P26008-N25 Towers of p-class fields over algebraic number fields
Project Publications
Further Activities
Final Report


Daniel C. MAYER, Ph. D.,
Austrian citizen, born in Graz, Styria, Austria … 28.05.1956
Address: Naglergasse 53, 8010 Graz, Austria
Web site: www.algebra.at

Research interests:
Towers of p-class fields with non-abelian higher p-class groups as Galois groups, Golod-Shafarevich cohomology theory, generator rank and relation rank of p-groups, Galois action, Schur σ-groups and generalizations, bivariate polynomial ideals as annihilator modules of the main commutator of two-generated metabelian p-groups, uniserial p-adic space groups, infinite pro-p groups, coclass graphs of finite p-groups, periodicity of pruned branches of coclass trees, parametrized presentations for infinite sequences of p-groups, p-group generation algorithm, quadratic, cubic, quartic, quintic, dihedral, metacyclic and Frobenius extensions of algebraic number fields, Voronoi's algorithms for computing lattice minima, fundamental systems of units, and class group structure, multiplicities of discriminants, ring and ray class fields, differential principal factorizations and ambiguous principal ideals.

See also: Wikiscientist.

Curriculum vitae:

Ph. D. studies of mathematics, physics, and chemistry
Faculty of natural sciences, Karl-Franzens-University, Graz
… 01.10.1975 - 05.07.1983
Ph. D. Thesis in functional analysis and differential equations
… 05.1983
Rigorous exams in mathematics, physics, and philosophy
… 06.1983

The theory of formal power series mappings between Banach spaces and its application to formal vector fields, Jordan decompositions and group embeddings of formal transformations,
Austrian National Library, 1983.

Promotion to Ph. D. (Doctor of Philosophy) … 05.07.1983

In my graduate years of study at the Karl-Franzens-University in Graz, I was introduced to modular and automorphic functions by Ludwig Reich, who was the supervisor of my Ph. D. thesis from 1981 to 1983. Alexander Aigner and his former student Franz Halter-Koch acquainted me with analytic and algebraic number theory.

University assistant:
Institute for mathematics, Karl-Franzens-University, Graz

Studying assistant … 01.03.1982 - 14.07.1983
Assistant with contract … 15.07.1983 - 31.10.1983
University assistant … 01.11.1983 - 31.10.1990

Conference presentations:
  1. Jordan decompositions and analytic iterations of formal power series mappings,
    3. Austrian Mathematicians Meeting, University of Salzburg … 28.09.1983
  2. Sharp bounds for the partition function of integer sequences,
    11. Austrian Math. Congress and 95. Annual Meeting of the DMV 1985, University of Graz … 19.09.1985
  3. On differential principal factors among the lattice minima in the Minkowski image of orders in simply real cubic number fields,
    4. Austrian Math. Meeting, Bressanone, Italy
    … 21.09.1987
  4. Ambiguous principal ideals in non-Galois number fields with dihedral normal field,
    12. Austrian Math. Congress and 99. Annual DMV Meeting 1989, University of Technology, Vienna
    … 18.09.1989
  5. Multiplicities of cubic discriminants,
    5. Austrian Math. Meeting, University of Klagenfurt
    … 24.09.1991
Teaching experience in university lectures and courses:
  1. Elliptic and Automorphic Functions … fall term 1983/84
  2. Functional Analysis and Spectral Theory
    … spring term 1984
  3. Linear Algebra 1 and 2
    … fall term 1984/85 and spring term 1985
  4. Algebraic Number Theory … fall term 1985/86
  5. Linear Algebra … spring term 1986
  6. Analysis 1 -- 2 (Differential and Integral Calculus)
    … fall term 1986/87 and spring term 1987
  7. Linear Algebra and Analytic Geometry
    … fall term 1987/88 and spring term 1988
  8. Higher Algebra … fall term 1988/89
  9. Arithmetically Equivalent Number Fields
    … spring term 1989
  10. Modular Functions and Complex Multiplication
    … fall term 1989/90
  11. Voronoi's 1- and 2-Dimensional Unit Algorithms
    … spring term 1990
My main working areas as a University Assistant from 1983 to 1990 were generalized continued fraction algorithms for number fields of unit rank 1 and 2 by G. F. Voronoi and A. Brentjes. At conferences I became acquainted with Michael Pohst and Johannes Buchmann in 1986. Together with Hugh C. Williams, whom I met in 1987, I investigated conditions for generators of ambiguous principal ideals and integral multiples of units to appear among the lattice minima of the geometric Minkowski image of orders in quadratic and cubic fields, e. g. in
Differential principal factors and units in pure cubic number fields,
Inst. for Math., Karl-Franzens-Univ. Graz, 1988,
Lattice minima and units in real quadratic number fields,
Publ. Math. Debrecen 39 (1991), 19 - 86.

