a relevant and unique proposal."
Read the full review in MAA Reviews
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"Overall the book is a valuable one; it is always nice to see a new approach to an old subject, particularly when the material is handled as deftly as it is here. Instructors teaching geometry, or just interested in the subject for their own pleasure, should definitely look at this book."
Read the full review in The Mathematical Gazette
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Artem Zvavitch's review of Asymptotic Geometric Analysis, Part II, by Shiri Artstein-Avidan, Apostolos Giannopoulos, and Vitali D. Milman
"The book is very well written, and proofs of the theorems are presented in a most natural and accessible way. Moreover, one can see the authors' fresh and personal touch on many of them. Geometric, analytic and probabilistic views are given in parallel and connections between those subjects are beautifully presented. The book is simply pleasant to read and easy to navigate. After each chapter the authors give a good list of notes with all possible references, which definitely helps with further reading and study. The book contains an outstanding collection of references, which in itself is a great treasure. Thus the book, and indeed the series, provide an excellent source for specialists working in the field as well as for people who wish to join the game and for graduate students."
Read the full review at MathSciNet
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"Teaching with Number Theory through the Eyes of Sophie Germain is a delight. There is truly something for everyone. The newcomer to number theory is given a motivated and accessible path to discovering all the essentials. The experienced number theorist is treated to an exciting journey through Germain’s attempt to crack Fermat’s Last Theorem. Having used this text for an independent study, I can say that this wonderful book is not for the casual reader. However, the rewards of the effort are well worth it. How often do you find your heart racing while reading a math textbook, eager to discover what happens next?"
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"Students and others who pick up this volume will find so many life stories that surely they will find one to whom they can relate. Indeed, Zitarelli strongly believed in the value of history of mathematics for humanizing mathematics. His genuine concern for other people carried over into how he treated students and colleagues, and that trait is also present in this book in his conversational, storytelling-oriented writing style."
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June 2023
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"For almost 50 years the source to turn to for the basics of ultrafilters has been W. W. Comfort and S. Negrepontis [The theory of ultrafilters. Berlin-Heidelberg-New York: Springer-Verlag (1974; Zbl 0298.02004)]; it still is but its main objects of study were the ultrafilters themselves: their properties as combinatorial objects and as points in topological spaces. The book under review goes way beyond this and shows where ultrafilters may be used in other branches of mathematics."
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"These notes constitute the most influential text ever written on the topology and geometry of three-manifolds. Soon after they were released, they dramatically transformed the field of geometric topology in the sense that they gave it a completely new and unpredicted direction which turned out to be highly fecund, providing the necessary tools and techniques for the field development in this direction. Today, these notes remain fresh and the ideas they contain constitute an inexhaustible source of inspiration."
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"Every professional mathematician or an university mathematics lecturer must have faced the question of ‘where is all these math that you are doing/teaching come up in real life?’. The answer to this type of question is not always easy and sometimes we may not be able to think of an ‘easy’ on the spot application of the material we are teaching. The book under review gives several really good answers to this question in less than 200 pages. The author takes the view that mathematics is literally everywhere and backs this thesis with a wide range of applications of mathematics, ranging from pizza-eating to epidemiology. Some old-timers like coin tossing, magic squares and card tricks makes their appearences with newer subjects like COVID-19 and vaccines. The book is peppered with humorous story telling which makes it much more accessible than many other books on the same subject. A very enjoyable read, this is recommended for both the novice and the professional."
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"The two most exciting pedagogical innovations of the last 30 years in college math instruction—inquiry-based learning and the reading of primary historical sources—are masterfully combined here. Students who take a course using this text are in for a revelatory, potentially life-changing, experience. So, too, for instructors who might teach out of it."
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May 2023
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"Although the authors primarily see this book, “encouraged by appreciative and constructive feedback from faithful readers, as a compilation and revision of the 2020 and 2021 volumes of the online journal, Mathematical Reflections", this reviewer would emphasize two other aspects of the book: i) a multi-national statement of human resilience in response to the pandemic, and ii) a collection of 16 beautiful articles succinctly summarizing current and interesting mathematical topics not easily accessible."
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John D. Cook's MAA review of A Panoply of Polygons, by Claudi Alsina and Roger B. Nelsen
"The recent resolution to the long-standing question about the existence of aperiodic monotiles shows that there are deep questions one can ask about simple shapes. Panoply is a more advanced book than you might expect if your last exposure to polygons was in a high school geometry class, though the book is accessible to readers without much more than a high school level background mathematics."
