The William Lowell Putnam Mathematics Competition is the most prestigious undergraduate mathematics problem-solving contest in North America, with thousands of students taking part every year. This volume presents the contest problems for the years 2001–2016. The heart of the book is the solutions; these include multiple approaches, drawn from many sources, plus insights into navigating from the problem statement to a solution. There is also a section of hints, to encourage readers to engage deeply with the problems before consulting the solutions. The authors have a distinguished history of engagement with, and preparation of students for, the Putnam and other mathematical competitions. Collectively they have been named Putnam Fellow (top five finisher) ten times. Kiran Kedlaya also maintains the online Putnam Archive.

A collection of problems from the William Lowell Putnam Competition which places them in the context of important mathematical themes.

The first comprehensive, unified development of the theory of p-adic differential equations.

Introduced by Peter Scholze in 2011, perfectoid spaces are a bridge between geometry in characteristic 0 and characteristic p, and have been used to solve many important problems, including cases of the weight-monodromy conjecture and the association of Galois representations to torsion classes in cohomology. In recognition of the transformative impact perfectoid spaces have had on the field of arithmetic geometry, Scholze was awarded a Fields Medal in 2018. This book, originating from a series of lectures given at the 2017 Arizona Winter School on perfectoid spaces, provides a broad introduct.

The authors describe a new approach to relative $p$-adic Hodge theory based on systematic use of Witt vector constructions and nonarchimedean analytic geometry in the style of both Berkovich and Huber. They give a thorough development of $\varphi$-modules over a relative Robba ring associated to a perfect Banach ring of characteristic $p$, including the relationship between these objects and etale ${\mathbb Z}_p$-local systems and ${\mathbb Q}_p$-local systems on the algebraic and analytic spaces associated to the base ring, and the relationship between (pro-)etale cohomology and $\varphi$-cohomology. They also make a critical link to mixed characteristic by exhibiting an equivalence of tensor categories between the finite etale algebras over an arbitrary perfect Banach algebra over a nontrivially normed complete field of characteristic $p$ and the finite etale algebras over a corresponding Banach ${\mathbb Q}_p$-algebra. This recovers the homeomorphism between the absolute Galois groups of ${\mathbb F}_{p}((\pi))$ and ${\mathbb Q}_{p}(\mu_{p}\infty)$ given by the field of norms construction of Fontaine and Wintenberger, as well as generalizations considered by Andreatta, Brinon, Faltings, Gabber, Ramero, Scholl, and, most recently, Scholze. Using Huber's formalism of adic spaces and Scholze's formalism of perfectoid spaces, the authors globalize the constructions to give several descriptions of the etale local systems on analytic spaces over $p$-adic fields. One of these descriptions uses a relative version of the Fargues-Fontaine curve.

Abstract: "In 1985 Lenstra and van der Hulst [2] demonstrated that sufficiently strong explicit measures of algebraic independence can be used to provide fast factorization of multivariate integer polynomials. Motivated by their result, we adapt an old approach of Mahler [6] to provide the first explicit measure of algebraic independence for e[superscript [alpha]1] ..., e[superscript [alpha][subscript m], where [alpha]1 ..., [alpha][subscript m] are algebraic numbers linearly independent over the rationals. We also show that, because of the amount of precision required, these explicit measures of algebraic independence are not strong enough, even in the most promising cases, to produce fast factorizations of multivariate integer polynomials."

Abstract: "Consider the system of two generalized Pell equations in three integer unknowns: ax2 - by2 = c dz2 - ey2 = f. If the above system is nontrivial, it is well known that deep results of Baker can be used to effectively bound the absolute value of all integer solutions. Motivated by an idea of Cohn, we present a method for solving many of these systems in an elementary fashion. Using the method, new and old results of P[subscript t]-sets can be obtained systematically. Furthermore, the method can be applied to the problem of determining integer points on certain elliptic curves."