Basic Topology 2: Topological Groups, Topology of Manifolds and Lie Groups

這本三卷書的第二卷針對數學本科和研究生的學生，作為拓撲學的基礎課程。它聚焦於拓撲學的許多變體及其在現代分析、幾何、代數和數論中的應用。本卷提供了有關拓撲學、分析和代數的適當背景，討論了拓撲群和拓撲向量空間，這些提供了許多有趣的幾何對象，將代數與幾何和分析相關聯。本卷以系統和全面的初等方法探討了與流形相關的拓撲，強調微分拓撲。它進一步介紹了導致拓撲群、流形以及李群作為數學主題的概念的歷史背景和動機。這本書將以平衡統一的方式涵蓋廣泛的理論和應用，促進這一學科的範圍、能力和積極學習。

AVISHEK ADHIKARI, Ph.D., M.Sc. (Gold Medalist), is Professor at the Department of Mathematics, Presidency University, Kolkata, India. A recipient of the President of India Medal, ISCA Young Scientist Award, and the NANUM Fund by International Mathematical Union, Professor Adhikari did his Ph.D. from the Indian Statistical Institute, India, under the guidance of Professor Bimal Roy, the former Director of the Indian Statistical Institute. Earlier, Professor Adhikari was a faculty member at the Department of Pure Mathematics, University of Calcutta, Kolkata, from July 2006-January 2019. He is the founder Secretary of the Institute for Mathematics, Bioinformatics, Information Technology & Computer Science (IMBIC), India, having branches in Sweden and Japan, and the Treasurer of the Cryptology Research Society of India. He was the former Eastern Zonal coordinator of the M.Sc. and Ph.D. scholarship examinations by the National Board of Higher Mathematics (NBHM). He was a Post-Doctoral fellow at INRIA-Rocquencourt, France, and Visiting Scientist at Linkoping University, Sweden; Indian Statistical Institute, Kolkata. Professor Adhikari delivered invited talks at various universities and institutions abroad.

He has authored four textbooks on mathematics including Basic Modern Algebra with Applications (Springer, 2014) and edited two volumes including Mathematical and Statistical Applications in Life Sciences and Engineering (Springer, 2017). His research papers have been published in reputed international journals, conference proceedings and contributed volumes. Professor Adhikari is one of the investigators of the twelve sponsored research projects funded by agencies like DRDO, WESEE (the Ministry of Defense), DST, DIT, NBHM of the Government of India, including two international collaborative projects supported by DST-JSPS and DST-JST (both Indo-Japan projects). Five of his Ph.D. students have already been awarded degrees, and two of the students have submitted their theses and, currently, three Ph.D. scholars are conducting research under his supervision.

MAHIMA RANJAN ADHIKARI, Ph.D., M.Sc. (Gold Medalist), is the founder president of the Institute for Mathematics, Bioinformatics and Computer Science (IMBIC), Kolkata, India. He is a former professor at the Department of Pure Mathematics, University of Calcutta, India. His research papers are published in national and international journals of repute, including the Proceedings of American Mathematical Society. He has authored nine textbooks and is the editor of two, including: Basic Modern Algebra with Applications (Springer, 2014), Basic Algebraic Topology and Applications (Springer, 2016), and Mathematical and Statistical Applications in Life Sciences and Engineering (Springer, 2017).

Twelve students have been awarded Ph.D. degree under his guidance on various topics such as algebra, algebraic topology, category theory, geometry, analysis, graph theory, knot theory and history of mathematics. He has visited several universities and research institutions in India, USA, UK, Japan, China, Greece, Sweden, Switzerland, Italy, and many other counties on invitation. A member of the American Mathematical Society, Prof. Adhikari is on the editorial board of several journals of repute. He was elected as the president of the Mathematical Sciences Section (including Statistics) of the 95th Indian Science Congress, 2008. He has successfully completed research projects funded by the Government of India.

AVISHEK ADHIKARI博士，M.Sc.（金牌得主），是印度加爾各答總統大學數學系的教授。Adhikari教授獲得了印度總統獎章、ISCA青年科學家獎和國際數學聯盟的NANUM基金。Adhikari教授在印度統計研究所師從印度統計研究所前所長Bimal Roy教授的指導下獲得了博士學位。之前，Adhikari教授是加爾各答加爾各答大學純數學系的教職成員，任職時間為2006年7月至2019年1月。他是印度數學、生物信息學、信息技術和計算機科學研究所（IMBIC）的創始秘書，該研究所在瑞典和日本設有分支機構，並且是印度密碼學研究協會的財務主管。他曾擔任國家高等數學委員會（NBHM）碩士和博士獎學金考試的東部區域協調員。他曾在法國INRIA-Rocquencourt擔任博士後研究員，並在瑞典林雪平大學和印度統計研究所擔任訪問科學家。Adhikari教授在國外的各個大學和機構發表了邀請演講。

他撰寫了四本數學教科書，包括《基礎現代代數及其應用》（Springer，2014），並編輯了兩卷，包括《生命科學和工程中的數學和統計應用》（Springer，2017）。他的研究論文發表在知名的國際期刊、會議論文集和貢獻專輯中。Adhikari教授是由印度國防研究和發展組織（DRDO）、國防部（WESEE）、科技部（DST）、信息技術部（DIT）、印度政府國家高等數學委員會（NBHM）等機構資助的十二個研究項目的調查人之一，其中包括由DST-JSPS和DST-JST（兩個印日合作項目）支持的兩個國際合作項目。他的五名博士生已經獲得學位，其中兩名學生已經提交了論文，目前有三名博士研究生在他的指導下進行研究。

MAHIMA RANJAN ADHIKARI博士，M.Sc.（金牌得主），是印度加爾各答IMBIC數學、生物信息學和計算機科學研究所的創始人兼主席。他曾任職於印度加爾各答大學純數學系的教授。他的研究論文發表在國內外知名期刊，包括《美國數學學會會議論文集》。他撰寫了九本教科書，並編輯了兩本，包括《基礎現代代數及其應用》（Springer，2014），《基礎代數拓撲及其應用》（Springer，2016）和《生命科學和工程中的數學和統計應用》（Springer，2017）。

在他的指導下，已有十二名學生在各種主題上獲得了博士學位，包括代數、代數拓撲、范畴論、幾何、分析、圖論、結繩理論和數學歷史等。他曾應邀訪問印度、美國、英國、日本、中國、希臘、瑞典、瑞士、意大利和其他許多國家的多所大學和研究機構。Adhikari教授是美國數學學會的成員，並擔任多個知名期刊的編輯委員會成員。他曾當選為第95屆印度科學大會（包括統計學）數學科學部門的主席。他成功完成了由印度政府資助的研究項目。