"A Panoply of Polygons is a fun read. The topics are nontrivial, but accessible, and can be read one or two pages at a time. It is not a textbook, at least not a conventional textbook, though it does have challenges at the end of each chapter that could be used as exercises. I think of it more as a book to keep on your nightstand, if, like the reviewer, you are the kind of person who keeps mathematics books on your nightstand."
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April 2023
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"Pengelley is a deft teacher, a gifted expositor, and highly skilled at IBL instruction. The student is led, gently, to investigate illuminating examples through more or less elementary computations, and then asked to consider what Germain must have thought when performing these same computations. By the end, the students are operating at quite a sophisticated level both mathematically and historically. There is no other book like this; students in a course taught from it will be taken on an extraordinary intellectual journey."
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"Congratulations to both the author for writing this valuable book, and the AMS for its publication as a volume in the prestigious series ‘Pure and Applied Undergraduate Texts’. There are all good reasons to strongly recommended the book to the thousands of students worldwide studying stochastic calculus, in particular to students following MSc and PhD programs in the area of ‘mathematical finance’. Teachers of courses in stochastic calculus can efficiently combine this book with other sources."
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"Sage for Undergraduates reads like a well-written textbook, not like software documentation. Readers with more experience and less patience may prefer to read the Sage documentation, but this book provides more of a gentle on-ramp to Sage. The book has a conversational tone, and many examples, making it far more welcoming than a software reference manual.”
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"…As a whole, it’s remarkable how Dr. Bonato has distilled a huge body of literature into an extensively referenced 250 page book that could have easily been at least twice that length. Researchers will surely find this a helpful reference text, with lots of important results and proofs conveniently organized in one place, but they are also likely to encounter something new and interesting, and probably an open problem or two to ponder. Meanwhile, those who are newer to exploring these topics will almost certainly find at least one motivating idea—a particular application or an entry point into the theory—that gets them hooked into reading more.”
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"…I think this is an excellent book for anyone who wants to know how to compute the homology of finite simplicial complexes using combinatorial methods without caring much about topology in general.”
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"This 4-volume set is a fantastic resource. It provides complete access to material that was as easily available before. . . . There is much to learn here. One can also be impressed by the amazing breadth of this collection, and by the prodigious number of great results that it includes. It is a great tribute to the genius of one of the greatest mathematicians of all time, as well as a great resource for today’s and tomorrow’s mathematicians.”
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"The mathematical theory of tilings has three ingredients that make it attractive to a wide readership: Many difficult problems are easy to formulate. Many mathematical subfields meet here, such as algebra, topology, discrete geometry, and computability theory. Practically no other mathematical topic can be illustrated so beautifully. This book offers all these ingredients and combines them into a well-rounded whole.”
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March 2023
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"The present book under review entitled Algebraic Geometry – Notes on a Course by Michael Artin is, according to my very subjective viewpoint, one of the best textbooks devoted to basics on algebraic geometry. I am aware of the fact that this is a rather bold statement, but I will try to justify my claim here. There are many textbooks devoted to the foundations of algebraic geometry, and it seems that there is no room for new ideas or strategies in writing such books. Everything has been tried. This was also my first prediction before receiving this book, and I can honestly say that I am very happy to have been wrong.
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"...With this in mind, I tell you that this is a book filled with stories of love. It is, of course, more than that. It is a book about resilience, struggle, persistence, obstacles, grit, barriers, determination, failure, success, and so many other things. But at its core, I see the common theme of love. I read these as stories of people who love mathematics so much that they have endured heartbreaking struggle within systems that are built to keep them out. They have overcome obstacles that many of us can only imagine, just for the privilege to do math. These are also stories of people who helped them to succeed, through large and small acts of kindness, generosity, and thoughtfulness; in other words: love. And this is why I think you should read this book."
Read the full review at MathSciNet
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Jeno Lehel's MathSciNet review of Combinatorial Convexity, by Imre Bárány
"It is a real gift for students and the much larger readership if they can learn firsthand from an active researcher in a subject. Imre Bárány is one of them; in particular his work has been a driving force behind the recent progress of combinatorial convexity. His book of the highest standard can be used as a textbook for graduate or undergraduate courses. The short chapters are suitable for one- or two-hour lectures. At the end of each chapter various exercises complete the material and help deepen understanding. Basic linear algebra, linear programming, and some experience in graph and hypergraph theory, that is, certain mathematical maturity, are expected from the reader."
