Research
activities
The members of our department do research in the following topics.
Algebra
The details of the results and the lists of publications can
be found on the individual homepages of our faculty. We
now give a short survey of these research activities.
Group
Theory and Generalizations
Péter
Pál
Pálfy investigates several problems in finite group theory.
 Is every finite lattice isomorphic to an interval
sublattice in the subgroup lattice of a finite group? The question is
originally related to a general representation problem in universal
algebra (P. Pudlák),
sometimes reduces to some hard, currently intractable questions about
finite simple groups (R. Baddeley, A. Lucchini), and
recently proved relevant for functional analysis as well (M. Aschbacher,
Y.
Watatani).
 CIgroups: These are the groups (introduced by L. Babai)
for which two Cayleygraphs are isomorphic if and only if the
generating subsets correspond to each other by some automorphism of the
group. One of the basic open questions is:which elementary abelian
pgroups are CIgroups.
 Generalization
of the class number problem of the group of upper unitriangular
matrices to the unit group of similar semigroupalgebras.
Piroska
Csörgő
is
mainly interested in finite loops that come up when investigating
finite groups. Questions on solvability and nilpotence, like
establishing a bound for the nilpotence class using abelian inner
mapping groups are
considered in collaboration with M. Niemenmaa
(Finland), A. Drápal and T. Kepka (Czech Republic). Some special
classes of loops like Buchsteiner loops, LCC loops and the
nuclear and central nilpotence of Moufang loops are investigated
with M. Kinyon (USA)
and A. Drápal.
Additional fields of research in our department are finite
supersolvable groups, pgroups, and the study of the
structure of classes of finite groups with some embedding properties.
The results in this area were achieved by Piroska Csörgő, partly with
coauthors M. Asaad, M. Ramadan, M. Ezzat,
A. A. Heliel (Egypt) and M. Herzog (Israel), and by Péter Hermann with K.
Corrádi, L.
Héthelyi and E. Horváth
(Budapest).
Rings
and modules
Ring theoretic research in
Hungary started with such
wellknown mathematicians as Tibor Szele, Andor Kertész and
László Rédei. From later generations, we should mention the
name of Richárd Wiegandt
and Ferenc Szász, who worked mainly in radical theory. Furthermore, in
general
ring theory the accomplishments of László Márki and Pham Ngoc Ánh should be noted.
In the 1990's new topics started to gain ground in Hungary: in Debrecen
Béla Bódi (A.
A. Bovdi)
and his students started to work in the theory of group rings, in
Miskolc Jenő Szigeti
and his
collaborators in the theory of polynomial identities, and another new
direction was brought into Hungary by introducing research in
representation theory, in particular research in the theory of
quasihereditary algebras.
The notion of quasihereditary algebras was defined in 1987 by Ed
Cline, Brian
Parshall and Leonard
Scott in connection
with their work in the theory of Lie algebras and algebraic groups. The
first
basic results in this field were obtained by Vlastimil Dlab and Claus Michael Ringel.
István Ágoston
from our
department entered this research area, together with Erzsébet Lukács from the
Budapest University of Technology and Economics and Piroska
Lakatos from the University of Debrecen. In their joint
work, Ágoston, Dlab and Lukács proved theorems concerning
the structural and homological properties of
quasihereditary algebras. Later, they extended these
investigations to the class of standardly stratified
algebras  a class of algebras arising naturally in the
representation theory of Lie algebras, which generalizes the
concept of quasihereditary algebras. Furthermore,
Ágoston, Dieter
Happel, Lukács and Luise
Unger obtained results concerning the finitistic dimension of
standardly
stratified algebras. This area of ring theory is studied, among others,
by Karin Erdmann,
Steffen König,
Volodymyr Mazorchuk
and Changchang
Xi. Mátyás Domokos
(who formerly
worked at our department, too) and Pham Ngoc Ánh work in areas
that is close to representation theory.
General
algebras and algorithms
Ervin Fried, who
is a Professor Emeritus at our Department, together with Tamás
Schmidt and Béla
Csákány, were internationally
acclaimed, active participants of the research in general algebras, and
they created the school of Universal Algebraists in Hungary. Some
prominent researchers in this area are Ágnes
Szendrei, Gábor
Czédli, László
Zádori and their students in Szeged, László
Márki, in Budapest, Sándor
Radeleczki in Miskolc. Around 1985, Ralph
McKenzie, jointly with other mathematicians, has discovered
the deep Tame Congruence Theory, using some results of Péter
Pál Pálfy, who has a part time position at our
Department. The primary research area of Emil
Kiss is the application and further development of this
theory. Recently, the famous Constraint
Satisfaction Problem
provided a bridge to algorithm theory as well. Csaba
Szabó,
using algebraic and combinatorial methods, works on problems that are
related to both algorithm theory and general algebra, and has many
bright students. This Hungarian research group has a working
relationship with some important centers of general algebra around the
world, characterized by such names, besides Ralph McKenzie, as George
Gratzer, Ralph
Freese, Matthew
Valeriote, Keith
Kearnes and Rudolf
Wille.
