Special Issue "Topological Groups: Yesterday, Today, Tomorrow"
Deadline for manuscript submissions: closed (30 September 2015) | Viewed by 26067
A printed edition of this Special Issue is available here.
2. Department of Mathematical and Physical Sciences, La Trobe University, Melbourne, VIC 3086, Australia
Interests: topological groups, especially locally compact groups; pro-Lie groups; topological algebra; topological vector spaces; Banach spaces; topology; group theory; functional analysis; universal algebra; transcendental number theory; numerical geometry; history of mathematics; information technology security; health informatics; international education; university education; online education; social media in the teaching of mathematics; stock market prediction; managing scholarly journals
Special Issues, Collections and Topics in MDPI journals
Special Issue in Axioms: Topological Algebra
Special Issue in Axioms: 10th Anniversary of Axioms: Geometry and Topology
In 1900, David Hilbert asked whether a locally euclidean topological group is in fact a Lie group. This was the fifth of his famous 23 questions which foreshadowed much of the mathematical creativity of the twentieth century. It required half a century of effort by several generations of eminent mathematicians until it was settled in the affirmative. These efforts resulted over time in the Peter-Weyl Theorem, the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups, and finally of the solution of Hilbert 5 and the structure theory of locally compact groups, through the combined work of Andrew Gleason, Kenkichi Iwasawa, Deane Montgomery, and Leon Zippin. For a presentation of Hilbert 5 see the 2014 book “Hilbert’s Fifth Problem and Related Topics” by the winner of a 2006 Fields Medal and 2014 Breakthrough Prize in Mathematics, Terence Tao.
It is not possible to describe briefly the richness of topological group theory and the many directions taken since Hilbert 5. The 900 page reference book in 2013 “The Structure of Compact Groups” by Karl Hofmann and I, deals with one aspect of compact group theory. There are several books on profinite groups including those written by John S. Wilson (1998) and by Luis Ribes and Pavel Zalesskii (2012). The 2007 book “The Lie Theory of connected pro-Lie groups” by Karl Hofmann and me, demonstrates how powerful Lie Theory is in exposing the structure of infinite-dimensional Lie groups.
The study of free topological groups initiated by S. Kakutani, A.A. Markov and M.I. Graev has resulted in a wealth of interesting results, in particular those of A.V. Arkhangelʹskiĭ and many of his former students who developed this topic and its relations with topology.
Compactness conditions in topological groups has been another direction which has proved very fruitful to the present day.
In this Special Issue, we particularly seek contributions of the following two kinds:
- survey articles which present significant (new or not so new) open questions;
- new results on topological groups presented in a historical context and with open questions.
Prof. Dr. Sidney A. Morris
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- topological groups
- compact groups
- profinite groups
- locally compact groups
- Lie groups
- pro-Lie groups
- almost periodic
- semitopological groups
- structure theory
- transformation groups
- free topological groups and free products
- variety of topological groups
- Hilbert’s 5th problem
- (locally) minimal topological groups
- compactness conditions in topological groups
- duality and reflexivity
- covering theory for topological groups
- suitable sets for topological groups
- algebraic topology and topological groups