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.

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.

Distance problems in discrete geometry include fascinating examples of questions that are easy to state and hard to solve. Three of the best known problems of this type, raised in the 40s, are the Erd\H{o}s Unit Distance Problem, his Distinct Distances Problem, and the Hadwiger-Nelson Problem about the chromatic number of the unit distance graph in the plane. I will describe surprisingly tight recent solutions of the analogs of all three problems for typical norms, settling, in a strong form, questions and conjectures of Matou\v{s}ek, of Brass, of Brass, Moser and Pach, and of Chilakamarri. I will also discuss a related work about ordering points according to the sum of their distances from chosen vantage points. The proofs combine Combinatorial, Geometric and Probabilistic methods with tools from Linear Algebra, Topology, and Algebraic Geometry. Based on recent joint works with Matija Buci\'c and Lisa Sauermann, and with Colin Defant, Noah Kravitz and Daniel Zhu.