by Kâzım İlhan İkeda (Yeditepe University, Istanbul, Turkey)

In the late 1970s, using higher algebraic K-theory, class field theory was generalized to higher-dimensional local and global fields. Thus, it is a very natural question to ask and seek, although conjectural in nature, for the higher-dimensional version of the reciprocity principle of Langlands and, more generally, the higher-dimensional analog of the functoriality principle of Langlands. However, there are very few works addressing this very important, open and exciting problem of current mathematics.

This book aims at providing an introductory, detailed and up-to-date study of Kapranov's seminal work, “Analogies between topological quantum field theory and Langlands correspondence”, published in 1995, which is the first paper in literature discussing the formal framework and the formulation of higher-dimensional Langlands correspondence, together with closely related recent works of Kazhdan, Parshin and others.

This book will be of interest to researchers and graduate students in number theory and related areas.

• Introduction — Review of Langlands Functoriality Principle
• Lang Density Theorem
• Formulation of Non-Abelian Multi-dimensional Reciprocity Laws
• Review of K-Theoretic Class Field Theory
• Kapranov Reciprocity Principle
• Kapranov Functoriality Principle