This book discusses major topics in measure theory, Fourier transforms, complex analysis and algebraic topology. It presents material from a mature mathematical perspective. The text is suitable for a two-semester graduate course in analysis and will help students prepare for a research career in mathematics. After a short survey of undergraduate analysis and measure theory, the book highlights the essential theorems that have now become ubiquitous in mathematics. It studies Fourier transforms, derives the inversion theorem and gives diverse applications ranging from probability theory to mathematical physics. It reviews topics in complex analysis and gives a synthetic, rigorous development of the calculus of residues as well as applications to a wide array of problems. It also introduces algebraic topology and shows the symbiosis between algebra and analysis. Indeed, algebraic archetypes were providing foundational support from the start. Multivariable calculus is comprehended in asingle glance through the algebra of differential forms. Advanced complex analysis inevitably leads one to the study of Riemann surfaces, and so the final chapter gives the student a hint of these motifs and underlying algebraic patterns.

Autorentext

M. RAM MURTY is Queen's Research Chair and A.V. Douglas Distinguished University Professor at Queen's University, in Kingston, Ontario, Canada. He obtained his Ph.D. from Massachusetts Institute of Technology, USA, in 1980, and subsequently held positions at the Institute for Advanced Study in Princeton, Tata Institute for Fundamental Research in Mumbai, and McGill University in Montreal. He is Fellow of the Royal Society of Canada, the American Mathematical Society and the Indian National Science Academy, India. He has authored more than 250 research papers and written more than a dozen mathematical textbooks. His monograph, Non-vanishing of L-functions and Applications, written jointly with Professor V. Kumar Murty, won the 1996 Balaguer Prize. He also teaches the Indian philosophy at Queen's University and has authored a book, Indian Philosophy: An Introduction.

Inhalt

1. Background.- 2. Measure Theory.- 3. Fourier Transforms.- 4. Complex Analysis.- 5. Introduction to Algebraic Topology.- Index.