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.

A period is a certain kind of complex number which can be written as an integral of algebraic quantities. Kontsevich and Zagier conjectured that all identities between periods can be obtained from the elementary rules of calculus. After discussing several examples I will focus on the case of multiple zeta values which were first introduced in a special case by Euler, and now occur in numerous branches of mathematics. They satisfy many families of relations which are the subject of several open conjectures.

We investigate the behavior of the interface between two fluids. We are interested in the singularities which develop when the bridge connecting two pieces of fluid goes to zero thickness. One physical situation is Hele-Shaw flow: two fluids are trapped between parallel glass plates and feel frictional forces. Another situation is a cylindrically symmetrical stream, for example, in a dripping faucet. In a third case, a fluid in an electric or magnetic field which pulls off a piece of fluid ending in a sharp point. The last case has an interface between a light and a heavier fluid being sucked up as in a drinking straw. At a critical value of the sucking, a very thin bridge of the heavy fluid is formed. Analytical, numerical, and experimental methods are used to describe what happens right around the pinchoff of these bridges. Much of the resulting behavior can be understood via scaling arguments.

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.

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.