99 variations on a proof

An exploration of mathematical style through 99 different proofs of the same theoremThis book offers a multifaceted perspective on mathematics by demonstrating 99 different proofs of the same theorem. Each chapter solves an otherwise unremarkable equation in distinct historical, formal, and imagi...

A course on surgery theory

An advanced treatment of surgery theory for graduate students and researchersSurgery theory, a subfield of geometric topology, is the study of the classifications of manifolds. A Course on Surgery Theory offers a modern look at this important mathematical discipline and some of its applications. ...

A primer on mapping class groups

The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time g...

Characteristic classes

The theory of characteristic classes provides a meeting ground for the various disciplines of differential topology, differential and algebraic geometry, cohomology, and fiber bundle theory. As such, it is a fundamental and an essential tool in the study of differentiable manifolds.In this volume...

Elliptic curves

An elliptic curve is a particular kind of cubic equation in two variables whose projective solutions form a group. Modular forms are analytic functions in the upper half plane with certain transformation laws and growth properties. The two subjects--elliptic curves and modular forms--come togethe...

Introduction to toric varieties

Toric varieties are algebraic varieties arising from elementary geometric and combinatorial objects such as convex polytopes in Euclidean space with vertices on lattice points. Since many algebraic geometry notions such as singularities, birational maps, cycles, homology, intersection theory, and...

Lectures on kähler groups

An introduction to the state of the art in the study of Kähler groupsThis book gives an authoritative and up-to-date introduction to the study of fundamental groups of compact Kähler manifolds, known as Kähler groups. Approaching the subject from the perspective of a geometric group theorist, Pie...

Moduli stacks of étale (ϕ, γ)-modules and the existence of crystalline lifts

A foundational account of a new construction in the p-adic Langlands correspondenceMotivated by the p-adic Langlands program, this book constructs stacks that algebraize Mazur’s formal deformation rings of local Galois representations. More precisely, it constructs Noetherian formal algebraic sta...

Multiplicative hitchin fibrations and the fundamental lemma

A proof of the endoscopic fundamental lemma for spherical Hecke algebras using a new comprehensive framework of multiplicative Hitchin fibrationsThis book develops a comprehensive framework of multiplicative Hitchin fibrations, which is a novel analogue of the classical Hitchin fibrations, and th...

Quasi-projective and formal-analytic arithmetic surfaces

A milestone in the geometric understanding of algebraization theorems that also provides an introduction to Arakelov geometryMotivated by questions of transcendental number theory, arithmetic, and Diophantine geometry, this book provides a thorough study of a new kind of mathematical object—forma...

Spin geometry

The arithmetic of polynomial dynamical pairs

New mathematical research in arithmetic dynamicsIn The Arithmetic of Polynomial Dynamical Pairs, Charles Favre and Thomas Gauthier present new mathematical research in the field of arithmetic dynamics. Specifically, the authors study one-dimensional algebraic families of pairs given by a polynomi...

The rising sea

An accessible, motivated introduction to one of the most dynamic areas of mathematicsDecades ago, Mumford wrote that algebraic geometry “seems to have acquired the reputation of being esoteric, exclusive, and very abstract, with adherents who are secretly plotting to take over all the rest of mat...

Unipotent ideals and harish-chandra bimodules

A groundbreaking book that applies new geometric tools to one of the oldest problems in representation theory, expanding the field and paving the way for further progressIn the 1920s, Hermann Weyl gave a complete classification of the irreducible unitary representations of a compact Lie group G. ...

What determines an algebraic variety?

A pioneering new nonlinear approach to a fundamental question in algebraic geometryOne of the crowning achievements of nineteenth-century mathematics was the proof that the geometry of lines in space uniquely determines the Cartesian coordinates, up to a linear ambiguity. What Determines an Algeb...