New and Noteworthy Titles on our Bookshelf

Polynomials, Dynamics, and Choice

The Price We Pay for Symmetry

By Scott Crass. Chapman and Hall/CRC, 2022, 190 pp.

Reproduced by permission of Taylor & Francis Group.

The fundamental theorem of algebra states that a polynomial of degree $n$ with coefficients in $\mathbb{C}$ will have $n$ roots in $\mathbb{C}$, counting multiplicity. However, it is often the case that knowing these roots exist gives no insight into how to locate them. The book makes the crucial observation that the roots are an unordered set and one can exploit the symmetry that arises from this fact. For instance, the solution set of the polynomial $x^2-14x+40$ could be thought of as either of the ordered pairs $(4,10)$ or $(10,4)$. Thus the line $y=x$ provides symmetry in this small case. Higher dimensional cases have symmetry as well.

In Polynomials, Dynamics, and Choice, the author exploits symmetry to solve higher order polynomials. In Part I, he compares and contrasts two algorithms based on algebra, symmetry, geometric structures, and dynamical processes to solve certain polynomials up to degree 6. Though the text doesn’t assume a reader has seen group theory, a course in abstract algebra would be helpful prior to working through this book. In Part II, Crass explains the effect of symmetry and choice in our everyday decision making.

This book would make a good independent study text for an advanced undergraduate or could be used in an introduction to geometry or dynamics graduate course. Alternatively, a group of graduate students could work through the book as part of a reading group, seminar, or independent study, especially if they use the works cited to recreate the algorithms. Be warned there are no exercises included; however, Crass presents plenty of examples. The book also contains colorful figures of many symmetries and dynamics to aid in one’s understanding.

Mathematical Tools for Real-World Applications

A Gentle Introduction for Students and Practitioners

By Alexandr Draganov. MIT Press, 2022, 306 pp.

Courtesy of MIT Press.

Many mathematics students will end up in industry, but math courses do not always contain the techniques professionals need to use. Instead of being required to regurgitate an algorithm or apply the correct equation, students will need to analyze a situation and have an idea of how to evaluate models and improve them. The real world does not always require “the right answers,” and we need our students to be able to adapt and think critically about complex phenomena.

The first six sections of Mathematical Tools for Real-World Applications illustrate the techniques of checking units, limiting cases, symmetry, scaling, order of magnitude estimates, and successive approximations. The last two sections present two real-world problems the author offers as an extra challenge. Throughout the book, problems are solved in more than one way to illustrate how various methods can be used to achieve or check a solution. Examples range from typical, like the intersection of a circle and a line, to unusual, such as designing satellite antenna. Every section includes key points, a summary, and a long list of exercises.

I imagine one could use this text as part of an undergraduate mathematical modeling class or a course for future engineers. A mathematics department that offers a “getting to know your major” course or even an advanced high school student preparing for an engineering degree could find insight here. I think this text strikes a balance between teaching technical knowledge and building critical thinking skills that students desperately need upon graduation. I found the material in this book to be interesting and insightful; I think you and your students will, too.