11 research outputs found
On the essential logical structure of inter-universal TeichmĂźller theory in terms of logical AND ââ§â/logical OR ââ¨â relations: Report on the occasion of the publication of the four main papers on inter-universal TeichmĂźller theory
The main goal of the present paper is to give a detailed exposition of the essential logical structure of inter-universal TeichmĂźller theory from the point of view of the Boolean operators --such as the logical AND ââ§âlogical OR ââ¨â operators-- of propositional calculus. This essential logical structure of inter-universal TeichmĂźller theory may be summarized symbolically as follows: A ⧠B = A ⧠(Bâ â¨Ë Bâ â¨Ë...) â A ⧠(Bââ¨ËBââ¨Ë...â¨Ë BĚâ â¨Ë BĚâ â¨Ë...) -- where ¡ the ââ¨Ëâ denotes the Boolean operator exclusive-OR, i.e., âXORâ; ¡ A, B, Bâ, Bâ, BĚâ, BĚâ, denote various propositions; ¡ the logical AND ââ§'sâ correspond to the Î-link of inter-universal TeichmĂźller theory and are closely related to the multiplicative structures of the rings that appear in the domain and codomain of the Î-link; ¡ the logical XOR ââ¨Ë'sâ correspond to various indeterminacies that arise mainly from the log-Kummer-correspondence, i.e., from sequences of iterates of the log-link of inter-universal TeichmĂźller theory, which may be thought of as a device for constructing additive log-shells. This sort of concatenation of logical AND ââ§'sâ and logical XOR ââ¨Ë 'sâ is reminiscent of the well-known description of the âcarry-additionâ operation on TeichmĂźller representatives of the truncated Witt ring â¤/4⤠in terms of Boolean addition ââ¨Ëâ and Boolean multiplication ââ§â in the field Fâ and may be regarded as a sort of âBoolean intertwiningâ that mirrors, in a remarkable fashion, the âarithmetic intertwiningâ between addition and multiplication in number fields and local fields, which is, in some sense, the main object of study in inter-universal TeichmĂźller theory. One important topic in this exposition is the issue of âredundant copiesâ, i.e., the issue of how the arbitrary identification of copies of isomorphic mathematical objects that appear in the various constructions of inter-universal TeichmĂźller theory impacts-- and indeed invalidates-- the essential logical structure of inter-universal TeichmĂźller theory. This issue has been a focal point of fundamental misunderstandings and entirely unnecessary confusion concerning inter-universal TeichmĂźller theory in certain sectors of the mathematical community. The exposition of the topic of âredundant copiesâ makes use of many interesting elementary examples from the history of mathematics
Groups, operator algebras and approximation
Two main objects of the research in this thesis are countable discrete groups and their operator algebras (C*-algebras and von Neumann algebras). Discrete groups are often succesfully studied using geometric and ergodic-theoretic methods, the corresponding areas of mathematics being called geometric resp. measured group theory.
This thesis has a cumulative form: each chapter is a research article, and therefore has its own abstract and bibliography. The majority of these publications have been peer-reviewed and published in various journals
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic
Dagstuhl News January - December 2011
"Dagstuhl News" is a publication edited especially for the members of the Foundation "Informatikzentrum Schloss Dagstuhl" to thank them for their support. The News give a summary of the scientific work being done in Dagstuhl. Each Dagstuhl Seminar is presented by a small abstract describing the contents and scientific highlights of the seminar as well as the perspectives or challenges of the research topic
Proceedings of the 1st International Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020)
International audienceOriginating in arithmetics and logic, the theory of ordered sets is now a field of combinatorics that is intimately linked to graph theory, universal algebra and multiple-valued logic, and that has a wide range of classical applications such as formal calculus, classification, decision aid and social choice.This international conference âAlgebras, graphs and ordered setâ (ALGOS) brings together specialists in the theory of graphs, relational structures and ordered sets, topics that are omnipresent in artificial intelligence and in knowledge discovery, and with concrete applications in biomedical sciences, security, social networks and e-learning systems. One of the goals of this event is to provide a common ground for mathematicians and computer scientists to meet, to present their latest results, and to discuss original applications in related scientific fields. On this basis, we hope for fruitful exchanges that can motivate multidisciplinary projects.The first edition of ALgebras, Graphs and Ordered Sets (ALGOS 2020) has a particular motivation, namely, an opportunity to honour Maurice Pouzet on his 75th birthday! For this reason, we have particularly welcomed submissions in areas related to Mauriceâs many scientific interests:⢠Lattices and ordered sets⢠Combinatorics and graph theory⢠Set theory and theory of relations⢠Universal algebra and multiple valued logic⢠Applications: formal calculus, knowledge discovery, biomedical sciences, decision aid and social choice, security, social networks, web semantics..
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum