The proposed research is located at the intersection of metric geometry and analysis on metric spaces. The principal aim of the project is to gain a deeper understanding of the large scale geometry of various classes of metric spaces through the study of higher-dimensional isoperimetric inequalities. The project's first and main part focuses on simply connected geodesic metric spaces of non-positive curvature in the sense of Alexandrov, called Hadamard spaces. In this setting, a particular goal is to establish connections between the behavior of higher-dimensional isoperimetric functions (and other filling invariants) and various notions of rank. If successful, the research will result in an answer to Gromov's conjecture on linear isoperimetric inequalities which asserts Euclidean behavior in the dimensions below and linear behavior in the dimensions above the Euclidean rank in a proper cocompact Hadamard space. The proposed approach relies on methods from geometric measure theory and analysis on metric spaces, in particular on the notion of metric currents, the theory of which has recently been developed by Ambrosio and Kirchheim. An important second part of the project will thus be to further develop the theory of metric currents, which provides a suitable notion of surfaces in a metric space and also gives powerful tools in the study of many other problems in geometric analysis, e.g. such involving energy minimization. Isoperimetric inequalities also play a fundamental role in geometric group theory. For finitely presented groups one-dimensional isoperimetric functions measure the complexity of the word problem and have been extensively studied over the past 15 years. The study of higher-dimensional isoperimetric functions, on the other hand, has only recently become a very active field of research. A last part of the project investigates filling problems in the context of nilpotent Lie groups.
The study of isoperimetric problems has a long history. Not only has it inspired many mathematicians, but it has also led to many new theories in mathematics. In the most classical setting-known already to the Ancient Greeks-the isoperimetric inequality asserts that a closed curve in the plane encloses an area no larger than that of a disc of the same circumference. Higher-dimensional analogues are concerned with the question of how well a k-dimensional closed surface can be filled with a (k+1)-dimensional surface. Filling problems appear in many different fields of mathematics, including analysis, geometry, probability theory and group theory. The aim of the present project is to study the effects that curvature has on the isoperimetric filling problem. The investigation will in particular lead to a deeper understanding of the geometry of non-positively curved singular spaces. Such arise for example in the mathematical study of billiard trajectories. The project proposes to use methods from the field of geometric measure theory, dealing with the study of singular surfaces. This theory was to a large extent inspired by Plauteau's problem, which asks for the existence of a minimal surface (or a soap film) with prescribed boundary. Higher-dimensional analogues of this question were the starting point in the development of the theory of currents, which nowadays has many applications going far beyond Plateau's problem.
