All the Math You Missed (But Need to Know for Graduate School)

All the Math You Missed (But Need to Know for Graduate School)

By Thomas A. Garrity. Cambridge University Press, 2021, 416 pp.

Students entering graduate school in mathematics come from a diversity of backgrounds and experiences. It’s rare for a beginning graduate student to already have all the mathematical knowledge they need to be successful; students frequently find themselves needing to quickly learn new areas of mathematics they had not been exposed to before. The new edition of All the Math You Missed (But Need to Know for Graduate School) by Thomas A. Garrity is intended to help students do exactly this. It covers twenty topics but also highlights the commonalities between them, such as the fundamental nature of equivalence problems and the broad use of functions. It seeks to provide a beginning graduate student in mathematics, or other readers of a similar background, with the key definitions, problems, results, and approaches in each of these areas. Throughout, the book has an informal, friendly tone, focusing on motivation and the big picture rather than rigor. It puts things in an understandable order intended to promote learning and intuition—for example, at times it uses a lemma before providing a proof, which gives the reader an understanding of the lemma’s usefulness and a motivation to learn more about it before diving into the details of its possibly quite technical proof.

Each chapter is intended to be a first touchpoint for students to begin to explore that topic. Research shows that when people learn, they build knowledge structures in their brain, and seek to attach new information onto their existing knowledge structures. A novice in an area will frequently have sparse, disconnected knowledge structures, while an expert’s knowledge structures will be dense, well connected, and well-organized 1. This book serves the purpose of helping students build an initial knowledge structure for a new area: while not covering every detail, it highlights the key ideas and connections between them and gives students an idea of the important features of a field. Once students have begun to understand an area, they are then well-equipped to seek out additional sources to make their knowledge more robust; having the initial knowledge structure provided by this book will make these future explorations easier. Towards this end, each chapter features pointers to additional references students can seek out to learn more as well as some exercises students can attempt to expand their understanding. These range from looking something up in a different textbook to proving one of the results in the chapter to diving deeper into some aspect of the topic.

For example, the book begins with a chapter on Linear Algebra and the sentence, “Though a bit of an exaggeration, it can be said that a mathematical problem can be solved only if it can be reduced to a calculation in linear algebra.” The chapter highlights the basic important points of linear algebra (vector spaces, linear transformations, determinants, eigenvalues), beginning with the concrete example of $\mathbb{R}^n$ before moving on to the general definition of a vector space. In addition to definitions and examples, a big emphasis is placed on intuition and conceptual understanding. For example, the author gives three different ways of understanding the determinant: with an inductive definition, as the unique function on matrices satisfying three important properties, and in terms of volumes of transformed unit cubes. The author also doesn’t shy away from going slightly out-of-order when doing so makes the most sense for the conceptual understanding of the novice reader. For example, when he states a key theorem of linear algebra—a list of equivalent conditions for a matrix to be invertible—he includes a condition about eigenvalues even though they haven’t been discussed yet, along with a clear caveat that definitions of eigenvalues are coming. This allows the complete theorem to be stated at a logical point in the narrative (after discussion linear transformations and determinants) while also motivating the later discussion of similar matrices and eigenvalues. I also particularly like that the author makes it clear when his examples are carefully engineered to work out nicely: “I did not just suddenly ‘see’ that A and B are similar. No, I rigged it to be so.”

The chapter on Probability would have been a particularly good resource for me, had I known about it when I needed it: as a first year graduate student, I found myself taking a class in probabilistic combinatorics and beginning research in randomized algorithms, despite never having studied probability before. The chapter includes basic, clear explanations of sample spaces and variance, including the two formulas for variance, which would have been extremely useful for me at the time. I particularly liked that the chapter focused on the central limit theorem just for Bernoulli random variables – the simple case encapsulates all the concepts and ideas of the more general central limit theorem while also being more straightforward for a beginner to comprehend. The chapter includes the clever proof that the integral of a normal random variable is one, a straightforward proof of the central limit theorem, and a proof I was unfamiliar with for Stirling’s approximation for $n!$. While I’ve used Stirling’s formula many times, I’d actually never seen this proof, which uses the ideas of the central limit theorem within the proof. I like that the chapter highlighted the connection between what I typically understand as two entirely different tools, the central limit theorem and Stirling’s formula; it certainly helped me connect these two concepts in a way I hadn’t before. Learning about such connections will be extremely useful to beginning graduate students as they build their knowledge in these fields.

