The Best Mathematical Research Books of All Time

“This man could plainly do for mathematics what Brian Cox has done for physics” — Sunday Times. What goes on inside the mind of a rock-star mathematician? Where does inspiration come from? With a storyteller’s gift, Cédric Villani takes us on a mesmerising journey as he wrestles with a new theorem that will win him the most coveted prize in mathematics. Along the way he encounters obstacles and setbacks, losses of faith and even brushes with madness. His story is one of courage and partnership, doubt and anxiety, elation and despair.

We discover how it feels to be obsessed by a theorem during your child’s cello practise and throughout your dreams, why appreciating maths is a bit like watching an episode of Columbo, and how sometimes inspiration only comes from locking yourself away in a dark room to think. Blending science with history, biography with myth, Villani conjures up an inimitable cast of characters including the omnipresent Einstein, mad genius Kurt Godel, and Villani’s personal hero, John Nash. Birth of a Theorem combines passion and imagination to take us on a fantastical adventure through the beautiful, mysterious world of mathematics. Maths has never seemed so magical or so exciting.

From the reviews of the first edition: "This book exposes the beautiful confluence of deep techniques and ideas from mathematical physics and the topological study of the differentiable structure of compact four-dimensional manifolds, compact spaces locally modeled on the world in which we live and operate... The book is filled with insightful remarks, proofs, and contributions that have never before appeared in print. For anyone attempting to understand the work of Donaldson and the applications of gauge theories to four-dimensional topology, the book is a must.

" #"Science"#1 "I would strongly advise the graduate student or working mathematician who wishes to learn the analytic aspects of this subject to begin with Freed and Uhlenbeck's book." #"Bulletin of the American Mathematical " "Society"#2.

This book gives a treatment of exterior differential systems including both the general theory and various applications. Topics include: a review of exterior algebra, simple exterior differential systems, the generation of integral manifolds through the solution of a succession of initial- value problems, involution, linear differential systems, tableau and torsion, the characteristic variety of a differential system, prolongation, the Algebra of a linear Pfaffian system, and an introduction to Spencer Theory. Much emphasis is placed on the general theory while many examples are given.