An Introduction to Automorphic Representations: With a View Toward Trace Formulae

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【簡介】 The goal of this textbook is to introduce and study automorphic representations, objects at the very core of the Langlands Program. It is designed for use as a primary text for either a semester or a year-long course, for the independent study of advanced topics, or as a reference for researchers. The reader is taken from the beginnings of the subject to the forefront of contemporary research. The journey provides an accessible gateway to one of the most fundamental areas of modern mathematics, with deep connections to arithmetic geometry, representation theory, harmonic analysis, and mathematical physics. The first part of the text is dedicated to developing the notion of automorphic representations. Next, it states a rough version of the Langlands functoriality conjecture, motivated by the description of unramified admissible representations of reductive groups over nonarchimedean local fields. The next chapters develop the theory necessary to make the Langlands functoriality conjecture precise. Thus supercuspidal representations are defined locally, cuspidal representations and Eisenstein series are defined globally, and Rankin-Selberg L-functions are defined to give a link between the global and local settings. This preparation complete, the global Langlands functoriality conjectures are stated and known cases are discussed. This is followed by a treatment of distinguished representations in global and local settings. The link between distinguished representations and geometry is explained in a chapter on the cohomology of locally symmetric spaces (in particular, Shimura varieties). The trace formula, an immensely powerful tool in the Langlands Program, is discussed in the final chapters of the book. Simple versions of the general relative trace formulae are treated for the first time in a textbook, and a wealth of related material on algebraic group actions is included. Outlines for several possible courses are provided in the Preface.

TOWARD 6G - A NEW ERA OF CONVERGENCE

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【簡介】 The latest developments and recent progress on the key technologies enabling next-generation 6G mobile networks Toward 6G: A New Era of Convergence offers an up-to-date guide to the emerging 6G vision by describing new human-centric services made possible by combinations of mobile robots, avatars, and smartphones, which will be increasingly replaced with wearable displays and haptic interfaces that provide immersive extended reality (XR) experiences. The authors―noted experts on the topic―include a review of their work and information on the recent progress on the Tactile Internet and multi-sensory haptic communications. The book highlights decentralized edge computing in particular via Ethereum blockchain technologies, most notably the so-called decentralized autonomous organization (DAO) for crowdsourcing of human skills to solve problems that machines (such as autonomous artificial intelligence agents and robots) alone cannot solve well. The book also contains a review of the most recent and ongoing work on XR (including virtual/augmented/mixed reality). Specifically, the book describes the implications of the transition from the current gadgets-based Internet to a future Internet that is evolving from bearables (such as smartphones), moves towards wearables (for example Amazon's recently launched voice-controlled Echo Loop ring, glasses, and earbuds), and then finally progresses to nearables with embedded computing technologies and intelligent provisioning mechanisms for the delivery of human-intended services, including sixth-sense perceptions, in a 6G post-smartphone era. This important text: Offers a review of the 6G network architectures and key enabling technologies Explains why 6G should not be a mere exploration of more spectrum at high-frequency bands, but rather a convergence of upcoming technological trends Describes the Tactile Internet's human-in-the-loop centric design principles and haptic communications models Includes analytical frameworks to estimate the fluid orchestration of human + machine co-activities across unified communication network infrastructures Explores the performance gains of cooperative computation offloading with communications and computation limitations in both fronthaul and backhaul Written for students, network researchers, professionals, engineers, and practitioners, Toward 6G: A New Era of Convergence explores the most recent advances on the key technologies enabling next-generation 6G mobile networks, with an emphasis on their seamless convergence. 【目錄】