In a departure from many math textbooks, there is also a chapter about theoretical computer science. Though the chapter is titled “Algorithms,” it discusses a bit of complexity as well. As someone who works on the boundary of math and theoretical computer science, it was nice to see this included. This chapter is motivated by the distinction between existence proofs and constructive proofs: sometimes one can prove something exists without knowing how to find it, while sometimes the main motivation is being able to find the desired object. The chapter certainly takes a much more mathematical approach to algorithms and complexity than computer science textbooks usually do, but this makes sense for the intended audience, which is math graduate students. The selection of topics also reflects the interests of the intended audience of mathematicians, covering graphs, sorting lower bounds, P vs. NP, and Newton’s method for finding zeros of functions.

While I’ve highlighted the chapters that are most relevant to my interests and experiences, the book is comprehensive, including twenty chapters on topics ranging from Fourier Analysis to Category Theory:

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Linear Algebra

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$\varepsilon$ and $\delta$ Real Analysis

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Calculus for Vector-Valued Functions

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Point Set Topology

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Classical Stokes’ Theorem

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Differential Forms and Stokes’ Theorem

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Curvature for Curves and Surfaces

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Geometry

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Countability and the Axiom of Choice

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Elementary Number Theory

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Algebra

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Algebraic Number Theory

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Complex Analysis

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Analytic Number Theory

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Lebesgue Integration

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Fourier Analysis

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Differential Equations

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Combinatorics and Probability Theory

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Algorithms

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Category Theory

Whatever topic a math graduate student wants to learn about, they’ll be able to find some relevant sections in this book. Beyond graduate students, it can also be a useful resource for experts in other areas who want to learn about various mathematical topics; if I found myself needing to learn about a new area of mathematics, this book is where I’d start.

The field of mathematics is often beset by the “myth of genius,” a belief that big advances in the field come from a select few who have some innate talents the rest of us cannot hope to learn 4. However, that’s rarely the case. As the author says in the preface, “I know of no serious mathematician who finds math easy. In fact, most, after a few beers, will confess how slow and stupid they are.” Math is not easy, but it can be learned. Pedagogical research at the university level shows that students are more likely to succeed when they have positive expectancies, that is, when they believe they can successfully achieve their desired goals and outcomes 1. This book is an extraordinarily helpful tool that can make it possible for students to believe they are capable of learning a new topic and exploring a new area, by explaining key ideas and approaches in a friendly, accessible way.

Work has shown that in scientific fields where the “myth of genius” is more pervasive, academic departments tend to have fewer numbers of women 5. Counteracting this myth and empowering students to believe they are capable of success is thus critical for improving gender diversity across mathematics. For example, seminal work of Jane Margolis and Allan Fisher studying the experiences of female students in a male-dominated undergraduate computer science program connected female students’ success to having a growth mindset, a belief that even if they don’t immediately understand something now, with hard work and effort they’ll be able to in the future 6. Related work looking at high school math students had similar findings: students with a growth mindset were on average more successful, and this effect was stronger for female students 3. Though not specific to math, interventions focused on encouraging students to see intelligence as malleable rather than fixed also led to improved outcomes among African American college students, with more improvements seen than in white students given similar interventions 2. According to these studies, a growth mindset is a critical tool for fostering student success. By helping all kinds of students believe they are capable of learning, this book has the potential to increase diversity in mathematics.

Overall, this book is a valuable resource for beginning graduate students. By introducing students to new areas of mathematics in an intuitive way with a friendly, accessible tone, it enables them to build on their mathematical knowledge and gain confidence in their abilities in new areas. This makes it more possible for a student who has gaps in their mathematical knowledge to be successful in graduate school, an admirable goal that helps us bring more people into our mathematical community.