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In the study of geometric objects that arise naturally, the main tools are either groups or equations. In the rst case, powerful algebraic methods are available and enable one to solve many deep problems. While algebraic methods are still important in the second case, analytic methods play a dominant role, especially when the dening equations are transcendental. Indeed, even in the situation where the geometric object is homogeneous or algebraic, analytic methods often lead to important contributions. In this talk, we shall discuss a class of problems in differential geometry and the analytic methods that are involved in solving such problems.

Riemannian geometry is characterized, and research is oriented towards and shaped by concepts (geodesics, connections, curvature,...) and objectives, in particular to understand certain classes of (compact) Riemannian manifolds defined by curvature conditions (constant or positive or negative curvature,...). By way of contrast, geometric analysis is a perhaps somewhat less systematic collection of techniques, for solving extremal problems naturally arising in geometry and for investigating and characterizing their solutions. It turns out that the two fields complement each other very well; geometric analysis offers tools for solving difficult problems in geometry, and Riemannian geometry stimulates progress in geometric analysis by setting ambitious goals.

In this paper we discuss the change in contact structures as their supporting open book decompositions have their binding components cabled. To facilitate this and applications we define the notion of a rational open book decomposition that generalizes the standard notion of open book decomposition and allows one to more easily study surgeries on transverse knots. As a corollary to our investigation we are able to show there are Stein fillable contact structures supported by open books whose monodromies cannot be written as a product of positive Dehn twists. We also exhibit several monoids in the mapping class group of a surface that have contact geometric significance.

A longstanding question in the dual Brunn–Minkowski theory is “What are the dual analogues of Federer’s curvature measures for convex bodies?” The answer to this is provided. This leads naturally to dual versions of Minkowski-type problems: What are necessary and sufficient conditions for a Borel measure to be a dual curvature measure of a convex body? Sufficient conditions, involving measure concentration, are established for the existence of solutions to these problems.