Research activities

Péter Pál Pálfy investigates several problems in finite group theory.

1. 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).
2. CI-groups: These are the groups (introduced by L. Babai) for which two Cayley-graphs 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 p-groups are CI-groups.
3. Generalization of the class number problem of the group of upper unitriangular matrices to the unit group of similar semigroup-algebras.

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, p-groups, 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 co-authors 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).

Ring theoretic research in Hungary started with such well-known 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 quasi-hereditary algebras.

The notion of quasi-hereditary 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 quasi-hereditary 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 quasi-hereditary 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.

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.

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 (1910-1976), 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.

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, Sidon-type 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 Jean-Louis 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 Erdős-Heilbronn conjecture.
The Hungarian research in algebraic number theory was famed by such well-known 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 Iwasawa-theory. One of the founders and most well-known 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 p-adic representations of p-adic groups. To understand the - by now perhaps less mystical - relationship of these to Galois-representations is a newly emerging task in algebraic number theory.

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, Jean-Louis 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.

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