OAT 2019: Strength in Numbers – Mathematics

1. 100 years of math milestones: the Pi Mu Epsilon centennial collection
Garcia, Stephan Ramon. by Stephan Ramon Garcia and Steven J. Miller American Mathematical Society, 2019

“Dedicated to the promotion of mathematics and recognition of students who successfully pursue mathematical understanding,” Pi Mu Epsilon (PME) regularly includes challenging problems in its journal to encourage engagement with members. For PME’s centennial in 2013, the journal’s problem editors created a special collection of 100 problems. Miller (Williams College), one of the editors, and Garcia (Pomona College) collected those problems, each one celebrating a year since 1913 with a discussion of a significant event in the development of mathematics, and revised and expanded them to include cross references where possible. The new collection is a tour de force of exposition and invention, allowing readers to delve deeply into some topics by weaving concepts from one entry with others. For example, the 1913 entry features Paul Erdös and his various problems on arithmetic progressions among the natural numbers. The 2004 entry returns to this idea with an account of the Green-Tao theorem, which settles Erdös’s conjecture about progressions among the primes. The authors make many more wonderful choices, producing a remarkable contribution to mathematical literature.
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2. 99 variations on a proof
Ording, Philip. Princeton, 2019

Virtually every mathematics student is required to take an introduction to mathematics or an introduction to mathematical proofs class. These courses vary widely in the topics covered, but all attempt to help students develop the ability to write a “good” mathematical proof. Students often find this intimidating, imagining that, when dealing with mathematics, there is only one correct answer. In this text Ording (Sarah Lawrence College) provides 99 different proofs of the same theorem. The theorem itself is a fairly simple statement about the roots of a particular cubic equation. However, in reading the numerous proofs given for this theorem, one begins to appreciate both the aesthetics of proofs and the number of acceptable approaches to writing them. The proofs presented range from ancient to the Renaissance to the modern, even including a physics-based proof using null points of an electric field. The reader will gain insight into mathematical literature as it has evolved over time and as it has been influenced by various philosophies and cultures.
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3. Critical statistics: seeing beyond the headlines
Vries, Robert de. Red Globe Press, 2018

With Critical Statistics, de Vries (Univ. of Kent, UK) has given a significant gift to statistics literature. The book is not a first text in statistical computation and methods. Instead, it is a work in statistics numeracy: numbers in the news, numbers in life, numbers to explain the world. In ten chapters, the author discusses how newsworthy numbers are generated, how samples work, measurement methods, averages, inference, graphics, contexts for numeric statements, and the use of numbers to reach conclusions. The text is written as a straightforward and engaging narrative, but chapters helpfully conclude with summaries, terminology, key thoughts, and exercises. The graphics are intuitive and enhance understanding of the concepts discussed. The publisher maintains a website which provides supplementary information, new studies, more exercises, reader submissions, and links to data sets.
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4. Extremal problems for finite sets
Frankl, Peter. by Peter Frankl and Norihide Tokushige American Mathematical Society, 2018

Where a generic set X (always finite in this book) contains elements, its powerset P(X) contains all subsets of X, and then P(P(X)) comprises all families of subsets of elements of X. Families possess various numerical invariants (i.e., statistics), such as the maximum size of any pairwise disjoint subfamily. Extremal set theory studies, essentially, natural constraints on the possible simultaneous values of such invariants, with the subject’s name emphasizing the important role of extreme cases. The subject has themes (e.g. Erdos’s probabilistic method), but also, characteristically, many very clever basically ad hoc arguments. The largely independent chapters typically present details of one major result or technique each. Frankl (Rényi Institute, Hungary) long ago established himself as a leader in the field; his union-closed conjecture is now one of the most notorious open problems in mathematics. “Ad hoc” makes this essential book attractive in principle to undergraduates, albeit only the stouthearted.
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5. History of cryptography and cryptanalysis: codes, ciphers, and their algorithms
Dooley, John F. Springer, 2018

While providing a history of cryptology, this book excels—it documents, illustrates, instructs, reveals, and entertains. It is comparable to David Kahn’s classic The Codebreakers: A History of Secret Writing (1967). Dooley (emer., Knox College) provides an excellent overview of cryptographic devices and algorithms, starting with ancient Rome and up to WW II, and covering monoalphabetic ciphers, polyalphabetic substitutions, the Vigenére cipher, and cipher machines such as Enigma. The author then ably extends this known history into the computer age via the creation of public-key cryptography, web security, cyber warfare (e.g., worms carrying a virus), the security of household devices (e.g., Amazon Echo), and quantum computing. Most of the chapters introduce and motivate new ideas—techniques, concerns—via a historical account. When necessary, aspects of discrete-level mathematics are introduced to illustrate the principles and processes of both cryptography and cryptoanalysis. Short biographies of individuals who contributed to cryptologic advances are included, woven in with stories of the people who used and broke transmitted codes.
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6. The model thinker: what you need to know to make data work for you
Page, Scott E. Basic Books, 2018

Page, a prominent social science researcher and complex systems scholar (Univ. of Michigan), has written a tremendously significant book embracing a creative, innovative approach for thinking about the complex mechanisms of social and natural phenomena. Page emphasizes a many-model thinking approach rather than a stand-alone model to understand and solve various real-world problems. His many-model thinking approach has certain parallels to multiscale mathematical modeling and ensemble modeling in data science. Page introduces models from various disciplines to facilitate many-model thinking. For each individual model, he thoroughly describes all levels of components, from related assumptions, to mathematics equations, to applications and implications. At the end of the book, he ultimately demonstrates how opioid pandemics and income inequality can be explained by the many-model approach. The Model Thinker could serve as a complementary textbook to mathematically and computationally intensive courses in econometrics, statistics, and data science.
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