AlgNTh

Research group for Algebra and Number Theory

Our current research interests focus on

NON-UNIQUE FACTORIZATIONS IN MONOIDS AND INTEGRAL DOMAINS

The main objective of factorization theory is a systematic treatment of phenomena related to the non-uniqueness of factorizations in structures of arithmetical interest. The main focus areas are principal and non-principal orders in algebraic number fields and function fields, Mori domains, Krull monoids and congruence monoids. The methods are algebraic (key word: multiplicative ideal theory), combinatorial (key word: zero-sum problems) and analytic (key word: abstract analytic number theory).
For more information see a recent survey paper or a monograph on factorization theory.

STRUCTURE AND IDEAL THEORY OF COMMUTATIVE RINGS AND MONOIDS

The main objective of multiplicative ideal theory is the description of the multiplicative structure of commutative monoids and domains by means of ideals or certain systems of ideals. At present we investigate the (arithmetic and ideal theoretic) structure of congruence monoids, C-monoids and their various generalizations. For a modern treatment of multiplicative ideal theory, valid for both commutative rings and monoids, in the language of ideal systems oncommutative monoids we refer to the monograph of ideal theory.

MODULES OVER COMMUTATIVE RINGS

We are interested in the structure of finitely generated modules over commutative Noetherian rings, with a strong emphasis on the structure of indecomposable modules and non-uniqueness of direct-sum decompositions. One of the goals is to obtain new results on the structure of finitely generated indecomposable modules over local rings. For instance, the question which local rings admit finitely generated indecomposable modules having arbitrarily prescribed rank at the minimal primes was almost completely answered in the course of recent investigations.
A second objective is to investigate direct-sum cancellation of modules over (global) commutative Noetherian rings. A third goal is to study connections between the failure of the Krull-Remak-Schmidt-Azumaya Theorem and the theory of non-unique factorizations in Krull monoids.

DIOPHANTINE EQUATIONS

We apply methods from algebraic number theory (arithmetic of algebraic integers, structure of the unit group) and analytic number theory (lower bounds for linear forms in logarithms, the hypergeometric method) to solve parametrized families of Thue equations and Thue inequalities. Currently we use methods from algebraic function fields to investigate the structure of the set of solutions of families of Thue equations.
Another focus of research tries to understand how representations of integers by binary quadratic forms ("solvability of Pellian equations") can be detected from the continued fraction expansion of the underlying quadratic irrationalities.

COMBINATORIAL AND ADDITIVE NUMBER THEORY

We focus on the structure theory of set addition and zero-sum problems over abelian groups. The investigation of problems with immediate (or indirect) applications to factorization theory is emphasized.
For instance, Davenport asked for the maximal length of a sequence in a finite abelian group without a subsequence whose terms sum to zero; Erdös, Ginzburg and Ziv considered the analogous problem for subsequences of length equal to the order or exponent of the group. These problems are only solved for special types of groups, yet their values and the structure of the extremal sequences are important controlling factors for the behaviour of factorizations over Krull domains.
For more information see a recent survey paper and a recent book.

Current and recent research projects