Language and the Rise of the Algorithm by 
A Graduate Course in Probability by
This book grew out of the notes for a one-semester basic graduate course in probability. As the title suggests, it is meant to be an introduction to probability and could serve as textbook for a year long text for a basic graduate course. It assumes some familiarity with measure theory and integration so in this book we emphasize only those aspects of measure theory that have special probabilistic uses.The book covers the topics that are part of the culture of an aspiring probabilist and it is guided by the author's personal belief that probability was and is a theory driven by examples. The examples form the main attraction of this subject. For this reason, a large book is devoted to an eclectic collection of examples, from classical to modern, from mainstream to 'exotic'. The text is complemented by nearly 200 exercises, quite a few nontrivial, but all meant to enhance comprehension and enlarge the reader's horizons.While teaching probability both at undergraduate and graduate level the author discovered the revealing power of simulations. For this reason, the book contains a veiled invitation to the reader to familiarize with the programing language R. In the appendix, there are a few of the most frequently used operations and the text is sprinkled with (less than optimal) R codes. Nowadays one can do on a laptop simulations and computations we could only dream as an undergraduate in the past. This is a book written by a probability outsider. That brings along a bit of freshness together with certain "naiveties."
How to Cheat with Statistics - And Get away With it : From Data Snooping over Kitchen Sink Regression to "Creative Reporting" by
How to Cheat with Statistics explains how to identify and catch statistical cheaters. Meissner encountered many weaknesses and flaws in statistics through 30 years of teaching. These weaknesses allow a malevolent researcher to manipulate the inputs, the calculations, and the reporting of results to derive a desired outcome. This book should be valuable to everyone who wants to gain a deeper understanding of the weaknesses in statistics and learn how to evaluate statistical research to catch a statistical cheater! The math is explained in simple terms and is easy to follow. In addition, the book comes with 18 Excel spreadsheets and 7 Python codes. There are also questions and problems at the end of each chapter, which should facilitate the usage in a classroom.
Linear Algebra : Vector Spaces and Linear Transformations by
This textbook is directed towards students who are familiar with matrices and their use in solving systems of linear equations. The emphasis is on the algebra supporting the ideas that make linear algebra so important, both in theoretical and practical applications. The narrative is written to bring along students who may be new to the level of abstraction essential to a working understanding of linear algebra. The determinant is used throughout, placed in some historical perspective, and defined several different ways, including in the context of exterior algebras. The text details proof of the existence of a basis for an arbitrary vector space and addresses vector spaces over arbitrary fields. It develops LU-factorization, Jordan canonical form, and real and complex inner product spaces. It includes examples of inner product spaces of continuous complex functions on a real interval, as well as the background material that students may need in order to follow those discussions. Special classes of matrices make an entrance early in the text and subsequently appear throughout. The last chapter of the book introduces the classical groups.
Finite Fields, with Applications to Combinatorics by
This book uses finite field theory as a hook to introduce the reader to a range of ideas from algebra and number theory. It constructs all finite fields from scratch and shows that they are unique up to isomorphism. As a payoff, several combinatorial applications of finite fields are given: Sidon sets and perfect difference sets, de Bruijn sequences and a magic trick of Persi Diaconis, and the polynomial time algorithm for primality testing due to Agrawal, Kayal and Saxena. The book forms the basis for a one term intensive course with students meeting weekly for multiple lectures and a discussion session. Readers can expect to develop familiarity with ideas in algebra (groups, rings and fields), and elementary number theory, which would help with later classes where these are developed in greater detail. And they will enjoy seeing the AKS primality test application tying together the many disparate topics from the book. The pre-requisites for reading this book are minimal: familiarity with proof writing, some linear algebra, and one variable calculus is assumed. This book is aimed at incoming undergraduate students with a strong interest in mathematics or computer science.
