The theory of elliptic PDEs stands apart from many other areas of PDEs because sharp results are known for very general elliptic PDEs (linear, fully nonlinear, boundary value problems, …). Many of the classical techniques from harmonic analysis were first developed to prove these sharp results; and the study of elliptic PDEs is intimately intertwined with the Fourier transform and Riemannian geometry.
 
Starting with the work of Hörmander, Kohn, Folland, Stein, and Rothschild in the 60s and 70s, a far-reaching generalization of ellipticity was introduced: now known as maximal hypoellipticity. A wide program was initiated to generalize the elliptic theory to these possibly much more degenerate partial differential operators.
 
Where elliptic operators are connected to Riemannian geometry, maximally hypoelliptic operators are connected to Carnot-Carathéodory geometry. The Fourier transform is no longer a central tool but can often be replaced with more modern tools from harmonic analysis.
 
In this talk we describe some results regarding general maximally hypoelliptic PDEs, viewed through the lens of elliptic PDEs and classical harmonic analysis.

We define a systematic way of regularising a sub-Riemannian structure by a sequence of Riemannian manifolds and we study the associated Laplace operators. We also apply this method to other related singular operators.

Sternin and Shatalov introduced for a closed submanifold of N of M a calculus to study so-called relative elliptic problems. Their calculus is generated by pseudodifferential operators on both manifolds, the restriction operator from M to N and the extension operator from N to M. In this talk, I will explain a geometric construction of this calculus using groupoids. Following the work of van Erp and Yuncken, the operators of the calculus can be characterized using essentially homogeneous distributions on appropriate deformation spaces. One advantage of this approach is that it is also applicable to filtered manifolds and we obtain new results towards the study of relative problems for hypoelliptic operators.

In recent years there was a significant progress in the theory of pseudo-differential operators on filtered manifolds. In my talk I will introduce a wider class of manifolds which I call manifolds with a tangent Lie structure. I will explain an approach to pseudo-differential theory which gives a simplified (coarse) pseudo-differential calculus containing only operators of order 0 and negative order. This coarse calculus easily leads to the Atiyah-Singer type index theorem for operators of order 0 on manifolds with a tangent Lie structure. In this approach the index is an element of the K-homology group of the manifold. The coarse calculus agrees with the known Hörmander and van Erp - Yuncken calculi, which allows to extend the index theorem to operators of any order on filtered manifolds.

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In the pseudodifferential calculus developped for filtered manifolds (e.g. subriemannian) the symbols appear as convolution operators on the so called osculating tangent groups. Even if they share a lot of properties with symbols in the usual pseudodifferential calculus (linerarity, product, vanishing conditions...) their product is not commutative anymore. We can however show that the C*-algebra of order zero principal symbols is solvable. This means that we can decompose the algebra into a finite sequence of nested ideals for which every subquotient is a commutative algebra of functions (tensored with compact operators). This decomposition generalizes a result of Epstein and Melrose in the case of contact manifolds. I will explain how to obtain such a decomposition, using the representation theory of nilpotent groups.

We consider hypersurfaces in contact sub-Riemannian manifolds as well as in quasi-contact sub-Riemannian manifolds. We analyse the structures canonically induced on such hypersurfaces and we study different types of singular points which can arise. We further discuss model cases for these settings.

THE TALK IS ON ZOOM (please contact organizers for the zoom link)

Principal symbol mapping on a Heisenberg group is a *-homomorphism from the C*-algebra generated by multiplication operators and Heisenberg-Riesz transforms. Its co-domain is the minimal tensor product of C_0-functions and the C*-algebra generated by the Heisenberg-Riesz transforms. Contact manifold can be equivalently defined as the one equipped with atlas with transition mappings being Heisenberg diffeomorphisms. Principal symbol on the Heisenberg group behaves equivariantly with respect to Heisenberg diffeomorphisms. This allows to define a principal symbol mapping on contact manifold as *-homomorphism of C*-algebras. A version of Connes trace formula for sub-Riemannian contact manifolds is presented as an application. This is a joint work with Yuri Kordyukov and Fedor Sukochev (published in Lecture Notes in Mathematics).

