## Harmonic and Geometric Analysis

The Harmonic and Geometric Analysis group of the Department of Mathematics and Applications presents a considerable spectrum of research interests between Harmonic Analysis and Global Analysis.

As regards Harmonic Analysis, research topics include classical Harmonic Analysis, Representation Theory of discrete groups, the analysis of a wide variety of problems in Global Analysis and Geometric Measurement Theory on Riemannian varieties, the analysis of sub-laplacians on nilpotent groups and CR varieties, the analysis of fine phenomena related to representations of semi-simple groups, discrepancy theory, the convergence of developments in autofunctions of elliptic differential operators on compact varieties, the study of the Fourier transform on the Heisenberg group. Although the intersections between the research undertaken by the members of the group have empty intersections, the fundamental method of Harmonic Analysis, which consists, in the wake of the tradition opened up by Fourier in the study of heat diffusion, in analysing objects of an analytical nature by breaking them down into simpler objects that are better suited to being studied, and then deducing properties of the original objects, is present in all the above-mentioned strands of research, and constitutes the essential element of cohesion of the group.

It is important to mention that current research also uses methods from Algebra and Geometry: this is evident in research involving the analysis of varieties, where analytical and geometrical methods act synergistically, or in research involving the analysis of hyperbolic groups where analytical methods and algebraic properties are used in conjunction. Mention must also be made of research involving some of us, concerning the analysis of a classical problem, which consists in estimating the number of integer points of a domain of Euclidean space in dependence on a parameter describing its dilatation, and the analogous problem in the geometric domain, which consists in drawing a "good" distribution of a discrete set of points on a Riemannian variety.

The group has been enriched by the recent addition of several members whose research activity focuses more specifically on questions of Global Analysis and Geometric Measurement Theory. Geometric Analysis can be thought of as a collection of analytical techniques developed to deal geometric and topological problems. Typically, these problems are formulated on a smooth space equipped with a geometric structure (e.g. Riemannian) and involve functions of geometric content that are subject to systems of equations or, more generally, differential inequations, very frequently non-linear. In more recent years, the approach of Geometric Analysis has been broadly extended to include singular spaces, important for example as limits of smooth spaces for appropriate notions of convergence. These spaces are studied both intrinsically (Metric Geometry) and, if they appear as non-regular subsets of an ambient space, from an extrinsic point of view (Geometric Measurement Theory). The extension to general geometric contexts (smooth and singular) of the typical tools of analysis in Euclidean spaces has opened the way to Global Analysis, which, by reversing the point of view, is interested in studying how the geometry of the underlying space, for example its curvature, influences the qualitative and quantitative global properties of classical analytical objects from the sphere of differential equations.

Topics:

Harmonic Analysis

• Classical Harmonic Analysis
• Reticles representations of Lie groups
• Analysis of sub-laplaciani on nilpotent groups and CR varieties
• Representations of semi-simple groups
• Discrepancy theory
• Developments in autofunctions of elliptic differential operators on compact varieties
• Fourier transform on the Heisenberg group

Geometric Analysis

• Differential (Dis)equations and integral inequalities on Riemannian varieties
• Geometric measurement theory
• Harmonic and p-harmonic maps: regularities and Dirichlet problems
• Metric geometry: surfaces with bounded integral curvature (BIC) and Alexandrov spaces