Origins and Varieties of Logicism : On the Logico-philosophical Foundations of Mathematics by
This book offers a plurality of perspectives on the historical origins of logicism and on contemporary developments of logicist insights in philosophy of mathematics. It uniquely provides up-to-date research and novel interpretations on a variety of intertwined themes and historical figures related to different versions of logicism. The essays, written by prominent scholars, are divided into three thematic sections. Part I focuses on major authors like Frege, Dedekind, and Russell, providing a historical and theoretical exploration of such figures in the philosophical and mathematical milieu in which logicist views were first expounded. Part II sheds new light on the interconnections between these founding figures and a number of influential other traditions, represented by authors like Hilbert, Husserl, and Peano, as well as on the reconsideration of logicism by Carnap and the logical empiricists. Finally, Part III assesses the legacy of such authors and of logicist themes for contemporary philosophy of mathematics, offering new perspectives on highly debated topics-neo-logicism and its extension to accounts of ordinal numbers and set-theory, the comparison between neo-Fregean and neo-Dedekindian varieties of logicism, and the relation between logicist foundational issues and empirical research on numerical cognition-which define the prospects of logicism in the years to come. This book offers a comprehensive account of the development of logicism and its contemporary relevance for the logico-philosophical foundations of mathematics. It will be of interest to graduate students and researchers working in philosophy of mathematics, philosophy of logic, and the history of analytic philosophy.
Amplitudes, Hodge Theory and Ramification by
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 Matemáticas-ICMAT (Institute of Mathematical Sciences) in Madrid, Spain. It covers the presentations by S. Bloch, by M. Marcolli and by L. Kindler and K. Rülling. 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.
K3 Surfaces by
A K3 surface is a connected compact 2-dimensional complex manifold that is simply connected and whose canonical line bundle is trivial. Given this definition, it looks like a rather special class of objects. Why should one be interested in them and even read a whole book about them?
Since the 19th century, K3 surfaces showed up in many very different contexts in complex geometry, algebraic geometry, arithmetic geometry, and they continue to show up in sometimes quite surprising and unexpected places, such as in spacetime compactifications in mathematical physics. As such, K3 surfaces connect very different fields and provide stimulation for conjectures and further research. From a modern perspective, they form an important part of the so-called Enriques-Kodaira classification of compact 2-dimensional complex manifolds. They are non-trivial, yet still accessible, which is why they are also an important test class for conjectures.
Number Theory: Step by Step by
Number theory is one of the oldest branches of mathematics that is primarily concerned with positive integers. While it has long been studied for its beauty and elegance as a branch of pure mathematics, it has seen a resurgence in recent years with the advent of the digital world for itsmodern applications in both computer science and cryptography. Number Theory: Step by Step is an undergraduate-level introduction to number theory that assumes no prior knowledge, but works to gradually increase the reader's confidence and ability to tackle more difficult material. The strength of the text is in its large number of examples and the step-by-step explanation of each topic as it is introduced to help aid understanding the abstract mathematics of number theory. It is compiled in such a way that allows self-study, with explicit solutions to all the set of problems freely available online via the companion website. Punctuating the text are short and engaging historical profiles that add context for the topics covered and provide a dynamic background for the subject matter.
Hyperidentities : Boolean and De Morgan Structures by
Hyperidentities are important formulae of second-order logic, and research in hyperidentities paves way for the study of second-order logic and second-order model theory. This book illustrates many important current trends and perspectives for the field of hyperidentities and their applications, of interest to researchers in modern algebra and discrete mathematics. It covers a number of directions, including the characterizations of the Boolean algebra of n-ary Boolean functions and the distributive lattice of n-ary monotone Boolean functions; the classification of hyperidentities of the variety of lattices, the variety of distributive (modular) lattices, the variety of Boolean algebras, and the variety of De Morgan algebras; the characterization of algebras with aforementioned hyperidentities; the functional representations of finitely-generated free algebras of various varieties of lattices and bilattices via generalized Boolean functions (De Morgan functions, quasi-De Morgan functions, super-Boolean functions, super-De Morgan functions, etc); the structural results for De Morgan algebras, Boole-De Morgan algebras, super-Boolean algebras, bilattices, among others. While problems of Boolean functions theory are well known, the present book offers alternative, more general problems, involving the concepts of De Morgan functions, quasi-De Morgan functions, super-Boolean functions, and super-De Morgan functions, etc. In contrast to other generalized Boolean functions discovered and investigated so far, these functions have clearly normal forms. This quality is of crucial importance for their applications in pure and applied mathematics, especially in discrete mathematics, quantum computation, quantum information theory, quantum logic, and the theory of quantum computers.