One of earliest developments in Noncommutative Geometry was the introduction of Connes' integral, the definition of the integral of a smooth function with respect to the Lebesgue measure  in terms  of the singular trace of a certain operator associated with this function. It turned out later that this result is a particular case of earlier considerations by Birman-Solomyak about eigenvalue asymptotics of weighted polyharmonic operators. We discuss recent developments on eigenvalue properties for pseudodifferential operators  involving singular weights, which leads to the extension of  Connes' approach to integration with respect to some classes of singular measures.

Classical pseudodifferential operators on a manifold can be characterised via convolution kernels on Connes’ tangent tangent groupoid which are homogeneous, modulo smooth kernels, for the zoom action of Debord-Skandalis. The same holds for the Heisenberg calculus on a contact manifold or (something akin to) Melin’s calculus for a Lie filtered manifold if one replaces the tangent bundle in the tangent groupoid by the bundle of osculating nilpotent Lie groups.  For more singular situations, such as appear in the Helffer-Nourrigat conjecture or subriemannian geometry, the osculating groups need to be replaced by osculating groupoids due to Mohsen. We will explain the structure of these groupoids and the associated pseudodifferential calculi. (Joint work with I. Androulidakis and O. Mohsen.)

Androulidakis, Mohsen and Yuncken defined a pseudodifferential calculus adapted to a filtration of the tangent bundle of a manifold. I will explain how to prove that order zero operators in their calculus are bounded on Lp for 1<p<infinity. This is a special case of a more general result of Brian Street, but the nature of the operators in the Androulidakis-Mohsen-Yuncken calculus enables a short and conceptual proof. Based on arXiv:2410.13701. I will also discuss issues related to Gårding's inequality.

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We generalize previous work on X-ray tomography in sub-Riemannian geometry. In particular, we show that an integrable function on an H-type Lie group is determined by its integrals over sub-Riemannian geodesics. Our argument uses the group Fourier transform to block-diagonalize the X-ray transform. We also describe partial data results and observe improvements when the codimension of the sub-Riemannian distribution is greater than one. These results provide explicit answers to injectivity in a class of geometries with abundant conjugate points and mixed curvature.

For a domain in M, the relative heat content is defined as the total amount of heat contained in the domain at time t, allowing the heat to flow outside the domain. We study the small-time asymptotics of the relative heat content associated with smooth non-characteristic domains of a general rank-varying sub-Riemannian structure, equipped with an arbitrary smooth measure. We establish the existence of an asymptotic series, up to order 4. Significant difficulties emerges, as the boundary behavior of the temperature function is not known: we use an “asymptotic” symmetry argument of the heat diffusion to obtain information on the small-time behavior of temperature at the boundary of the domain. This is a joint work with A. Agrachev and L. Rizzi.

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We consider the linear wave equation where the Laplacian is replaced by a sub-Laplacian, which is an hypoelliptic operator. We discuss the propagation of singularities in such equations : the main new phenomenon that we describe is that singularities can propagate along abnormal curves at any speed between 0 and 1. This general result extends an idea due to R. Melrose, and we then illustrate it on an example, the Martinet case, following a joint work with Y. Colin de Verdière. Our statements are part of a classical/quantum correspondance between sub-Riemannian geometry (on the classical side) and hypoelliptic operators (on the quantum side).

We study the semiclassical spectral asymptotics of the degenerate second-order differential operator \nabla D(x)\nabla in a planar domain \Omega in the plane, where D(x) vanishes on the boundary. The operator is defined on L2 as the Friedrichs extension of the associated minimal operator. Focusing on a nearly integrable case, we apply the uniformization method developed by V. E. Nazaikinskii and collaborators, which resolves the degeneracy by lifting the problem to a higher-dimensional configuration space. This geometric approach yields asymptotic spectral series, including eigenvalues and quasimodes, with explicit formulas involving Airy and Bessel functions. The method is not only computationally effective but also extends naturally to a broader class of hypoelliptic operators in geometric analysis.

The Steklov problem is the spectral problem for the Dirichlet-to-Neumann map (DtN) for the Laplacian–an operator which appears, for example, in the Calderon problem and the Sloshing problem. Motivated by the Steklov problem on higher dimensional, piecewise smooth Lipschitz domains, we consider the massive DtN map (i.e. the DtN map for -\Delta +M^2) on infinite, asymptotically conic, domains. We develop a scattering theory for the DtN map by proving a limiting absorption principle for this operator and construct the associated scattering matrix. Using this as a tool, we describe the spectrum of the massive DtN on a piecewise smooth, bounded, Lipschitz domain in two dimensions. Based on joint work with Jeffrey Galkowski and Ruoyu P. T. Wang.

