Deformations of Path Algebras of Quivers With Relations - Lecture I

Explore the foundations of deformation theory in path algebras of quivers with relations in this first lecture of a four-part minicourse. Delve into the fundamental concepts that underpin representation theory and algebraic geometry, as presented by lecturers Severin Barmeier and Zhengfang Wang. Discover how path algebras naturally arise in various contexts, including the representation theory of finite-dimensional algebras, coordinate rings of affine varieties, and endomorphism algebras of tilting bundles. Learn about the explicit description of deformation theory for finite quivers and finitely generated ideals of relations, focusing on the use of reduction systems to systematically deform relations. Gain insights into related topics such as noncommutative Gröbner bases, L-infinity algebras, algebraization of formal deformations, and deformation quantization of Poisson structures. Benefit from an accessible approach designed for non-experts, complete with illustrative examples to reinforce understanding of these complex mathematical concepts.