Postgraduate visiting research assistant, conducting the research project Galois number fields with S3-group on my own responsibility:
Department of Computer Science, University of Manitoba, Winnipeg, Canada … 01.09.1990 -- 28.08.1991
supported by Erwin Schrödinger Grant Nr. J0497-PHY from the Austrian Science Fund

Presentations at international conferences:
  1. Classification of cubic number fields,
    20th Manitoba Conference on Numerical Mathematics and Computing, University of Manitoba, Winnipeg, Manitoba, Canada … 29.09.1990
  2. Multiplicities and classification of dihedral discriminants,
    1990 Westcoast Number Theory, Asilomar Conference Grounds, Pacific Grove, Monterey, California, USA
    … 20.12.1990
  3. Dihedral fields from quadratic infrastructure,
    101. Annual Meeting of the DMV 1991, University of Bielefeld, Germany … 19.09.1991
  4. Dihedral fields from quadratic invariants,
    1992 Westcoast Number Theory Conference, Oregon State University, Corvallis, Portland, Oregon, USA
    … 18.12.1992
University lectures and courses:
  1. Galois Cohomology and Class Field Theory
    … fall term 1990/91
  2. Selected Topics on Dihedral Field Extensions
    … spring term 1991
Aside from fulfilling standard teaching duties I tried to offer access to advanced areas for students. However, due to a low number of participants, cancellation of special courses aiming at research topics became inavoidable, repeatedly. Therefore, I asked H. C. Williams for the opportunity to launch a research project with title Galois number fields with S3-group at the University of Manitoba, Winnipeg, Canada, and I applied for an Erwin Schrödinger grant at the Austrian Science Fund in 1990. Inspired by errors in the table of complex cubic fields by G. W. Fung and H. C. Williams, I developed my theory of multiplicities of discriminants in the course of this Schrödinger Project. In collaboration with Pierre Barrucand at Paris, I extended the multiplicity theory from dihedral to metacyclic fields.

Secondary university studies:
Teachership in mathematics and physics
Faculty of natural sciences, University of Regensburg
… 12.04.1994 - 18.12.1995

and authorization to take the title Lehramts-Assessor:
1st State Exam for teachership at high schools … 18.12.1995
2nd State Exam for teachership at high schools … 09.12.1997

Several deficiencies in the daily routine of lectures at universities, a considerable gap between teaching and research, and the lack of impressive graphical visualizations convinced me to start second studies at the University of Regensburg, Germany. In particular, I paid attention to the cognitive psychology of perception, attention, motivation, memory, learning, and language, and to technical didactics, from elementary school to high school and university. With Theodor Bröcker, Klaus Jänich, and Ernst Kunz I probed deeper into differential topology, algebraic topology and algebraic geometry.

Professional qualification:
Exam in Austrian School Laws … 28.06.1999
Approval of Bavarian State Exams by the
Federal Ministry of Education and Cultural Matters, Vienna
… 07.09.1999

Teaching experience at high schools:
computer science, electrotechnology, mechanical engineering, projective geometry, applied mathematics, applied physics
… 1996 - 2013

Recent research:
Due to my research on the capitulation problem for number fields I was increasingly attracted by the theory of finite p-groups and infinite pro-p groups.

