Curriculum vitae.

A) General data.

`Born in Bucharest-Romania, 3.03.1940, graduate of the Faculty of
Electronics of the Technical University of Bucharest in 1961 and of the
Faculty of Mathematics of the University of Bucharest in 1969 with a diplom
thesis devoted to the Galois Cohomology, having the late Prof.Ionel Bucur
as superviser. In 1977 I defended the PhD thesis with the title "Arithmetic
and model theory" at the Faculty of Mathematics of the University
of Bucharest under the supervision of Acad. Octav Onicescu, who replaced
Prof.Ionel Bucur after his premature death. `

`During the period 1979-1982, with some intrerruptions, I activated
as visiting professor at the Institute of Mathematics of the University
of Heidelberg thanks to a two years fellowship granted by the Alexander
von Humboldt Foundation. As a member of the research group of Algebra and
Number Theory, I have been decisively influenced by the personality of
my academic mentor Prof. Peter Roquette. `

`In 1983 I obtained a four months fellowship to visit the Universities
of Firenze and Camerino-Italy, but unfortunatelly I was prevented from
honouring the invitation by the communist Romanian authorities. `

`In 1993 I visited the Universities of Wales-Bangor, Queen Marry-London,
and Oxford Mathematical Institute under a three months fellowship granted
by the European Communities. `

`This year I obtained from the the Alexander von Humboldt Foundation
a three months fellowship. The visit in Germany will take place in the
period 1.10-31.12.1998, including research stages at the Universities of
Heidelerb g(Prof.Peter Roquette), Konstanz (Prof.Alexander Prestel), and
MPIM-Bonn. `

`In the present I am Senior researcher 1 and the scientific secretary
of the Institute of Mathematics of the Romanian Academy. I am also associate
professor at the University "Ovidius"-Constanta,Romania. `

`Beginning with 1990 I was accepted as superviser of doctorands.
In June 1995 my doctorand Mihai Caragiu defended his PhD thesis entitled
"Applications of finite fields:power distribution,finite semiplanes,
arithmetical physics". `

`I am a member of the editorial board and the scientific secretary
of "Revue Roumaine de Mathematiques Pures et Appliquees". `

`I am a reviewer to "Mathematical Reviews","Zentralblatt
fur Mathematik und ihre Grenzgebiete","Journal of Symbolic Logic".
`

`I am the delegate for Mathematics in the executive secretariate
of the Humboldt Club Romania founded in 1990. `

`In 1991 I received the "Gheorghe Lazar" prize for Mathematics
of the Romanian Academy. `

B) The research activity.

`The results of my scientific activity are presented in about 40
papers published in the following mathematical journals: Journal of Algebra,
Journal fur reine und angewandte Mathematik,Annals of pure and applied
Logic, Journal of Symbolic Logic, Journal of pure and applied Algebra,
Communications in Algebra,Comptes Rendus Acad.Sci.Paris, Manuscripta Mathematica,Results
of Mathematics, Fundamenta Informaticae, Revue Roumaine de Mathematiques
pures et appliquees, Studii si cercetari matematice,etc. `

`The main contributions are described in the following. `

1)Model theoretic Algebra. 1.1)Henselian valued fields.

`Having as starting point the fundamental works of James Ax, Simon
Kochen and Jurii Ershov from 1965-1966 concerning some diophantian problems
over local fields,I developed in in my PhD thesis and in a series of papers[4,12-15]a
systematic model theoretic study of the Henselian valued fields of characteristic
zero, positive residue characteristic and finite ramification index. The
main results are model theoretic classification criteria (elementary equivalence,
model-completeness, etc) for such valued fields in terms of some elementary
invariants of the value groups and the residue rings. An application to
the theory of integrally defined functions on valued fields is given in
[15], where an interesting class of valued fields, called prehenselian,
is introduced and investigated. `

`Using algebraic and model theoretic techniques, I proved in [36]
a theorem on the relative elimination of quantifiers for Henselian valued
fields of characteristic zero ,extending the results of Angus MacIntyre,and
Prestel-Roquette on quantifier elimination for p-adically closed fields,
as well as the results of primitive-recursive nature of V.Weispfenning.
Related to I mention the isomorphism criterion for Henselian valued fields,
algebraic over a given common valued subfield, presented in the joint paper
[41] with F.V.Kuhlmann. `

`Devoted to the same field of interest,the paper [30]extends a classical
theorem of Abraham Robinson on quantifier elimination for algebraically
closed valued fields as well as a result from 1973 of Lipshitz and Saracino,and
Carson concerning the model completion of the elementary theory of regular(in
the sense of von Neumann) commutative rings. `

1.2)Formally p-adic fields.

