Fame For Austria 2018

Fame and honor for Austria

and for the capital Vienna

Austria - Coat Of Arms

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

Vienna - Parliament - Pallas Athena Fountain

Progressive Innovations and Outstanding Scientific Achievements:

  1. 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

  2. Co-periodicity Isomorphisms between co-class forests
    arising from infinitely repeated Multifurcations

  3. Artin Limit Patterns for the Successive Approximation
    of the stages of p-class towers and their Galois groups

  4. Fork Topologies on Structured Descendant Trees and
    quantitative measures of Pattern Search Complexity

  5. 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


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

Daniel C. Mayer

Principal Investigator and
Project Leader of several

International Scientific Research Lines:

Web master's e-mail address:

Prime Number 2017
IPAD and IPOD 2016
29ièmes Journées Arithmétiques 2015
Fame For Styria 2014
Research Frontier 2013
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