Algebra Minicourses

Lecturer: Johan Öinert

Blekinge Institute of Technology, Sweden

Title: Epsilon-strongly graded rings - theory and applications

Abstract: In this lecture series, we will introduce the audience to the theory of group-graded rings and more precisely epsilon-strongly graded rings. Some key results will be dissected and explained, and applications  e.g. partial crossed prudcts and Leavitt path algebras will be put on display.

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Lecturer: Sujit Kumar Sardar

Jadavpur University, India

Title: Understanding groupoid approach in the study of inverse semigroups and some

combinatorial algebras

Abstract: A groupoid is a small category consisting of isomorphisms only. Groupoids generalize groups and have close relationship with inverse semigroups which capture partial symmetries on objects. In this talk we will understand groupoid and its components from a purely algebraic point of view. The study of topological groupoids is quite classical in literature. It appears in several areas of mathematics including operator algebras, theory of inverse semigroups, ergodic theory etc. Among the interesting connections between groupoids and inverse semigroups one can talk about the non-commutative generalization of Stone duality due to Lawson which interrelates ample groupoids (or Stone groupoids) with Boolean inverse semigroups. We will try to explore this. One of the interesting and perhaps the most striking fact about topological groupoids is that they can be used to model algebras constructed from combinatorial objects. Once this is done the structural properties of the algebra can be characterized via the algebraic and topological properties of the modeling groupoid. This idea was pioneered for graph C∗-algebras by Paterson and for discrete inverse semigroup algebras by Steinberg. In this talk we will witness groupoid approaches in the study of (i) Leavitt path algebras of directed graphs over commutative unital rings, commutative semirings and Clifford semifields, (ii) Kumjian-Pask algebras of higher-rank graphs over commutative unital rings.

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Lecturer: Wolfgang Bock

Linneaus University, Sweden

Title: On the Growth of Groups and Algebras 

Abstract: This course introduces the study of growth in algebraic structures, with particular emphasis on finitely generated groups and filtered associative algebras. Central to the course is the analysis of growth functions and their classification into polynomial, exponential, and intermediate types. The Gelfand–Kirillov dimension is discussed as a key invariant for filtered algebras, while the growth of groups is examined through classical results such as Milnor’s and Gromov’s theorems. Special attention is given to the concept of algebraic entropy, that measures exponential growths. In special cases one can show that the algebraic and topological entropy coincide.  As an outlook, ideas for the growths of vertex algebras are given.