Have you ever wondered about the explicit formulas in analytic number theory? This short book provides a streamlined and rigorous approach to the explicit formulas of Riemann and von Mangoldt. The race between the prime counting function and the logarithmic integral forms a motivating thread through the narrative, which emphasizes the interplay between the oscillatory terms in the Riemann formula and the Skewes number, the least number for which the prime number theorem undercounts the number of primes. Throughout the book, there are scholarly references to the pioneering work of Euler. The book includes a proof of the prime number theorem and outlines a proof of Littlewood's oscillation theorem before finishing with the current best numerical upper bounds on the Skewes number.
This book is a unique text that provides all the mathematical background for understanding the Skewes number. Many exercises are included, with hints for solutions. This book is suitable for anyone with a first course in complex analysis. Its engaging style and invigorating point of view will make refreshing reading for advanced undergraduates through research mathematicians.
Undergraduate and graduate students interested in analytic number theory.
Student Mathematical Library Volume: 92
2020; 152 pp; Softcover
MSC: Primary 11;
Print ISBN: 978-1-4704-6257-4
Discrete Differential Geometry (DDG) is an emerging discipline at the boundary
between mathematics and computer science. It aims to translate concepts
from classical differential geometry into a language that is purely finite
and discrete, and can hence be used by algorithms to reason about geometric
data. In contrast to standard numerical approximation, the central philosophy
of DDG is to faithfully and exactly preserve key invariants of geometric
objects at the discrete level. This process of translation from smooth
to discrete helps to both illuminate the fundamental meaning behind geometric
ideas and provide useful algorithmic guarantees.
This volume is based on lectures delivered at the 2018 AMS Short Course “Discrete Differential Geometry,” held January 8?9, 2018, in San Diego, California.
The papers in this volume illustrate the principles of DDG via several recent topics: discrete nets, discrete differential operators, discrete mappings, discrete conformal geometry, and discrete optimal transport.
Graduate students and researchers interested in discrete differential geometry and its applications.
Proceedings of Symposia in Applied Mathematics Volume: 76; 2020; 140 pp; Softcover
MSC: Primary 53; 52; 65;
Print ISBN: 978-1-4704-4662-8
Borel's Conjecture entered the mathematics arena in 1919 as an innocuous remark about sets of real numbers in the context of a new covering property introduced by Emile Borel. In the 100 years since, this conjecture has led to a remarkably rich adventure of discovery in mathematics, producing independent results and the discovery of countable support iterated forcing, developments in infinitary game theory, deep connections with infinitary Ramsey Theory, and significant impact on the study of topological groups and topological covering properties.
The papers in this volume present a broad introduction to the frontiers of research that has been spurred on by Borel's 1919 conjecture and identify fundamental unanswered research problems in the field. Philosophers of science and historians of mathematics can glean from this collection some of the typical trends in the discovery, innovation, and development of mathematical theories.
Graduate students and research mathematicians interested in topology, set theory, and infinitary Ramsey theory.
Contemporary Mathematics Volume: 755; 2020; 242 pp; Softcover
MSC: Primary 03; 22; 54; 91;
Print ISBN: 978-1-4704-5099-1
A co-publication of the AMS and Clay Mathematics Institute
This is the first volume of the lectures presented at the Clay Mathematics Institute 2014 Summer School, “Periods and Motives: Feynman amplitudes in the 21st century”, which took place at the Instituto de Ciencias Matematicas?ICMAT (Institute of Mathematical Sciences) in Madrid, Spain. It covers the presentations by S. Bloch, by M. Marcolli and by L. Kindler and K. Rulling.
The main topics of these lectures are Feynman integrals and ramification theory. On the Feynman integrals side, their relation with Hodge structures and heights as well as their monodromy are explained in Bloch's lectures. Two constructions of Feynman integrals on configuration spaces are presented in Ceyhan and Marcolli's notes. On the ramification theory side an introduction to the theory of l-adic sheaves with emphasis on their ramification theory is given. These notes will equip the reader with the necessary background knowledge to read current literature on these subjects.
