Reproving Friedlander’s inequality with the de Rham complex

An eigenvalue problem for vibrating strings with concave densities

Trace formulas for non-self-adjoint periodic Schrödinger operators and some applications

We prove that for any domain in the Heisenberg group the (k+1)'th Neumann eigenvalue of the sub-Laplacian is strictly less than the k'th Dirichlet eigenvalue. As a byproduct we obtain similar inequalities for the Euclidean Laplacian with a homogeneous magnetic field.

On conformal manifolds of even dimension n ≥ 4 we construct a family of new conformally invariant differential complexes, each containing one coboundary operator of order greater than 1. Each bundle in each of these complexes appears either in the de Rham complex or in its dual (which is a different complex in the non-orientable case). Each of the new complexes is elliptic in case the conformal structure has Riemannian signature. We also construct gauge companion operators which (for differential forms of order k ≤ n/2) complete the exterior derivative to a conformally invariant and (in the case of Riemannian signature) elliptically coercive system. These (operator, gauge) pairs are used to define finite dimensional conformally stable form subspaces which are are candidates for spaces of conformal harmonics. This generalizes the n/2-form and 0-form cases, in which the harmonics are given by conformally invariant systems. These constructions are based on a family of operators on closed forms which generalize in a natural way Branson's Q-curvature. We give a universal construction of these new operators and show that they yield new conformally invariant global pairings between differential form bundles. Finally we give a geometric construction of a family of conformally invariant differential operators between density-valued differential form bundles and develop their properties (including their ellipticity type in the case of definite conformal signature). The construction is based on the ambient metric of Fefferman and Graham, and its relationship to the tractor bundles for the Cartan normal conformal connection. For each form order, our derivation yields an operator of every even order in odd dimensions, and even order operators up to order n in even dimension n. In the case of unweighted (or true) forms as domain, these operators are the natural form analogues of the critical order conformal Laplacian of Graham et al., and are key ingredients in the new differential complexes mentioned above.

In the history of finite elements, the earliest Lagrange finite elements, consisted of scalar-valued functions. To approximate fluxes, vector-valued finite elements with continuous normal (n) components across element interfaces, or n-continuous elements, were developed later. The finite element toolkit was then supplemented by t-continuous vector-valued Nedelec elements with continuous tangential (t) components, now routinely used for Maxwell equations. Although these elements were developed separately, today we understand them together as fitting into a cochain subcomplex of a de Rham complex of Sobolev spaces. Other tensor-valued finite elements are now being viewed with increasing interest and they form the main subject of this talk. The earliest of these consists of matrix-valued functions whose normal-normal (nn) component varies continuously across element interfaces: these are the nn-continuous matrix fields of the Hellan-Herrmann-Johnson element. More recently, nt-continuous matrix-valued finite elements were developed to approximate viscous stress in incompressible flows: they have continuous shear, or normal-tangential (nt) components. To add to this picture, matrix-valued elements with continuous tangential-tangential (tt) components, called Regge elements, are finding increasing utility: they are key to approximating the metric tensor of Riemannian manifolds. This talk delves into the details of these developments. How does one connect these disparate developments with nn-, nt-, and tt-continuous matrix finite elements? This does not appear to be as easy as the previous synthesis of vector-valued elements by the de Rham complex. The spaces in de Rham complexes are connected by fundamental first-order differential operators (grad, curl, and div in three dimensions), all derived from a single definition of the exterior derivative. In contrast, what is natural for the above-mentioned tensor finite elements are other second-order differential operators. We conclude grazing the frontiers of our understanding on potentially unifying connections. This initiative is part of the “Ph.D. Lectures” activity of the project "Departments of Excellence 2023-2027" of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to Ph.D. students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.

We generalize a classical inequality between the eigenvalues of the Laplacians with Neumann and Dirichlet boundary conditions on bounded, planar domains: in 1955, Payne proved that below the k-th eigenvalue of the Dirichlet Laplacian there exist at least $$k + 2$$ eigenvalues of the Neumann Laplacian, provided the domain is convex. It has, however, been conjectured that this should hold for any domain. Here we show that the statement indeed remains true for all simply connected planar Lipschitz domains.

There are two de Rham complexes in diffeology. The original one is due to Souriau and the other one is the singular de Rham complex defined by a simplicial differential graded algebra. We compare the first de Rham cohomology groups of the two complexes within the Čech–de Rham spectral sequence by making use of the factor map which connects the two de Rham complexes. As a consequence, it follows that the singular de Rham cohomology algebra of the irrational torus $T_\theta$ is isomorphic to the tensor product of the original de Rham cohomology and the exterior algebra generated by a non-trivial flow bundle over $T_\theta$.

Let G be a semisimple Lie group with finite center, [Formula: see text] a maximal compact subgroup, and [Formula: see text] a parabolic subgroup. Following ideas of P. Y. Gaillard, one may use G-invariant differential forms on [Formula: see text] to construct G-equivariant Poisson transforms mapping differential forms on G/P to differential forms on G/K. Such invariant forms can be constructed using finite-dimensional representation theory. In this general setting, we first prove that the transforms that always produce harmonic forms are exactly those that descend from the de Rham complex on G/P to the associated Bernstein–Gelfand–Gelfand (or BGG) complex in a well defined sense. The main part of this paper is devoted to an explicit construction of such transforms with additional favorable properties in the case that [Formula: see text]. Thus, G/P is [Formula: see text] with its natural CR structure and the relevant BGG complex is the Rumin complex, while G/K is complex hyperbolic space of complex dimension [Formula: see text]. The construction is carried out both for complex and for real differential forms and the compatibility of the transforms with the natural operators that are available on their sources and targets are analyzed in detail.