`The theory of formally p-adic fields was developed by Simon Kochen
and Peter Roquette as a p-adic analogue of the classical theory of formally
real fields initiated by Emil Artin and Schreier. An enlarged framework
for this theory is provided in [17] where the main results of the paper
"The Nullstellensatz over p-adically closed fields",Journal of
the Mathematical Society of Japan,32(1980),by M.Jarden and P.Roquette,are
proved in this more general context. In a prolongation of [17],the paper
[18]investigates some situations when certain objects associated to a field
extension F/K(places of F/K,the Kochen ring and the holomorphy ring of
F/K for a p-adically closed base field K)are obtained by the contraction
of the corresponding objects associated to a field extension N/K subject
to F<N. As an application,the existence of some recursive bounds in
the theory of fields and the theory of formally p-adic fields is proved.
`

`Devoted to the same field of interest,the paper [21]provides a generalization
of the preorders of higher level introduced in 1979 by E.Becker to extend
Artin-Schreier theory to arbitrary powers sums in fields.Using Kadison-Dubois
representation theorem for Archimedean partially ordered rings,an operator
theoretic description of the t-preorders of level n is given,recovering
an unpublished result of P.Roquette in the particular case n=2. `

1.3)Nullstellensa"tze.

`The remarkable fact noticed by A.Robinson,namely the equivalence
between Hilbert's Nullstellensatz and the model completeness of the "elementary"theory
of algebraically closed fields opened the way for using specific model
theoretic concepts and results in the approach of some problems of the
algebraic geometry(in particular,the Nullstellensa"tze)over base fields
which are not necessarily algebraically closed,however having suitable
arithmetical and model theoretical properties.To this field of interest
belong the papers [17,23,27,30]devoted to extensions of some Nullstellensa"tze
over ordered fields,p-adically closed fields, pseudoalgebraically closed
fields due to D.Dubois,G.Stengle,M.JardenP. Roquette,B.Jacob,K.McKenna.
`

1.4)Pseudoreal closed fields.

`In Ax's fundamental work from Annals of Mathematics(1968) devoted
to the "elementary" theory of finite fields,an important class
of fields,called later by G.Frey pseudoalgebraically closed fields,is introduced
and investigated.An order theoretic analogue of this concept was introduced
and studied in [25].Later A.Prestel extended this concept calling a field
pseudoreal closed if it is existentially closed in any regular and totally
real field extension;equivalently,in geometric terms,a field K is pseudoreal
closed iff every absolutely irreducible affine variety defined over K has
a rational point over K whenever it has a simple rational point over any
real closure of K.In [24],an alternative proof based on nonstandard arithmetic
techniques is given for Prestel's result on the recursive axiomatizability
of the class of pseudoreal closed fields. `

`The paper [26]is devoted to the algebraic and the model theoretic
investigation of an important subclass of pseudoreal closed Hilbertian
fields.A positive answer to a question raised in [26]is announced without
proof by J.Ershov in a note from Dokladi Akad.Nauk SSSR(1982). `

`The papers [28]and [31]are devoted to the absolute Galois group
of a pseudoreal closed field,while some model theoretic transfer principles
for pseudoreal closed fields are proved in [32]. `

1.5)Abelian groups.

`The models of the "elementary" theories of the classes
of finite,resp. profinite,resp. torsion Abelian groups are characterized
in [8]and [11]in terms of some specific elementary invariants.The paper
[8]is the starting point-the Abelian case-of Felgner's works on pseudofinite
groups,the group theoretic analogue of Ax's pseudofinite fields. `

2)Rings with approximation property.

`Using the model theoretic concept of existential completeness,certain
types of good approximation in rings are introduced in the joint paper
[19], and the general theory is applied to the particular case of rings
with approximation property. `

3)The p-adic spectrum of a commutative ring and compactification.

`By analogy with Zariski's spectrum and the real spectrum of a commutative
ring,the concept of a p-adically closed field induces through a natural
process of globalization the notion of a p-adic spectrum introduced and
investigated in [35].p-adic analogues of some results of the real algebraic
geometry(as Artin-Lang theorem and the finiteness theorem conjectured by
Brumfiel)are obtained.Finally these results together with Kuhlmann-Prestel
density theorem(Crelle's Journal,1984)provide a model theoretic unitary
approach of the compactification procedure of the affine algebraic varieties
defined over local fields of characteristic zero,introduced in 1984 by
J.W.Morgan and P.Shalen for the complex and the real case,avoiding in this
way the appeal to Hironaka's desingularization. `

4)Diophantian problems.

