Unitary Representations and Unitary Duals
暫譯: 單位表示與單位對偶

Einsiedler, Manfred, Ward, Thomas

Unitary Representations and Unitary Duals
  • 出版商: Springer
  • 出版日期: 2025-10-31
  • 售價: $3,560
  • 貴賓價: 9.5 折 $3,382
  • 語言: 英文
  • 頁數: 564
  • 裝訂: Hardcover - also called cloth, retail trade, or trade
  • ISBN: 3032038987
  • ISBN-13: 9783032038982
  • 相關分類: 離散數學 Discrete-mathematics

    • 海外代購書籍(需單獨結帳)

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商品描述

This graduate textbook introduces the unitary representation theory of groups, emphasizing applications in fields like dynamical systems.

It begins with the general theory and motivation, then explores key classes of groups. Abelian and compact groups are treated through Pontryagin duality and the Peter-Weyl theorem. Metabelian groups illustrate links to ergodic theory and lead to the Mackey machine. Weak containment and the Fell topology are introduced through examples. The final chapters apply the theory to special linear groups in dimensions two and three, covering smooth vectors, spectral gaps, and decay of matrix coefficients. The two-dimensional case is examined in depth, including the Kunze-Stein phenomenon, spectral decomposition on the hyperbolic plane, and the Weil representation. The book concludes with a full description of the unitary dual of SL(2, R) and its Fell topology, applying the theory to prove effective equidistribution of horocycle orbits.

With its focus on key examples and concrete explanations, this textbook is aimed at graduate students taking first steps in unitary representation theory. It builds the theory from the ground up, requiring only some familiarity with functional analysis beyond standard undergraduate mathematics.

商品描述(中文翻譯)

這本研究生教科書介紹了群的單位表示理論，強調其在動態系統等領域的應用。

本書首先介紹一般理論和動機，然後探討關鍵的群類別。阿貝爾群和緊湊群通過龐特里亞金對偶性和彼得-維爾定理進行處理。超阿貝爾群展示了與遍歷理論的聯繫，並引入了麥基機器。通過例子介紹了弱包含和費爾拓撲。最後幾章將理論應用於二維和三維的特殊線性群，涵蓋平滑向量、譜隙和矩陣係數的衰減。二維情況深入探討，包括昆茲-斯坦現象、在雙曲平面上的譜分解以及維爾表示。書末對SL(2, R)的單位對偶及其費爾拓撲進行了全面描述，並應用該理論證明了horocycle軌道的有效等分佈。

本教科書專注於關鍵例子和具體解釋，旨在幫助研究生在單位表示理論上邁出第一步。它從基礎開始構建理論，只需對功能分析有一定的了解，超出標準的本科數學範疇。

作者簡介

Manfred Einsiedler studied at the University of Vienna, with a Ph.D. in 1999 under Klaus Schmidt. He held research positions at the University of East Anglia, Penn State University, the University of Washington, and Princeton University as a Clay Research Scholar. After becoming a Professor at Ohio State University he joined ETH Zürich. In 2004 he won the Research Prize of the Austrian Mathematical Society, in 2008 he was an invited speaker at the European Mathematical Congress in Amsterdam, and in 2010 he was an invited speaker at the International Congress of Mathematicians in Hyderabad. He works on ergodic theory and its applications to number theory (especially dynamical and equidistribution problems on homogeneous spaces). He has collaborated with Grigory Margulis and Akshay Venkatesh. With Elon Lindenstrauss and Anatole Katok, Einsiedler proved that a conjecture of Littlewood on Diophantine approximation is "almost always" true.

Thomas Ward studied at the University of Warwick, with a Ph.D. in 1989 under Klaus Schmidt. He held research positions at the University of Maryland, College Park and at Ohio State University before joining the University of East Anglia in 1992. Between 2008 and retirement in 2023 he served on university executives, as Pro-Vice-Chancellor for Education at the Universities of East Anglia, Durham, and Newcastle and as Deputy Vice-Chancellor (Student Education) at the University of Leeds. He worked on the ergodic theory of algebraic dynamical systems, compact group automorphisms, and number theory. A long collaboration with Graham Everest on links between number theory and dynamical systems included the book Heights of Polynomials and Entropy in Algebraic Dynamics and a paper on Diophantine equations that won the 2012 Lester Ford Prize for mathematical exposition. With Einsiedler he has written the books Ergodic Theory with a view towards number theory in 2011 and Functional Analysis, Spectral Theory, and Applications in 2017.

作者簡介(中文翻譯)

曼弗雷德·艾因西德勒於維也納大學學習，1999年在克勞斯·施密特（Klaus Schmidt）指導下獲得博士學位。他曾在東安格利亞大學、賓州州立大學、華盛頓大學和普林斯頓大學擔任研究職位，並作為克萊研究學者（Clay Research Scholar）工作。成為俄亥俄州立大學教授後，他加入了蘇黎世聯邦理工學院（ETH Zürich）。2004年，他獲得奧地利數學學會的研究獎，2008年在阿姆斯特丹的歐洲數學大會上擔任邀請演講者，2010年在海得拉巴的國際數學家大會上擔任邀請演講者。他的研究領域包括遍歷理論及其在數論中的應用（特別是同質空間上的動態和等分佈問題）。他曾與格里戈里·馬爾古利斯（Grigory Margulis）和阿克謝·文卡特什（Akshay Venkatesh）合作。與伊隆·林登斯特勞斯（Elon Lindenstrauss）和阿納托爾·卡托克（Anatole Katok）一起，艾因西德勒證明了利特伍德（Littlewood）關於丟番圖逼近的猜想「幾乎總是」成立。

托馬斯·沃德於華威大學學習，1989年在克勞斯·施密特（Klaus Schmidt）指導下獲得博士學位。他曾在馬里蘭大學（University of Maryland, College Park）和俄亥俄州立大學擔任研究職位，並於1992年加入東安格利亞大學。2008年至2023年退休期間，他在大學行政部門任職，擔任東安格利亞大學、達勒姆大學和紐卡斯爾大學的教育副校長，以及利茲大學的副校長（學生教育）。他的研究領域包括代數動態系統的遍歷理論、緊群自同構和數論。他與格雷厄姆·埃弗雷斯特（Graham Everest）長期合作，研究數論與動態系統之間的聯繫，並共同撰寫了書籍《多項式的高度與代數動態中的熵》（Heights of Polynomials and Entropy in Algebraic Dynamics）以及一篇關於丟番圖方程的論文，該論文獲得2012年萊斯特·福特獎（Lester Ford Prize）以表彰其數學表述。與艾因西德勒合作，他於2011年撰寫了《面向數論的遍歷理論》（Ergodic Theory with a view towards number theory）和於2017年撰寫的《泛函分析、譜理論及其應用》（Functional Analysis, Spectral Theory, and Applications）。

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