The department is home to renowned research in the fields of algebra, number theory, and combinatorics, each in the frame of international collaboration.
András Pongrácz coordinates research in algebra. His interest lies on the border of algebra and model theory, such topics are, for instance, the classification of structures that can be defined or interpreted by means of a given structure in first-order, the construction of Ramsey extensions, or the study of algebraic invariants like the automorphism groups, the endomorphism monoid, and the polimorphism clone.
Gábor Nyul and his students take on research in combinatorial themes. Primarily, they focus on enumeration problems related to permutations and classification of sets such as generalizations and variants of Stirling numbers, Bell numbers, and other related numbers; in addition, they explore graph-theoretic aspects of these objects. Beside these they obtained results concerning Ramsey-type number-theoretical problems.
The number theory research group has been founded and still being led by Kálmán Győry, it is known as the number theory school of Debrecen. Research efforts are focused mainly on Diophantine equations and related topics. They obtained considerable results in the investigation of unit equations, Thue and Thue-Mahler equations, elliptic and superelliptic equations, discriminant equations, index form equations, power integral bases, norm form equations, decomposable form equations, resultant equations, and combinatorial-type equations. Members of the research group have studied the number of solutions, their distribution, and arithmetic properties. They gave effective and explicit bounds on both the number and size of the solutions. Based on their results they presented and implemented efficient algorithms to solve computationally feasible cases completely. Their results have found many applications in the fields of algebraic and Diophantine number theory, Diophantine approximation, and the theory of polynomials. Using number-theoretic tools they obtained various results concerning digital image processing and cryptography.