Robin Pemantle grew up in Berkeley in the 1960’s and 70’s. You can probablyguess what that was like; then double that. He became interested in mathematics when he was very young. By age four, according to family lore, mathematics was hiscareer plan (to the chagrin of NASA and the local fire stations). Robin’s PhD in probability theory got its start when Persi Diaconis, a mathematician and former magician, visited M.I.T. Robin was supposed to be working on a dissertation in combinatorics (a subject to which he later returned), but found himself working mostly on problems he heard from Persi. During a year off traveling in the South Pacific, he ended up working on probability theory in various youth hostels and on boats. Robin received his PhD in 1988, spent three years on post-doctoral fellowships in Berkeley, Ithaca and Oregon State, most of the 1990’s at UW-Madison, three years at Ohio State, and has been at Penn since 2003. Robin’s research focuses on two areas. Within Probability Theory, the research concerns discrete probability models, including random graph theory, processes with reinforcement, statistical models and random walks. The other research area, analytic combinatorics, is the subject of a textbook with Mark C. Wilson and Steve Melczer (2013, 2024). Robin has been interested in Mathematics Education from an early age, growing up with an insider’s view of an alternative school, and teaching mathematics to grades 5–8 during his college years and before. Robin has written books and articles on mathematics education, most recently having published a book on K-12 education, with co-author Henri Picciotto, entitled, “There is No One Way to Teach Math.”(Routledge, 2024). In direct and indirect ways, Project SEED has played a big role in Robin’s life as a math educator. Robin has held a Sloan Fellowship, a Presidential Faculty Fellowship, the Rollo Davidson Prize, and a Lilly Teaching Fellowship. He was a top five finisher in the Putnam Competition and is a Simons Fellow, a fellow of the American Mathematical Society and a fellow of the Institute for Mathematical Statistics. Most recently, he was inducted into the National Academy of Sciences as a member of the class of 2024.Julian Gould grew up in the suburbs of Minneapolis, Minnesota. His father’s advice was always, “Take math until you just can’t take it anymore.” The advice had a much larger impact on Julian’s life than intended. Julian received his bachelor’s and master’s degrees from Johns Hopkins University, and is hopefully no longer a PhD candidate in mathematics at the University of Pennsylvania. He originally planned to study homotopy type theory, but his research took a turn toward applied algebraic topology and cellular sheaf theory. Before beginning his PhD, Julian worked as an associate quantitative analyst at T. Rowe Price in Baltimore. Julian’s passion for teaching began during his freshman year at Johns Hopkins, tutoring the football team in economics. He has received multiple honors for his work in the classroom, including Penn’s university-wide Dean’s Award for Distinguished Teaching by Graduate Students. He serves as a Master TA in the Penn mathematics department and helps lead the annual training for first-time teaching assistants. His friendship with Robin began when he was a teaching assistant for Robin’s introductorycalculus course for Wharton students. Julian has mentored seven undergraduate students through Penn’s Directed ReadingProgram on topics ranging from descriptive set theory and categorical computer science to financial mathematics, topological data analysis, and game-theoretic models of international relations. His experiences mentoring students helped motivate and inspire many aspects of this book. When he isn’t doing math, Julian likes to write about philosophy, trade on prediction markets, and pick up heavy things.

• Author Bios List of Figures Acknowledgments Introduction Chapter 1 Irrationality Chapter 2 Generating functions Chapter 3 Polynomial systems Chapter 4 Transfinite induction Chapter 5 The Brownian zoo Chapter 6 Approximation by polynomials Chapter 7 Public-key cryptography Chapter 8 Asymptotic analysis Chapter 9 Fractional dimension Chapter 10 Ultrafilters Chapter 11 Scissors congruence Chapter 12 Brownian safari Index of Topics Index of Names

“…[A]n invaluable, exceptional, unique, and unreservedly recommended contribution to personal, professional, and college/university library Algebraic Studies and Mathematics Analysis collections and supplemental Advanced Mathematics curriculum studies lists.”—Midwest Book Review"This is a light undergraduate textbook with rather more text than would be common in a maths book, introducing some genuinely interesting topics: irrationality, generating functions, polynomial systems, transfinite induction, Brownian motion (and its application to option pricing), approximation by polynomials, public key cryptography, asymptotic analysis, fractional dimensions, ultrafilters, scissors congruence, and Brownian motion take 2."—Brian Clegg, Popular Science