## Felix Otto : Gergen Lecture - Speaker, Felix Otto

- Gergen Lectures ( 407 Views )In three specific examples, we shall demonstrate how the theory of partial differential equations (PDEs) relates to pattern formation in nature: Spinodal decomposition and the Cahn-Hilliard equation, Rayleigh-B\'enard convection and the Boussinesq approximation, rough crystal growth and the Kuramoto-Sivashinsky equation. These examples from different applications have in common that only a few physical mechanisms, which are modeled by simple-looking evolutionary PDEs, lead to complex patterns. These mechanisms will be explained, numerical simulation shall serve as a visual experiment. Numerical simulations also reveal that generic solutions of these deterministic equations have stationary or self-similar statistics that are independent of the system size and of the details of the initial data. We show how PDE methods, i. e. a priori estimates, can be used to understand some aspects of this universal behavior. In case of the Cahn-Hilliard equation, the method makes use of its gradient flow structure and a property of the energy landscape. In case of the Boussinesq equation, a ``driven gradient flow'', the background field method is used. In case of the Kuramoto-Sivashinsky equation, that mixes conservative and dissipative dynamics, the method relies on a new result on Burgers' equation.

## Andrei Zelevinsky : Cluster algebras via quivers with potentials

- Gergen Lectures ( 283 Views )This lecture ties together the strands developed in the first two lectures. We discuss a recent proof (due to H. Derksen, J. Weyman, and the speaker) of a series of conjectures on cluster algebras by means of the machinery of quivers with potentials. An important ingredient of our argument is a categorification of cluster algebras using quiver Grassmannians, a family of projective algebraic varieties that are a far-reaching generalization of ordinary Grassmannians. Generalizing an idea due to P. Caldero, F. Chapoton and B. Keller, we show that the Euler characteristics of these varieties carry crucial information about the structure of cluster algebras.

## Andrei Zelevinsky : Quivers with potentials, their representation and mutations

- Gergen Lectures ( 272 Views )A quiver is a finite directed graph. A quiver representation assigns a finite-dimensional vector space to each vertex, and a linear map between the corresponding spaces to each arrow. A fundamental role in the theory of quiver representations is played by Bernstein-Gelfand-Ponomarev reflection functors associated to every source or sink of a quiver. In joint work with H. Derksen and J. Weyman (based on an earlier joint work with R. Marsh and M. Reineke) we extend these functors to arbitrary vertices. This construction is based on a framework of quivers with potentials; their representations are quiver representations satisfying relations of a special kind between the linear maps attached to arrows. The motivations for this work come from several sources: superpotentials in physics, Calabi-Yau algebras, and cluster algebras. However, no special knowledge will be assumed in any of these subjects, and the exposition aims to be accessible to graduate students.