Min-max methods were introduced in Geometry by Birkhoff in the 1910's.  Recently  the theory has been used to solve several long standing open questions in Geometry and Topology. I will survey these developments and mention some new results.

Generating series of interesting arithmetic functions are often related to modular forms. For instance, representation numbers of quadratic forms are given by coefficients of theta functions, and class numbers of imaginary quadratic fields by certain Eisenstein series. After introducing the basic notions, we discuss analogues of such results in the geometry and arithmetic of moduli spaces. For instance, a famous theorem of Gross, Kohnen, and Zagier states that the generating series of Heegner divisors on a modular curve is an elliptic modular form of weight 3/2 with values in the Picard group. This result can be viewed as an elegant description of the relations among Heegner divisors. More generally, Kudla conjectured that the generating series of codimension g special cycles on an orthogonal Shimura variety of dimension n is a Siegel modular form of genus g and weight 1+n/2 with coefficients in the Chow group of codimension g cycles. We report on recent progress on this and related conjectures.

Many important problem classes are governed by anisotropic features such as singularities concentrated on lower dimensional embedded manifolds, for instance, edges in images or shear layers in solutions of transport dominated equations. While the ability to reliably capture and sparsely represent anisotropic structures is obviously the more important the higher the number of spatial variables is, principal difficulties arise already in two spatial dimensions. Since it was shown that the well-known (isotropic) wavelet systems are not capable of efficiently approximating such anisotropic features, the need arose to introduce appropriate anisotropic representation systems. Among various suggestions, shearlets are the most widely used today. Main reasons for this are their optimal sparse approximation properties within a model situation in combination with their unified  treatment of the continuum and digital realm, leading to faithful implementations. An additional advantage is the availability of stable compactly supported systems for high spatial localization.

In this talk, we will provide an introduction to the anisotropic representation system of shearlets, in particular, compactly supported shearlets, present the main theoretical results, and discuss applications to imaging science and adaptive numerical solution of partial differential equations. Finally, we will present a general framework for systems with optimal sparse approximation properties with respect to anisotropic features, coined parabolic molecules, which in particular includes compactly supported shearlet systems.