`In the paper "Zeros of polynomials over local fields.The Galois
action",Journal of Algebra(1970),J.Ax introduced the "diameter
of conjugates" as a measure of the "closeness to the base field"
of an algebraic element over a given Henselian valued field.This concept
is used in [6,10]to study the "closeness to rationality" of the
points of an elliptic curve defined over a local field of characteristic
zero and positive residue characteristic. `

`Some questions concerning the torsion points on elliptic curves
defined over local and global fields are discussed in [20],were some extensions
of certain results of Demianenco and Hellegouarch are obtained. `

`The techniques of the nonstandard arithmetic are used in [7,22]to
approach some questions concerning the class field theory and the diophantian
approximation. `

`Barry Mazur's distributions are powerful tools in the study of some
arithmetical problems on cyclotomic fields,modular functions and abelian
extensions.A purely algebraic approach of the distributions defined on
distributive lattices and profinite groups is developed in [34],where some
extensions of certain results of Sinnott,Kubert and Lang are obtained.
`

5)Arboreal group theory.

`In the last twenty years various extensions of the Bass-Serre theory
of group actions on simplicial trees have been the subject of much investigation
combining elementary geometric considerations with very sophisticated techniques.The
variety of topics and applications of the field is well reflected in the
proceedings "Arboreal Group Theory",ed.R.C. Alperin,Mathematical
Sciences Research Publications 19,Springer-Verlag,1991. `

`Reading by chance the paper of J.Morgan and P.Shalen,"Valuations,trees
and degenerations of hyperbolic structures.1",Annals of Mathematics,120(1984),
I became interested in lambda-trees and the combinatorial group theoretic
information carried by a group action on a lambda-tree.Stimulated by this
paper and also by the paper of R.C.Alperin and H.Bass,"Length functions
of group actions on lambda-trees",Annals of Mathematical Studies,111,Princeton
University Press,1987,I succeeded in [37] to extend to lambda-trees some
basic constructions and results contained in the first chapter of Serre's
book "Trees",while some model theoretic principles for lambda-trees
have been established in [33]. `

`The technique developed in [37] has two complementary aspects:a
group theoretic one concerning actions on groupoids,and a metric one concerning
Lyndon length functions on groupoids.The simultaneous approach of these
two aspects introduced some complications in the logical line of the exposition
in [37],so in [38] and [40],I considered more natural to treat them apart
and eventually relate them.Moreover,a more general concept of tree,including
distributive lattices,lambda-trees where lambda is a lattice ordered group,
and Tits'buildings as special cases,is introduced and investigated in [40],
while the dual of the category of these general trees is described in [39]
using a suitable extension of Stone's representation theorem for distributive
lattices.To my surprise I learned later that my general concept of tree
was known already from 50's to lattice theorists under the name of median
algebra, however totally unknown to group theorists.Thus,having as starting
point of my research the geometric point of view of group actions,I rediscovered
the significant concept of median algebra and some results concerning it.Fortunatelly,
the geometric motivation of my approach permitted me to obtain also some
new results.For instance,in [43] I considered two basic operators on the
class of generalized trees assigning to a generalized tree T the generalized
tree Dir(T) of the directions on T,resp. the directed generalized tree
Fold(T) of the foldings(retractions) of T.The main results of [44] show
that the two operators above commute,providing an interpretation of the
composite operator Fold*Dir=Dir*Fold in terms of the so called quasidirections
on generalized trees. `

`Interested in the way in which the theory of structured groupoids
interacts with various combinatorial and metric problems, Prof.R.Brown
invited me in 1993 to visit the University of Wales in Bangor under an
European Communities research fellowship.With this occasion I noticed and
investigated some interesting connections between generalized trees and
the mathematical objects called racks,well known for their applications
in the theory of knots,links and braids,singularity theory and set theory.Some
related results are included in [43]. `

`In the last three years I was mainly interested in applying the
general concepts and results of the theory of generalized trees to some
significant mathematical frameworks.Thus in [42],motivated by some difficult
problems concerning the model theory of free groups and free profinite
groups,I considered a class of groups called discrete hyperbolic arboreal
groups,showing that given a discrete hyperbolic arboreal group G and a
suitable family of Abelian discrete hyperbolic arboreal groups which are
convex extensions of maximal abelian arboreal subgroups of G,the corresponding
amalgamated sum has a canonical structure of discrete hyperbolic arboreal
group.The works [45-47]are devoted to a systematic study of the arboreal
structure of a class of groups including the free groups,the free Abelian
groups and the Coxeter groups whose relations involve only commuting generators.In
a work in preparation[48] I study the generalized trees of groups and their
applications to SL2 over global fields,including connexions with Hilbert
modular group and Bost-Connes phase transitions with spontaneous symmetry.
`

6)Other papers concern nonabelian cohomology[3,5], formal Moufang loops[9], logical design of circuits and technical applications[1,2].