The Biggest Number in the World : A Journey to the Edge of Mathematics by
From cells in our bodies to measuring the universe, big numbers are everywhere We all know that numbers go on forever, that you could spend your life counting and never reach the end of the line, so there can't be such a thing as a "biggest number." Or can there? To find out, David Darling and Agnijo Banerjee embark on an epic quest, revealing the answers to questions like: are there more grains of sand on Earth or stars in the universe? Is there enough paper on Earth to write out the digits of a googolplex? And what is a googolplex? Then things get serious. Enter the strange realm between the finite and the infinite, and float through a universe where the rules we cling to no longer apply. Encounter the highest number computable and infinite kinds of infinity. At every turn, a cast of wild and wonderful characters threatens the status quo with their ideas, and each time the numbers get larger.
Topology of Numbers by
This book serves as an introduction to number theory at the undergraduate level, emphasizing geometric aspects of the subject. The geometric approach is exploited to explore in some depth the classical topic of quadratic forms with integer coefficients, a central topic of the book. Quadratic forms of this type in two variables have a very rich theory, developed mostly by Euler, Lagrange, Legendre, and Gauss during the period 1750-1800. In this book their approach is modernized by using the splendid visualization tool introduced by John Conway in the 1990s called the topograph of a quadratic form. Besides the intrinsic interest of quadratic forms, this theory has also served as a stepping stone for many later developments in algebra and number theory. The book is accessible to students with a basic knowledge of linear algebra and arithmetic modulo n. Some exposure to mathematical proofs will also be helpful. The early chapters focus on examples rather than general theorems, but theorems and their proofs play a larger role as the book progresses.
Hyperbolicity Properties of Algebraic Varieties by
Since its introduction in the 1970s, the notion of hyperbolicity in Kobayashi's sense has attracted much attention in the mathematical community. Alongside its aspects of complex analysis in several variables, a very fascinating theme is that of its interactions with the algebraic, arithmetic, and geometro-differential properties of algebraic varieties. The study of these interactions is essentially the objective of this book.
Among the topics covered are: distribution of values of integer curves, algebraic analogues of the notion of hyperbolicity, properties of hyperbolicity of projective hypersurfaces and varieties of general type, hyperbolicity of moduli spaces, relations between hyperbolicity and negative curvature , distribution of rational points in hyperbolic (arithmetic) varieties, and connections between several types of natural fibrations in algebraic and hyperbolic varieties.
This volume aims to draw up the state of the art, each chapter dealing with a different aspect of the subject, while trying to keep a fairly simple language that can encourage in particular doctoral students as well as young researchers in complex geometry to enter in the most recent developments in the study of the hyperbolicity properties of algebraic varieties.
Curvature of Space and Time, with an Introduction to Geometric Analysis by
This book introduces advanced undergraduates to Riemannian geometry and mathematical general relativity. The overall strategy of the book is to explain the concept of curvature via the Jacobi equation which, through discussion of tidal forces, further helps motivate the Einstein field equations. After addressing concepts in geometry such as metrics, covariant differentiation, tensor calculus and curvature, the book explains the mathematical framework for both special and general relativity. Relativistic concepts discussed include (initial value formulation of) the Einstein equations, stress-energy tensor, Schwarzschild space-time, ADM mass and geodesic incompleteness. The concluding chapters of the book introduce the reader to geometric analysis: original results of the author and her undergraduate student collaborators illustrate how methods of analysis and differential equations are used in addressing questions from geometry and relativity. The book is mostly self-contained and the reader is only expected to have a solid foundation in multivariable and vector calculus and linear algebra. The material in this book was first developed for the 2013 summer program in geometric analysis at the Park City Math Institute, and was recently modified and expanded to reflect the author's experience of teaching mathematical general relativity to advanced undergraduates at Lewis & Clark College. This book is published in cooperation with IAS/Park City Mathematics Institute.
