We give a brief introduction to the representation theory of finite-dimensional algebras via quivers and Auslander–Reiten theory. We describe the knitting algorithm, which gives a way to compute the Auslander–Reiten quiver of many algebras of finite-representation type. We also present recent work on the description of the representation theory of gentle and skew-gentle algebras via the geometry of oriented surfaces with boundary. An application of these geometric models is given in the context of τ-tilting theory.

Motor neuroscience centers on characterizing human movement, and the way it is represented and generated by the brain. A key concept in this field is that despite the rich repertoire of human movements and their variability across individuals, both the behavioral and neuronal aspects of movement are highly stereotypical, and can be understood in terms of basic principles or low dimensional systems. Highlighting this concept, this chapter outlines three core topics in this research field: (1) Trajectory planning, where prominent theories based on optimal control and geometric invariance aim at describing end-effector kinematics using basic unifying principles; (2) Compositionality, and specifically the ideas of motor primitives and muscle synergies that account for motion generation and muscle activations, using hierarchical low-dimensional structures; and (3) Neural control models, which regard the neural machinery that gives rise to sequences of motor commands, exploiting dynamical systems and artificial neural network approaches.