`Citations of my works occur in papers and books of P.Roquette, J.Ershov,
A.Prestel, W.Hodges, D.Popescu, M.Jarden, U.Felgner, E.Becker, F.Pop, V.Weispfenning,
F.V.Kuhlmann, D.Haran, F.Delon, L.Belair, G.Georgescu, M.Roller, R.Transier,
R.Farre, A.Solian, etc.16 papers are mentioned in the "OmegaBibliography
of Mathematical Logic",vol.3-Model Theory,Springer-Verlag 1987. `

`Among the international conferences to which I had the opportunity
to participate giving lectures,I mention Oberwolfach sessions on Model
Theory,Algebraic Number Theory,p-adic Analysis,Mathematical Logic,Field
Arithmetic,Hannover 1979 International Congress on Logic,Philosophie and
the Methodology of Science,Firenze 1982 Logic Colloquium,Easter Conferences
of Model Theory-Humboldt University Berlin,AMS Conference on Logic,Local
Fields and Subanalytic Sets-Amherst 1990,1991 Banach Semester on Algebraic
Methods in Logic and Applications in Informatics,NATO Advanced Study Institute
on Semigroups,Formal Languages and Groups-York 1993,1996 Banach minisemester
in the memory of Helena Rasiowa,1997 4'th Franco-Touranien Colloquium on
Model Theory-Luminy,etc. `

List of publications.

`1.(with V.Constantinescu,A.Stuparu,and E.Procopovici)Analogical-digital
control equipment for the positioning of pressing screws(in Romanian),Automatica
si Electronica 4(1967)174-177. `

`2.The logical design of the control unit of the major cycle for
a medium size computer(in Romanian)Probleme de Automatizare 6(1969)25-38.
`

`3.Cofibrated categories and non-commutative H2(in Romanian)Studii
si Cercetari Matematice 5(1972)665-678. `

`4.Some metamathematical aspects of the theory of Henselian fields(in
Romanian)Studii si Cercetari Matematice 10(1973)1449-1559;MR52#8104,Zbl1295#
12106. `

`5.Cohomologie des petites categories,Revue Roumaine de Math.5(1974)
559-575. `

`6.The closeness to rationality of the points of an elliptic curve
defined over a local field(in Romanian)Studii si Cercetari Matematice 6(1974)783-792.
`

`7.Some remarks concerning nonstandard admissible morphisms,Revue
Roumaine Sciences Sociales,Serie de Philosophie et Logique(1975)205-210.
`

`8.The models of the elementary theory of finite Abelian groups(in
Romanian)Studii si Cercetari Matematice 4(1975)381-386;MR53#7770,Zbl335#
02036. `

`9.Commutative formal Moufang loops(in Romanian)Studii si Cercetari
Matematice(1976)259-265. `

`10.Le diametre des conjugues des points des courbes elliptiques,
C.R.Acad.Sci.Paris Math.282(1976)787-788. `

`11.On the elementary theories of Abelian profinite groups and Abelian
torsion groups,Revue Roumaine Math.3(1977)229-309;MR55#12508, Zbl388#03013.
`

`12.Some model theory for Henselian valued fields,J.Algebra 55:2(1978)191-212;MR
82m:03046,Zbl424#03016. `

`13.Quelques proprietees modele-theoriques des corps values henseliens,C.R.Acad.Sci.Paris
Math.287(1978)189-191;291-293;MR80a:12028, Zbl394#12008. `

`14.Model theoretic methods in the theory of Henselian valued fields:1,2(in
Romanian)Studii si Cercetari Matematice 1(1979)3-39;617-656; MR82e:12034,MR82e:12035,Zbl424#03015,Zbl446#03028,Zbl454#03013.
`

`15.A model theoretic transfer theorem for Henselian valued fields,J.Reine
Angew.Math.311/312,1-30;MR81h:03071,Zbl409#12030. `

`16.The diameter of the conjugates of a point on an elliptic curve
(in Romanian)Studii si Cercetari Matematice 2(1979)139-158. `

