商品描述
This book offers a clear and accessible pathway into information geometry for advanced undergraduate and graduate students. Readers will be guided from the fundamentals of topology and differentiable manifolds to the more advanced concepts of probability geometry and Frobenius manifolds in an intuitive manner, allowing them to build their knowledge gradually. Divided into three main parts, the first provides a concise introduction to differential topology and geometry, emphasizing the role of smooth manifolds, connections, and curvature in the formulation of geometric structures. Part II is then devoted to probability, measures, and statistics, where the notion of a probability space is refined into a geometric object, thus paving the way for a deeper mathematical understanding of statistical models. Finally, the third part introduces Frobenius manifolds, revealing their surprising connection to exponential families of probability distributions and, more broadly, their role in the geometry of information. Throughout all the chapters, there are exercises to test understanding, as well as solutions to some of the more challenging problems to aid learning.
商品描述(中文翻譯)
本書為高年級本科生及研究生提供了一條清晰且易於理解的信息幾何學入門之路。讀者將從拓撲學和可微流形的基本原理開始，逐步引導至更高級的概率幾何和Frobenius流形的概念，以直觀的方式讓他們逐漸建立知識。全書分為三個主要部分，第一部分簡要介紹了微分拓撲學和幾何學，強調了光滑流形、聯結和曲率在幾何結構形成中的作用。第二部分專注於概率、測度和統計，將概率空間的概念精煉為幾何對象，從而為統計模型的更深數學理解鋪平道路。最後，第三部分介紹了Frobenius流形，揭示了它們與指數族概率分佈之間的驚人聯繫，以及它們在信息幾何中的更廣泛角色。在所有章節中，均設有練習題以測試理解，並提供一些較具挑戰性的問題的解答以協助學習。
作者簡介
Noémie Combe is a mathematician and research professor at the Léonard de Vinci University Pole. She has secured several prestigious research grants, serving as principal investigator and group leader of the renowned Polonez Bis 3 programme--funded by Horizon 2020--at the University of Warsaw. Prior to this, she led a Minerva Group at the Max Planck Society for five years, where her research received significant recognition and sustained support. She earned her degrees from the University of Geneva, and completed her PhD at the University of Aix--Marseille and Sorbonne University (Paris 6). Philippe Combe is an emeritus professor of Mathematical Physics at the University of Aix-Marseille, as well as a long‐standing member of the Centre for Theoretical Physics (CPT). Over a distinguished career spanning more than four decades, he has made foundational contributions to functional analysis, group theory, operator theory, geometry, statistics, and stochastic process. His work is characterized by high‐impact collaborations across Europe and beyond. In addition to his research, Philippe Combe has provided key leadership in the scientific community. Hanna Nencka is a distinguished invited researcher in the Polonez Bis 3-sponsored Research Group at the University of Warsaw. She has held tenured professorships at Columbia University (USA), the Polish Academy of Science, and the University of Madeira (Portugal). Her prolific scholarship includes over one hundred peer-reviewed publications in leading journals. Her pioneering contributions to mathematical physics have earned her international recognition and multiple research awards, namely from the Polish Academy of Sciences for outstanding work.
作者簡介(中文翻譯)
Noémie Combe 是一位數學家及萊昂納多·達文西大學的研究教授。她獲得了多項著名的研究補助金，擔任華沙大學著名的 Polonez Bis 3 計畫的首席研究員及團隊領導，該計畫由 Horizon 2020 資助。在此之前，她在馬克斯·普朗克學會領導了一個 Minerva 團隊五年，其研究獲得了顯著的認可和持續的支持。她在日內瓦大學獲得學位，並在艾克斯-馬賽大學及索邦大學（巴黎 6）完成博士學位。
Philippe Combe 是艾克斯-馬賽大學的數學物理名譽教授，也是理論物理中心（CPT）的長期成員。在超過四十年的卓越職業生涯中，他對函數分析、群論、算子理論、幾何學、統計學和隨機過程做出了基礎性貢獻。他的工作以在歐洲及其他地區的高影響力合作為特徵。除了研究外，Philippe Combe 在科學社群中也提供了關鍵的領導。
Hanna Nencka 是華沙大學 Polonez Bis 3 贊助研究小組的傑出受邀研究員。她曾在哥倫比亞大學（美國）、波蘭科學院及馬德拉大學（葡萄牙）擔任終身教授。她的學術成果豐碩，包括在領先期刊上發表的超過一百篇同行評審的論文。她對數學物理的開創性貢獻使她獲得國際認可及多項研究獎項，特別是來自波蘭科學院的卓越工作獎。