Graduate students and researchers interested in mathematical aspects of Feynman integrals, in particular connections with Hodge theory and l-adic sheaves.
Clay Mathematics Proceedings Volume: 21
2020; 229 pp; Softcover
MSC: Primary 81; 14; 11;
Print ISBN: 978-1-4704-4329-0
This book provides an introduction to hyperbolic geometry in dimension three, with motivation and applications arising from knot theory. Hyperbolic geometry was first used as a tool to study knots by Riley and then Thurston in the 1970s. By the 1980s, combining work of Mostow and Prasad with Gordon and Luecke, it was known that a hyperbolic structure on a knot complement in the 3-sphere gives a complete knot invariant. However, it remains a difficult problem to relate the hyperbolic geometry of a knot to other invariants arising from knot theory. In particular, it is difficult to determine hyperbolic geometric information from a knot diagram, which is classically used to describe a knot. This textbook provides background on these problems, and tools to determine hyperbolic information on knots. It also includes results and state-of-the art techniques on hyperbolic geometry and knot theory to date.
The book was written to be interactive, with many examples and exercises. Some important results are left to guided exercises. The level is appropriate for graduate students with a basic background in algebraic topology, particularly fundamental groups and covering spaces. Some experience with some differential topology and Riemannian geometry will also be helpful.
Graduate students interested in hyperbolic geometry and knot theory.
Graduate Studies in Mathematics Volume: 209
2020; 369 pp; Softcover
MSC: Primary 57; Secondary 30
Print ISBN: 978-1-4704-5499-9
Number Theory is a newly translated and revised edition of the most popular introductory textbook on the subject in Hungary. The book covers the usual topics of introductory number theory: divisibility, primes, Diophantine equations, arithmetic functions, and so on. It also introduces several more advanced topics including congruences of higher degree, algebraic number theory, combinatorial number theory, primality testing, and cryptography. The development is carefully laid out with ample illustrative examples and a treasure trove of beautiful and challenging problems. The exposition is both clear and precise.
The book is suitable for both graduate and undergraduate courses with enough material to fill two or more semesters and could be used as a source for independent study and capstone projects. Freud and Gyarmati are well-known mathematicians and mathematical educators in Hungary, and the Hungarian version of this book is legendary there. The authors' personal pedagogical style as a facet of the rich Hungarian tradition shines clearly through. It will inspire and exhilarate readers.
Undergraduate and graduate students interested in number theory.
Pure and Applied Undergraduate Texts Volume: 48
2020; 552 pp; Softcover
MSC: Primary 11;
Print ISBN: 978-1-4704-5275-9
This book represents a novel approach to differential topology. Its main focus is to give a comprehensive introduction to the classification of manifolds, with special attention paid to the case of surfaces, for which the book provides a complete classification from many points of view: topological, smooth, constant curvature, complex, and conformal.
Each chapter briefly revises basic results usually known to graduate students from an alternative perspective, focusing on surfaces. We provide full proofs of some remarkable results that sometimes are missed in basic courses (e.g., the construction of triangulations on surfaces, the classification of surfaces, the Gauss-Bonnet theorem, the degree-genus formula for complex plane curves, the existence of constant curvature metrics on conformal surfaces), and we give hints to questions about higher dimensional manifolds. Many examples and remarks are scattered through the book. Each chapter ends with an exhaustive collection of problems and a list of topics for further study.
The book is primarily addressed to graduate students who did take standard introductory courses on algebraic topology, differential and Riemannian geometry, or algebraic geometry, but have not seen their deep interconnections, which permeate a modern approach to geometry and topology of manifolds.
Undergraduate and graduate students interested in teaching and learning the basics of algebraic and differential topology.
Graduate Studies in Mathematics Volume: 208
2020; Softcover
MSC: Primary 57; 53; 55; 30;
Print ISBN: 978-1-4704-6132-4