In this article, we construct two kinds of de Rham–like complexes which compute the cohomology of complete crystals on the higher-level q -crystalline site, which was introduced in a previous article by the author. One complex is the q -analog of the higher de Rham complex constructed by Miyatani, and another complex is the q -analog of the jet complex constructed by Le Stum–Quirós. The complexes we construct can also be regarded as the higher-level analogs of the q –de Rham complex.

Given a domain $$\Omega \subset \mathbb {R}^n$$ Ω ⊂ R n , the de Rham complex of differential forms arises naturally in the study of problems in electromagnetism and fluid mechanics defined on $$\Omega $$ Ω , and its discretization helps build stable numerical methods for such problems. For constructing such stable methods, one critical requirement is ensuring that the discrete subcomplex is cohomologically equivalent to the continuous complex. When $$\Omega $$ Ω is a hypercube, we thus require that the discrete subcomplex be exact. Focusing on such $$\Omega $$ Ω , we theoretically analyze the discrete de Rham complex built from hierarchical B-spline differential forms, i.e., the discrete differential forms are smooth splines and support adaptive refinements—these properties are key to enabling accurate and efficient numerical simulations. We provide locally-verifiable sufficient conditions that ensure that the discrete spline complex is exact. Numerical tests are presented to support the theoretical results, and the examples discussed include complexes that satisfy our prescribed conditions as well as those that violate them.

Spaces of discrete differential forms can be applied to numerically solve the partial differential equations that govern phenomena such as electromagnetics and fluid mechanics. Robustness of the resulting numerical methods is complemented by pointwise satisfaction of conservation laws (e.g., mass conservation) in the discrete setting. Here we present the construction of isogeometric discrete differential forms, i.e., differential form spaces built using smooth splines. We first present an algorithm for computing Bézier extraction operators for univariate spline differential forms that allow local degree elevation. Then, using tensor-products of the univariate splines, a complex of discrete differential forms is built on meshes that contain polar singularities, i.e., edges that are singularly mapped onto points. We prove that the spline complexes share the same cohomological structure as the de Rham complex. Several examples are presented to demonstrate the applicability of the proposed methodology. In particular, the splines spaces derived are used to simulate generalized Stokes flow on arbitrarily curved smooth surfaces and to numerically demonstrate (a) optimal approximation and inf–sup stability of the spline spaces; (b) pointwise incompressible flows; and (c) flows on deforming surfaces.

We use group cohomology and the de Rham complex on simplicial manifolds to give explicit differential forms representing generators of the cohomology rings of moduli spaces of representations of fundamental groups of 2-manifolds. These generators are constructed using the de Rham representatives for the cohomology of classifying spaces BK where K is a compact Lie group; such representatives (universal characteristic classes) were found by Bott and Shulman. Thus our representatives for the generators of the cohomology of moduli spaces are given explicitly in terms of the Maurer-Cartan form. This work solves a problem posed by Weinstein, who gave a corresponding construction (following Karshon and Goldman) of the symplectic forms on these moduli spaces. We also give a corresponding construction of equivariant differential forms on the extended moduli space X, which is a finite dimensional symplectic space equipped with a Hamiltonian action of K for which the symplectic reduced space is the moduli space of representations of the 2-manifold fundamental group in K.

A classical result in differential geometry states that for a free and proper Lie group action, the quotient map to the orbit space induces an isomorphism between the de Rham complex of differential forms on the orbit space and the basic differential forms on the original manifold. In this paper, this result is generalized to the case of a proper Lie groupoid, in which the orbit space is equipped with the quotient diffeological structure. As an application of this, we obtain a de Rham theorem for the de Rham complex on the orbit space.KeywordsDiffeologyLie groupoidde Rham complexBasic formsLinearizationMathematics Subject Classification (2010)58H0522A22

The purpose of the paper is to promote a new definition of cohomology, using the theory of non commutative differential forms, introduced already by Alain Connes and the author in order to study the relation between K K -theory and cyclic homology. The advantages of this theory in classical Algebraic Topology are the following: A much simpler multiplicative structure, where the symmetric group plays an important role. This is important for cohomology operations and the investigation of a model for integral homotopy types (Formes différentielles non commutatives et opérations de Steenrod, Topology, to appear). These considerations are of course related to the theory of operads. A better relation between de Rham cohomology (defined through usual differential forms on a manifold) and integral cohomology, thanks to a "non commutative integration". A new definition of Deligne cohomology which can be generalized to manifolds provided with a suitable filtration of their de Rham complex. In this paper, the theory is presented in the framework of simplicial sets. With minor modifications, the same results can be obtained in the topological category, thanks essentially to the Dold-Thom theorem (Formes topologiques non commutatives, Ann. Sci. Ecole Norm. Sup., to appear). A summary of this paper has been presented to the French Academy: CR Acad. Sci. Paris 316 (1993), 833-836.

On the geometry of forms on supermanifolds