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“This is an excellent self-explanatory and self-contained textbook for senior undergraduate or freshman graduate complex analysis and number theory students. It clearly explains how theoretical complex analysis can help mathematicians answer difficult questions about real numbers, in particular, questions about integers. Each of the six chapters starts with an easy explanation of the topic and then brilliantly moves to more involved problems and discoveries. Each section comes with both helpful solved examples as well as a comprehensive set of exercises with answers to most problems. These topics and their related exercises are beautifully illustrated using Mathematica graphing technology. What is special about this textbook is that at the end of each chapter, a history of the leading mathematicians and their discoveries is nicely narrated, which can be good motivation for curious young mathematicians to further explore and investigate these topics. These short, but informative, biographies explain how our current understanding of complex analysis evolves from three distinct lines of development, arising from the work of Riemann, Cauchy, and Weierstrass.”
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February 2023
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“The writing is very clear, and there are abundant cross-references and a good index in case you want to start in the middle of things rather than reading straight through. In particular the book is valuable if you already know about finite fields but would like to see some interesting applications. As abstract algebra texts go, this treatment is very concrete with lots of specific examples. The book has a strong number theory flavor and brings out how these abstract structures generalize the integers.”
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“Zaslavsky and Skopenkov’s Mathematics via Problems Part 2: Geometry is not your average textbook. Though it touches upon most high school geometry topics, it does go beyond the level required by the standards. In fact, it is not an instructive book for students’ first time engaging with relevant geometry content; rather, it is a collection of relevant problems meant to be pondered and discussed. The book serves as a problem-solving agenda for mathematics clubs and communities of practice called “math circles.” Math circles are vertical clubs of mathematicians, from elementary-aged students to professionals and researchers, that support engagement with mathematics via problems.”
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"There are exercises throughout, including many cases of “we leave the proof as an exercise.” The major change in this third edition is that full solutions are now included. Many of these were written by Roger Lipsett and then completed and revised by Cox. Inevitably, small errors and unclear spots were found in the course of preparing solutions, so one of the advantages of the new edition is that “small errors have been fixed and many hints have been clarified and/or expanded.” The solutions to the exercises fill 219 pages of the book, almost doubling its size."
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"... As with the first volume, this one is a very readable introduction into the history of mathematics starting from the 16th century. Appealing are the boxes summarizing important results. The volume is full of graphs, figures, and depiction and photos of mathematicians. It is obvious that the choice of topics concerning the sheer mass of material may be criticized in some cases but all in all the choices have been made wisely. According to its subtitle ‘A source-based approach’ there are very many excerpts of important sources so that the reader can ‘listen’ directly to the creators of the theories described."
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"In this textbook, the author provides a thorough introduction to pursuit-evasion games, a class of dynamic interactions that take place on a graph. In a pursuit-evasion game, there is a set of vertices and a set of edges linking them. Time is discrete and measured in rounds. One player, the evader, can move across vertices of the graph according to fixed rules. One or more other players, the pursuers, can move according to their own rules. The goal of the pursuers is to reach the same location as the evader, or to surround or locate the evader. A classic example is Cops and Robbers, where the evader and pursuers move in alternating rounds, and each may move along a single edge to a neighboring vertex."
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January 2023
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"There are a number of books available with intriguing problems for students (and others) to puzzle over, but you will want to make room in your bookshelf for one more. This collection, the 28th volume from MSRI’s Mathematical Circles Library, provides a unique set of resources for use in mathematical circles and similar formal and informal settings. The activities, chosen from those used in Julia Robinson Mathematics Festivals (founded in 2007 by Nancy Blackman and inspired by the Saint Mary’s College contests of the 1970s) have been carefully curated to optimize involvement (low threshold/high ceiling with ample opportunities for “tweaks” motivating new investigations)..."
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"This is really an excellent textbook. It covers a wealth of interesting material, it is written in a very clear and convincing style, and it explains ideas, rather than drowning the reader in technicalities, as many other books do. The author takes his task serious, by not restricting himself to just the presentation of definitions, theorems, and proofs, which makes reading often pretty dry, but also by giving some hints which should prevent unexperienced readers from walking into certain traps. (Such sections are called “Examples where this is illegal” and refer, for example, to differentiating under the integral sign.)"
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