In this talk we will present some recent results on functional calculus for certain subelliptic operators on nilpotent Lie groups and their nilmanifolds. As an application of the theory, we also obtain their semiclassical Weyl laws. This is joint work with Véronique Fischer.

In this talk, we will discuss the natural construction of a pseudo-differential calculus on filtered manifolds with symbols given in terms of the representations of the nilpotentization. In particular, we will see that the sub-calculus corresponding to poly-homogeneous symbols coincides with the calculus obtained from the groupoid approach.

Semiclassical analysis developed in the 1970’s in the Euclidean setting as an elegant manner to approach highly oscillating phenomena, in particular those described by solutions of partial differential equations. In this talk, we will explain how these ideas can be adapted in the non-commutative context of filtered manifolds, starting with graded Lie groups and nilmanifolds. We will focus on the semiclassical notion of coherent states and semiclassical measures, bringing them in a noncommutative context, developing examples and describing some of their properties.

The Schrödinger equation on the Heisenberg group is an example of a totally non-dispersive evolution equation: for this reason the classical approach that permits to obtain Strichartz estimates from dispersive estimates is not available. We obtain refined Strichartz estimates for the sub-Riemannian Schrödinger equation on Heisenberg and H-type Carnot groups reinterpreting Strichartz estimates as Fourier restriction theorems for noncommutative nilpotent groups. The same arguments permits to obtain refined Strichartz estimates for the wave equation.

We give a sharp estimate for the bottom of the spectrum of some negatively curved sub-Riemannian manifolds. The main ingredients are a generalization of the Cartan-Hadamard theorem and a sub-Laplacian comparison theorem.

We discuss isoperimetric inequalities for three analogues of the euclidean Gaussian measure on a (stratified) Lie group. Namely,  heat kernel of the sub-Laplacian, Gaussian measure associated with the Carnot-Carathéodory metric, and Gaussian measure associated with the Koranyi-Folland gauge. We will mostly focus on spectral theoretic methods but we shall also discuss a connection with some curvature dimension condition.

The Engel group is the lowest dimensional nilpotent Lie group of step 3. One can consider its natural subLaplacian and ask about the dispersive properties of the associated Schrödinger equation. These questions have already been considered in the case of nilpotent Lie groups of step 2: dispersive estimates are known and Strichartz inequalities have been proven in particular cases. One observes that such properties is best understood when using the natural Fourier theory of the group under study and are closely related to its subRiemannian structure. In this presentation, we will present a semiclassical pseudodifferential calculus adapted to the Engel group and in particular a two-microlocal approach, developped to give more insights on the dispersive nature of its subLaplacian. In particular, we will prove how singular geodesics on the Engel group can lead to obstruction to smoothing-type estimates, as well as obstruction to some family of Strichartz estimates.

The measure contraction property (MCP in short, introduced by Ohta and Sturm) is the only “classical” synthetic Ricci bound that can be satisfied by sub-Riemannian structures. It is related with weak quantitative estimates for the sub-Laplacian of the squared sub-Riemannian distance. Thanks to several contributions in the past 15 years, the qualitative picture about the validity of this property is well-understood for Carnot groups without non-trivial Goh abnormals. We present a work in collaboration with Samuel Borza, showing that MCP can fail in the presence of non-trivial Goh abnormals.

A (2,3,5) distribution is a maximally non-integrable tangent 2-plane field on a 5-manifold. These distributions are also known as generic rank two distributions in dimension five. They can equivalently be characterized as regular normal parabolic geometries of type (G_2,P) where G_2 denotes the split real form of the exceptional Lie group and P is a particular parabolic subgroup. The Rumin complex associated with a (2,3,5) distribution is a Rockland complex, the analogue of an elliptic complex in the Heisenberg calculus. In this talk we focus on the analytic torsion and the eta invariant of this Rumin complex, twisted by a unitary flat vector bundles. These spectral invariants are sub-Riemannian analogues of the Ray-Singer analytic torsion an the eta invariant of the odd signature operator. We present recent computations using harmonic analysis which lead to comparison results for (2,3,5) nilmanifolds.

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