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.

The wrinkling of thin elastic sheets is very familiar: our skin wrinkles, drapes have coarsening folds, and a sheet stretched over a round surface must wrinkle or fold.

What kind of mathematics is relevant? The stable configurations of a sheet are local minima of a variational problem with a rather special structure, involving a nonconvex membrane term (which favors isometry) and a higher-order bending term (which penalizes curvature). The bending term is a singular perturbation; its small coefficient is the sheet thickness squared. The patterns seen in thin sheets arise from energy minimization -- but not in the same way that minimal surfaces arise from area minimization. Rather, the analysis of wrinkling is an example of "energy-driven pattern formation," in which our goal is to understand the asymptotic character of the minimizers in a suitable limit (as the nondimensionalized sheet thickness tends to zero).

What kind of understanding is feasible? It has been fruitful to focus on how the minimum energy scales with sheet thickness, i.e. the "energy scaling law." This approach entails proving upper bounds and lower bounds that scale the same way. The upper bounds tend to be easier, since nature gives us a hint. The lower bounds are more subtle, since they must be ansatz-free; in many cases, the arguments used to prove the lower bounds help explain "why" we see particular patterns. A related but more ambitious goal is to identify the prefactor as well as the scaling law; Ian Tobasco's striking recent work on geometry-driven wrinkling has this character.

Lecture 1 will provide an overview of this topic (assuming no background in elasticity, thin sheets, or the calculus of variations). Lecture 2 will discuss some examples of tensile wrinkling, where identification of the energy scaling law is intimately linked to understanding the local length scale of the wrinkles. Lecture 3 will discuss our emerging undertanding of geometry-driven wrinkling, where (as Tobasco has shown) it is the prefactor not the scaling law that explains the patterns seen experimentally.

After the recent exciting progress in understanding the geometry of algebraic varieties over the complex numbers, it is natural to try to understand the geometry of varieties over an algebraically closed field of characteristic $p>0$. Many technical issues arise in this context. Nevertheless, there has been much recent progress. In particular, the MMP was established for 3-folds in characteristic $p>5$ by work of Birkar, Hacon, Xu and others. In this talk we will discuss some of the challenges and recent progress in this active area.

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.

The wrinkling of thin elastic sheets is very familiar: our skin wrinkles, drapes have coarsening folds, and a sheet stretched over a round surface must wrinkle or fold.

What kind of mathematics is relevant? The stable configurations of a sheet are local minima of a variational problem with a rather special structure, involving a nonconvex membrane term (which favors isometry) and a higher-order bending term (which penalizes curvature). The bending term is a singular perturbation; its small coefficient is the sheet thickness squared. The patterns seen in thin sheets arise from energy minimization -- but not in the same way that minimal surfaces arise from area minimization. Rather, the analysis of wrinkling is an example of "energy-driven pattern formation," in which our goal is to understand the asymptotic character of the minimizers in a suitable limit (as the nondimensionalized sheet thickness tends to zero).

What kind of understanding is feasible? It has been fruitful to focus on how the minimum energy scales with sheet thickness, i.e. the "energy scaling law." This approach entails proving upper bounds and lower bounds that scale the same way. The upper bounds tend to be easier, since nature gives us a hint. The lower bounds are more subtle, since they must be ansatz-free; in many cases, the arguments used to prove the lower bounds help explain "why" we see particular patterns. A related but more ambitious goal is to identify the prefactor as well as the scaling law; Ian Tobasco's striking recent work on geometry-driven wrinkling has this character.

Lecture 1 will provide an overview of this topic (assuming no background in elasticity, thin sheets, or the calculus of variations). Lecture 2 will discuss some examples of tensile wrinkling, where identification of the energy scaling law is intimately linked to understanding the local length scale of the wrinkles. Lecture 3 will discuss our emerging undertanding of geometry-driven wrinkling, where (as Tobasco has shown) it is the prefactor not the scaling law that explains the patterns seen experimentally.

In the first lecture I will give a brief discussion of local and non local diffusion and degenerate ellipticity and different local and non local models for compressible flows in porous media.
In the second and third lectures I will discuss some properties of the (infinitesimal) porous media equation, a non local in space model and equations with memory.

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.

A smooth one-parameter family F₀ : Mⁿx [0,T) ---> (Nⁿ⁺¹,g) of hypersurfaces in a Riemannian manifold (N⁽ⁿ⁺¹⁾,g) is said to move by its curvature if it satisfies an evolution equation of the form
(d/dt) F(p,t) = f(p,t) p Mⁿ, t [0,T),
such that at each point of the surface its speed in normal direction is a function $f$ of the extrinsic curvature of the hypersurface. Examples such as the flow by mean curvature, flow by Gauss curvature or flow by inverse mean curvature arise naturally both in Differential Geometry, where they exhibit fascinating interactions between the extrinsic curvature of the surfaces and intrinsic geometric properties of the ambient manifold, and in Mathematical Physics, where they serve as models for the evolution of interfaces in phase transitions. The first lecture gives a general introduction to the main examples and phenomena and highlights some recent results. The second lecture shows how parabolic rescaling techniques can be combined with a priori estimates to study and in some cases classify possible singularities of the mean curvature flow. The series concludes with applications of hypersurface families in General relativity, including a recent proof of an optimal lower bound for the total energy of an isolated gravitating system by Huisken and Ilmanen.

In this series of talks, I will focus on geometry and analysis of manifolds of dimension 2, 3 or 4. The first talk is a general introduction of this series. I will start the talk by reviewing some classical theories on Riemann surfaces and their recent variations in geometric analysis. Then we survey some recent progress on 3- and 4-manifolds. I hope that this talk will show some clues how geometric equations can be applied to studying geometry of underlying spaces. In the second talk, I will discuss recent works on the Ricci flow and its application to the geometrization of 3-manifolds, in particular, I will briefly discuss Perelman's work towards the Poincare conjecture. In last talk, I will discuss geometric equations in dimension 4 and how they can be applied to studying geoemtry of underlying 4-spaces. Some recent results will be discussed and some open problems will be given.