Number theory
The Hungarian number theory
school  founded by Pál Erdős and Pál Turán 
plays a leading role world wide in certain fields of combinatorial,
additive and multiplicative number theory (e.g., sequences,
probabilistic number theory, Turán's power sum method, prime number
theory).
Professor Pál
Turán (19101976), regular member of HAS (Hungarian Academy of
Sciences), founded the Department of Algebra and Number Theory and his
scientific works beside number theory (elementary and analytic number
theory, Diophantine approximation) and algebra (statistical theory of
groups, numerical algebra) had deterministic influence on the evolution
of graph theory, approximation theory and complex function theory. His
students and collaborators can be found in different departments of the
Eötvös University, the Debrecen University, in the Rényi Institute of
HAS and universities of many countries.
After Pál Turán the chair of the department was Professor János Surányi,
who worked in mathematical logic, number theory, algebra, combinatorics
and graph theory, approximation theory.
At present Professor András Sárközy,
regular member of HAS, leads the number theory research.
Research areas
András
Sárközy: combinatorial, additive and multiplicative number
theory, computational number theory, pseudorandomness and cryptographic
applications.
Gyula Károlyi:
discrepancy theory (irregularities of distributions), additive
combinatorics.
Mihály Szalay:
statistical theory of partitions, statistical theory of groups.
Róbert Freud,
Katalin P.
Kovács: characterization of additive arithmetic functions,
Sidontype problems, combinatorial number theory.
Katalin Gyarmati:
combinatorial number theory, Diophantine problems, pseudorandomness.
Gergely Zábrádi:
algebraic number theory.
One of the main research areas of András Sárközy
and Katalin Gyarmati is
the study of pseudorandom structures. They investigated this topic
jointly with French, Canadian, Austrian, German and Hungarian
mathematicians as Christian Mauduit, Pascal Hubert, Joël Rivat, Julien Cassaigne, Sébastien Ferenczi (all
from Marseille), Cécile Dartyge (Nancy), Cameron
L. Stewart (Waterloo), Harald
Niederreiter (Singapore), Arne
Winterhof (Linz), Attila
Pethő (Debrecen).
András
Sárközy and Mihály
Szalay studied partitions jointly with the French mathematicians JeanLouis Nicolas
(Lyon) and Cécile Dartyge (Nancy).
Gyula Károlyi's
main field of interest is the algebraic approach to combinatorial
number theoretic problems. His most remarkable result is a structure
theorem for extremal sequences concerning the ErdősHeilbronn
conjecture.
The Hungarian research in algebraic number theory was famed by such
wellknown mathematicians as Gyula Kőnig, József
Kürschák, Mihály
Bauer, and László Rédei
in the early 20 ^{th} century. Since then a very strong research
group was founded in Debrecen by Kálmán Győry in
the 1970s. Around the Millenium the research in algebraic number theory
was renewed in Budapest through the work of Tamás Szamuely.
One of the central questions in algebraic number theory is to
investigate arithmetic objects such as the class group of a number
field or the rational points on elliptic curves. One (very powerful)
method for studying these is Iwasawatheory. One of the founders and
most wellknown mathematicians in this area is John Coates
through his own research and his so many students.
He was Gergely Zábrádi's
research supervisor, as well.
At the University of Münster Gergely
Zábrádi's postdoctoral adviser was Peter
Schneider, who is a leading scientist in the theory of padic
representations of padic groups. To understand the  by now
perhaps less mystical  relationship of these to Galoisrepresentations
is a newly emerging task in algebraic number theory.
International connections
The number theory research
group has intensive international connections, in particular the
French, German and North American connections are specially active due
largely to the works of András Sárközy
with researchers of combinatorial, additive and multiplicative number
theory including leading researchers as Etienne Fouvry, JeanLouis Nicolas,
Rudolf
F. Ahlswede, Helmut Maier,
Cameron
L. Stewart, Peter
D.T.A. Elliott, Henryk
Iwaniec, Melvyn
B. Nathanson, Andrew
M. Odlyzko, Carl
Pomerance.
The present members of the department have many foreign coauthors  in
particular in the last 20 years we have had coauthors from 14 countries.