Member and referee of professional societies:
  1. Austrian Mathematical Society (ÖMG) … since 1983
  2. American Mathematical Society (AMS) … since 1990
  3. Reviewer for Mathematical Reviews (MR) and MathSciNet … since 1992
  4. Association for Computing Machinery (ACM)
    … since 2003
  5. Referee for International Journal of Number Theory (IJNT) … since 2011
  6. Referee for Asian-European Journal of Mathematics (AEJM) … since 2014
  7. Referee for Journal of Algebra and its Applications (JAA) … since 2014
Invited lectures:
  1. The p-class numbers of unramified cyclic extensions of a number field with p-class group of type (p,p),
    Journées de Théorie des Nombres, Algorithmique et Applications, Université Mohammed Premier, Faculté des Sciences d'Oujda (FSO), Oujda, Morocco
    … 27.09.2010
  2. The structure of the 3-class groups of the four unramified cyclic cubic extensions of a number field with 3-class group of type (3,3),
    Journées de Théorie des Nombres, Algorithmique et Applications, Université Mohammed Premier, FSO, Oujda, Morocco … 28.09.2010
  3. Quadratic p-ring spaces for counting dihedral fields,
    Workshop International NTCCCS 2012, Université Mohammed Premier, FSO, Oujda, Morocco
    … 26.04.2012
Presentations at international conferences:
  1. Multiplicities of discriminants of p-ring class fields over quadratic fields with modified p-class rank σ ≥ 2,
    15. Austrian Math. Congress and 111. Annual Meeting of the DMV 2001, University of Vienna, Vienna, Austria
    … 20.09.2001
  2. Classical and modern scope of two-stage metabelian 3-groups,
    8. Austrian Math. Meeting, Bolzano, Italy … 22.09.2003
  3. Two-stage towers of 3-class fields over quadratic number fields with triadic irregular discriminant,
    17. Austrian Math. Congress and 119. Annual Meeting of the DMV 2009, University of Technology, Graz, Austria
    … 25.09.2009
  4. The distribution of second p-class groups on coclass graphs,
    27ièmes Journées Arithmétiques 2011, Vilnius University, Vilnius, Lithuania … 01.07.2011
  5. Principalization algorithm via class group structure,
    Joint CSASC Conference 2011, Danube University, Krems, Austria … 25.09.2011
  6. First faultless disproof of Scholz and Taussky's claim concerning the 3-class field tower of Q((-9748)1/2),
    Workshop on Golod-Shafarevich groups and algebras and rank gradient, Erwin Schrödinger International Institute for Mathematical Physics, Vienna, Austria
    … 24.08.2012
  7. Number fields sharing a common discriminant,
    122. Annual Meeting of the DMV 2012, University of the Saarland, Faculty of Mathematics, Saarbrücken, Germany … 18.09.2012
  8. Finite 3-groups as viewed from class field theory,
    Groups St Andrews 2013, University of St Andrews, St Andrews, Fife, Scotland … 09.08.2013
    with coauthor Mike F. Newman
  9. 3-class field towers of exact length 3,
    18th ÖMG Congress 2013 and 123rd Annual DMV Meeting 2013, University of Innsbruck, Technology Campus, Innsbruck, Tyrol, Austria … 24.09.2013
    with coauthors Michael R. Bush and Mike F. Newman
  10. Class towers and capitulation over quadratic number fields,
    West Coast Number Theory 2013, Asilomar Conference Grounds, Pacific Grove, Monterey, California, USA
    … 18.12.2013
    with coauthors Michael R. Bush and Mike F. Newman
Skills in computer science and information technology:
Computer algebra systems: Magma, Pari/GP, GAP, Singular; Typesetting: TeX, LaTeX; Object Oriented Analysis, OO Design, and OO Programming: Pascal, Delphi, C, C++, Java, JSP, HTML, XML; usecase, class and sequence diagrams in UML, object-relational mappings of class hierarchies on database schemata, SQL, PL/SQL.

Professional qualification:
Sun Certified Java Programmer … 22.08.2003
Exam in object oriented programming in Java by Sun Microsystems, Inc. (now Oracle)

It turned out to be very useful to be able to write program scripts for the Computer Algebra Systems Pari/GP, GAP, Singular, and Magma and to write publications with TeX and LaTeX. I acquired additional skills in objectoriented analysis, design, and programming, and became a Sun Certified Java Programmer (now Oracle) … 22.08.2003

International Research Project P26008-N25:
Since neither algebraic number theory nor p-group theory is represented by Austrian universities in an amount comparable with my requirements, the present stand alone project P26008-N25 is intended to stabilize my well-known status in the international mathematical community, to establish an Austrian research centre for a multidisciplinary approach to problems which demand a high portion of expertise in both, number theory and group theory, and to strengthen my collaboration with international partners in the United States, the United Kingdom, Australia, Japan, and Morocco.

Invited lectures:
  1. La Théorie Algorithmique des Nombres,
    Corps de Nombres Algébriques et Corps de Classes,
    École de Recherche CIMPA UNESCO, Théorie des Nombres et ses Applications, Université Mohammed Premier, Faculté des Sciences d'Oujda (FSO), Oujda, Morocco … 18 - 29.05.2015
  2. Periodic sequences of p-class tower groups,
    1st International Conference on Groups and Algebras (ICGA) 2015, Guang Dong Hotel, Shanghai, China
    … 21.07.2015
  3. p-Capitulation over number fields with p-class rank two,
    2nd International Conference on Groups and Algebras (ICGA) 2016, Youngor Central Hotel, Suzhou, China
    … 26.07.2016
  4. Recent progress in determining p-class field towers,
    1st International Colloquium of Algebra, Number Theory, Cryptography and Information Security (ANCI) 2016, Taza, Morocco
    … 11.11.2016
Presentations at international conferences:
  1. Index-p abelianization data of p-class tower groups,
    29ièmes Journées Arithmétiques (JA) 2015, University of Debrecen, Hungary … 09.07.2015
  2. New number fields with known p-class tower,
    22nd Czech and Slovak International Conference on Number Theory (CSICNT) 2015, Hotel Sorea Máj, Liptovský Ján, Slovakia … 31.08.2015

With support from
FWFDer Wissenschaftsfonds