`17.Towards a general theory of formally p-adic fields,Manuscripta
Math.30(1980)279-327;MR81e:12028,Zbl451#12016. `

`18.Extension of places and contraction properties for function fields
over p-adically closed fields,J.Reine Angew.Math.326(1981)54-78; MR82j:03040,Zbl491#12025.
`

`19.(with V.Nica and D.Popescu)Approximation properties and existential
completeness for ring morphisms,Manuscripta Math.33(1981)227-282;MR82k:03047,
Zbl472#13013. `

`20.Some remarks concerning the torsion points of elliptic curves,
Revue Roumaine Math.6(1982)621-642. `

`21.On a class of preorderings of higher level,Manuscripta Math.37(1982)
163-210. `

`22.Roth's theorem:Nonstandard aspects(in Romanian)Studii si Cercetari
Matematice 2(1983)105-113;MR85b:11116,Zbl534#10050. `

`23.A Nullstellensatz over ordered fields,Revue Roumaine Math.7(1983)
553-566. `

`24.Axioms for pseudo real closed fields,Revue Roumaine Math.6(1984)
449-456;Zbl555#12009. `

`25.Definite functions on algebraic varieties over ordered fields,
Revue Roumaine Math.7(1984)527-535;MR85k:12006,Zbl578#12019. `

`26.On some classes of Hilbertian fields,Results Math.(Basel)7(1984)
1-34;MR86c:12002,Zbl547#12016. `

`27.A Nullstellensatz over some Henselian valued fields,An.Stiint.
Univ."Al.Cuza"Iasi Mat.31(1985)17-19. `

`28.The absolute Galois group of a pseudo real closed field with
finitely many orders,J.Pure Appl.Algebra 38(1985)1-18. `

`29.Diophantian problems in the perspective of the model theory(in
Romanian),in "Matematica in lumea de azi si de maine",Ed.Academiei
Romane (1985)22-25. `

`30.Transfer principles for monically closed valued regular rings,
Studii si Cercetari Matematice 1(1986)3-27. `

`31.Definite functions on algebraic varieties over ordered fields.2:
The absolute Galois group of a pseudo real closed field,Revue Roumaine
Math.34(1986)283-291. `

`32.Transfer principles for pseudo real closed e-fold ordered fields,J.Symbolic
Logic 4(1986)981-991. `

`33.Some model theory for generalized trees,J.Pure Appl.Algebra 57
(1989)109-125. `

`34.Distributions on distributive lattices and profinite groups,
Revue Roumaine Math.34:9(1989)793-815. `

`35.Morgan-Shalen compactification of affine algebraic varieties
over local fields,Seminarberichte Humboldt Univ.Berlin Math.104(1989)4-17.
`

`36.Relative elimination of quantifiers for Henselian valued fields,
Annals Pure Appl.Logic 53(1991)51-74. `

`37.On a problem raised by Alperin and Bass,in R.C.Alperin(ed.) "Arboreal
Group Theory",Springer-Verlag(1991)35-68. `

`38.On a problem raised by Alperin and Bass.1:Group actions on groupoids,J.Pure
Appl Algebra 73(1991)1-12. `

`39.The dual of the category of trees,Preprint Series IMAR,7(1992)21pp.
`

`40.On a problem raised by Alperin and Bass.2:Metric and order theoretic
aspects,Preprint Series IMAR,10(1992)30pp. `

`41.(with F.V.Kuhlmann)An isomorphism theorem for algebraic extensions
of valued fields,Manuscripta Math.77(1992)113-126;84(1994)113-114. `

`42.On discrete hyperbolic arboreal groups,Preprint Series IMAR,9(1995)
29p,to appear in Communications in Algebra. `

`43.Generalized trees as compatible families of semiracks,communicated
`

`to the York 1993 NATO Advanced Study Institute on Semigroups,Formal
Languages and Groups. `

`44.Directions and foldings on generalized trees,Fundamenta Informaticae,30:2(1997)125-149.
`

`45.Partially commutative Artin-Coxeter groups and their arboreal
structure,1,Preprint Series IMAR,5(1997)20pp. `

`46.Partially commutative Artin-Coxeter groups and their arboreal
structure,2,Preprint Series IMar,7(1997)26pp. `

`47.Partially commutative Artin-Coxeter groups and their arboreal
structure,3,Preprint Series IMAR,11(1997)42pp. `

`48.Median algebras of groups and SL2 over global fields,in preparation.
`

Serban A. Basarab

15.6.1998