Fame and honor for Austria

and for the capital Vienna

Our Mission: 1. To advance Austrian science to the forefront of international research and to stabilize this position. 2. To strengthen cooperation with international research centers: Université Mohammed Premier, Oujda, Morocco Australian National University, Canberra, Capital Territory Washington and Lee University, Lexington, Virginia, USA Aichi University of Education, Nagoya, Japan University of Calgary, Alberta, Canada International Conferences: January 13 - 15, 2018: 3rd International Conference on Groups and Algebras ICGA 2018, Sanya, China Invited Keynote: Deep Transfers of p-Class Tower Groups Conference Poster Progressive Innovations and Outstanding Scientific Achievements: 3-Class Field Towers with Three Stages and their Galois Groups During the last six years 2012 - 2018, many open questions in the context of finite p-class field towers, for an odd prime p, have been answered completely. The clarification commenced in 2012 with the first rigorous proof for the existence of imaginary quadratic fields K having a 3-class tower K = F(0) < F(1) < F(2) < F(3) = F(4) of precise length three, verifying a conjecture by Brink in 1984, respectively by Brink and Gold in 1987. This proof was established in my cooperation with Michael R. Bush and concerned fields K with 3-class group Cl3(K) ∼ C(3)×C(3) and Artin transfer pattern (τ(K),κ(K)) ∼ ((32,21,21,21),(2231)) for which a two-stage tower had been claimed erroneously by Scholz and Taussky in 1934 and by Heider and Schmithals in 1982. Our result was published in the Journal of Number Theory 2015 by Elsevier [1]. We intentionally abstained from most extensive generality in favour of quickly documenting our priority. However, there drowsed two essential capabilities for generalizations. Firstly, I extended the proof for imaginary quadratic fields K to infinitely many distinct Artin patterns (τ(K),κ(K)) with either τ(K) ∼ ((n+1,n),21,21,21) and κ(K) ∈ {(1231),(2231)} or τ(K) ∼ ((n+1,n),13,21,21) and κ(K) ∈ {(1122),(3122)}, where n > 1 denotes an arbitrary integer, based on the remarkable discovery of periodic bifurcations in descendant trees [2] which give rise to the required periodic sequences of Schur sigma-groups [4]. Secondly, I explored the behavior of real quadratic fields K with these Artin patterns [3] and found a striking difference because they admit 3-tower groups Gal(F(∞)/K) with less restrictive requirements for the relation rank, which necessitated the proof of criteria for distinguishing between two-stage and three-stage towers [5]. The final step in illuminating the periodic bifurcations in trees [2] and the periodic sequences of Schur sigma-groups [4] was completed 2018 in my collaboration with Mike F. Newman [6] by giving an explanation of all phenomena in terms of infinite limit groups whose finite quotients yield the periodic sequences of Schur sigma-groups. Since Elsevier denies open access to [1], all further papers [2,…,6] were published with gold open access and can be downloaded free of charge from the publisher's website: [1] 3-class field towers of exact length 3 [2] Periodic bifurcations in descendant trees of finite p-groups [3] Index-p abelianization data of p-class tower groups [4] Periodic sequences of p-class tower groups [5] Criteria for three-stage towers of p-class fields [6] Modeling rooted in-trees by finite p-groups Co-periodicity Isomorphisms between co-class forests arising from infinitely repeated Multifurcations Artin Limit Patterns for the Successive Approximation of the stages of p-class towers and their Galois groups Fork Topologies on Structured Descendant Trees and quantitative measures of Pattern Search Complexity Fundamental Principles: Polarization principle Mainline principle Monotony principle Background of Scientific Applications: Marius Sophus Lie, December 17, 1842, Nordfjordeid --- February 2, 1899, Kristiania (Oslo) Continuous transformation group --- Lie group, Tangent space at the identity --- Lie algebra Galilei transformation group  →  Classical mechanics in Euclidean 3-space Lorentz transformation group  →  Relativistic mechanics in Minkowski 4-space space - time - translations proper orthochronous transformations rotations in Euclidean 3-subspace spatial reflection, time inversion combined space - time - inversion Aims of Current Research: Symmetry groups  →  Noether Theorem: conservation laws for observables Poincaré group, Killing groups  →  General Relativity Maxwell, Feynman: Quantum Electro Dynamics (QED), Quantum Field Theories Gauge invariance, non-abelian Yang - Mills theory  →  Quantum Chromo Dynamics (QCD) Spectral class O Stars: Theta1 Orionis C1 (Trapezium Cluster) Luminous Blue Variables (LBV): Deneb, Eta Carinae, P-Cygni, LMC: S-Doradus Wolf-Rayet Stars (WR): Gamma Velorum, LMC: R136a1 White Dwarf Stars: Sirius B, Prokyon B (Chandrasekhar limit) Neutron Stars, Pulsars, Magnetars: RX J1856 (Tolman - Oppenheimer - Volkoff limit) Quark Stars: no proven example is known currently Black Holes: Cygnus X1 Supermassive Black Holes in Galactic Centers: Milky Way, Andromeda Galaxy Gamma Ray Bursts (GRB): GRB170817A, GW170817 Publications: Co-periodicity isomorphisms between forests of finite p-groups Successive approximation of p-class towers Deep transfers of p-class tower groups Modeling rooted in-trees by finite p-groups Recent progress in determining p-class field towers Principal Investigator and Project Leader of several International Scientific Research Lines: Towers of p-Class Fields over Algebraic Number Fields Structured Descendant Trees of Finite p-Groups Lie Groups in General Relativity, Astrophysics, and Particle Physics