Connes (College de France and Vanderbilt) and Marcolli (Max Planck Institute and Florida State) explain the relevance of noncommutative geometry in dealing with the construction of a theory of quantum gravity and the Riemann hypothesis. Also appropriate for physicists, the advanced mathematics textbook describes the Standard Model of elementary particles, a spectral realization of the zeros of the Riemann zeta function, quantum statistical mechanical systems, and a cohomological Lefschetz trace formula. Annotation ©2008 Book News, Inc., Portland, OR (booknews.com)