Related papers: Hilbert-Schmidt groups as infinite-dimensional Lie…

We derive sub-Riemannian Ricci curvature tensor for sub-Riemannian manifolds. We provide examples including the Heisenberg group, displacement group, and Martinet sub-Riemannian structure with arbitrary weighted volumes, in which we…

This is a preliminary version of a book on infinite-dimensional Lie groups. It covers the basics of calculus and manifolds in the context of locally convex spaces, based on Bastiani's notion of a smooth map. Starting from this concept, we…

We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4-manifold with a non-trivial Seiberg-Witten invariant. These allow one, for example, to exactly compute the infimum…

We introduce a special class of nilpotent Lie groups of step 2, that generalizes the so called $H$(eisenberg)-type groups, defined by A. Kaplan in 1980. We change the presence of inner product to an arbitrary scalar product and relate the…

All known examples of homogeneous Einstein metrics of negative Ricci curvature can be realized as left-invariant Riemannian metrics on solvable Lie groups. After defining a notion of maximal symmetry among left-invariant Riemannian metrics…

We consider the notion of discrete Ricci curvature for graphs defined by Schmuckenschl{\"a}ger \cite{shmuck} and compute its value for Bruhat graphs associated to finite Coxeter groups. To do so we work with the geometric realization of a…

We investigate analytic and geometric implications of non-constant Ricci curvature bounds. We prove a Lichnerowicz eigenvalue estimate and finiteness of the fundamental group assuming that $L+2 Ric$ is a positive operator where $L$ is the…

We generalize the concept of sub-Riemannian geometry to infinite-dimensional manifolds modeled on convenient vector spaces. On a sub-Riemannian manifold $M$, the metric is defined only on a sub-bundle $\calH$ of the tangent bundle $TM$,…

Kinematic algebras can be realised on geometric spaces and constrain the physical models that can live on these spaces. Different types of kinematic algebras exist and we consider the interplay of these algebras for non-relativistic limits…

A "2-group" is a category equipped with a multiplication satisfying laws like those of a group. Just as groups have representations on vector spaces, 2-groups have representations on "2-vector spaces", which are categories analogous to…

We develop a notion of rank one properly convex domains (or Hilbert geometries) in the real projective space. This is in the spirit of rank one non-positively curved Riemannian manifolds and CAT(0) spaces. We define rank one isometries for…

The geometric evolution equations provide new ways to address a variety of non-linear problems in Riemannian geometry, and, at the same time, they enjoy numerous physical applications, most notably within the renormalization group analysis…

This work addresses the questions: (i) Among all left-invariant Riemannian metrics on a given Lie group, is there any whose isometry group or isometry algebra contain that of all others? (ii) Do expanding left-invariant Ricci solitons…

We show that for certain arithmetic groups, geometrically finite subgroups are the intersection of finite index subgroups containing them. Examples are the Bianchi groups and the Seifert-Weber dodecahedral space. In particular, for…

For $1<p<\infty$, we prove the $L^p$-boundedness of the Riesz transform operators on metric measure spaces with Riemannian Ricci curvature bounded from below, without any restriction on their dimension. This large class of spaces include…

We consider Ricci flow on two classes of nilpotent Lie groups that generalize the three-dimensional Heisenberg group: the higher-dimensional classical Heisenberg groups, and the groups of real unitriangular matrices. Each group is known to…

In the infinite dimensional Heisenberg group, we construct a left invariant weak Riemannian metric that gives a degenerate geodesic distance. The same construction yields a degenerate sub-Riemannian distance. We show how the standard notion…

In this paper we study invariant rings arising in the study of finite dimensional algebraic structures. The rings we encounter are graded rings of the form $K[U]^{\Gamma}$ where $\Gamma$ is a product of general linear groups over a field…

This is partly an expository paper, where the authors' work on pseudoriemannian Einstein metrics on nilpotent Lie groups is reviewed. A new criterion is given for the existence of a diagonal Einstein metric on a nice nilpotent Lie group.…

In this paper we study expansions of infinite dimensional Hilbert spaces with a unitary representation of a discrete countable group. When the group is finite, we prove the theory of the corresponding expansion, regardless if it is…