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Topological Groups by Sidney A. Morris
English | PDF | 2019 | 162 Pages | ISBN : N/A | 1.20 MB
In 1900 David Hilbert presented an address at the International Congress of Mathematicians in Paris and formulated 23 problems that influenced much of the research in the 20th century. The fifth of these problems asked whether every locally euclidean group admits a Lie group structure.
This motivated a great amount of research on locally compact groups, culminating in the 1950s with the work of Gleason, Iwasawa, Montgomery, Yamabe, and Zippin, which gave a positive answer to Hilbert’s question and developed much structure theory of locally compact groups to boot. In the 1940s, the work on the free topological groups of Markov and Graev expanded the study of topological groups in a serious way to non-locally compact groups. In the early 21st century, pro-Lie groups were introduced as a natural and well-behaved extension of the notions of connected locally compact groups and compact groups focusing on their Lie theory. The Special Issue of Axioms called “Topological Groups: Yesterday, Today, Tomorrow” was published as book in 2016 and has had a tremendous reception. It addressed some of the significant research of this 115-year period. A wonderful feature of the book was the inclusion of surveys and a large number of open questions.
This second volume on topological groups, called “Topological Groups: Advances, Surveys, and Open Questions”, contains recent articles by some of the best scholars in the world on topological groups including Taras Banakh, Michael Megrelishvili, George A. Willis, Dmitri Shakhmatov, and O’lga V. Sipacheva, as well as a paper by the renowned scholar Saharon Shelah. A feature of the first volume was the surveys, and we continue that tradition in this second volume with three new surveys. These surveys are of interest, not only to the expert but also to those who are less experienced. Particularly exciting to active researchers, especially young researchers, is the inclusion of over three dozen open questions. This volume consists of 11 papers containing many new and interesting results and examples across the spectrum of topological group theory and related topics.
The first paper in this book is “Separability of Topological Groups: A Survey with Open Problems” by Arkady Leiderman and Sidney A. Morris. In recent years, Leiderman has been a leader in the study of the separability of topological groups. This paper alone states 20 open questions and puts them in context.
The second paper, “Categorically Closed Topological Groups”, is also a survey, this time by Taras Banakh. This paper surveys existing and new results on topological groups that are C-closed for various categories C of topologized semigroups. In particular, it analyzes solutions to a general problem consisting of 45 subproblems.
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