OpART workshop 2026 -- Saint-Dié-des-Vosges

26 mai 2026, 09:00 → 29 mai 2026, 13:00 Europe/Paris

Workshop for the members of the ANR project OpART and their collaborators.

Representation theory, noncommutative geometry, Lie groupoids, and their interactions

Mini-course by Brian Street on Subelliptic PDEs and singular integrals.  (Slides available at the bottom of this page.)

The goal of the workshop is to have a small number of talks, particularly by younger participants, and a large amount of free time for discussion and collaboration.

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Where : IUT Saint-Dié, at Saint-Dié-des-Vosges, 1h from Nancy.

Dates : 26-29 May 2026.

The conference will begin Tuesday 26/05 at 9h00, and finish Friday 29/05 at 12h00.  Participants can arrive on Monday 25/05 - note that it's a public holiday.

Talks will take place in room SC1 of the IUT Saint-Dié des Vosges. [map]

The conference dinner is Wednesday evening, 19h at Le Square. [map]

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Organisateurs : Bob Yuncken & Alexandre Afgoustidis

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Slides from Brian Street's mini-course on Singular Integral Operators

mar. 26 mai

1 Power counting in noncommutative Quantum Field Theory

I consider Callias operators (that is, Dirac-Schrödinger operators perturbed by an operator-valued potential $A$) on odd, $d$-dimensional hyperbolic space. Without imposing uniform invertibility of the potential outside a compact region, the operators are no longer Fredholm, so that the Witten index, a type of regularized index defined via an iterated trace limit, becomes the natural substitute for the classical notion of index.

An explicit trace formula, which leads to a new index formula, has already been found in the Euclidean case, in dimension $d=1$ by Pushnitski, Gesztesy et al. and others, and recently, by Fürst, for $d \geq 3$.
My approach combines Fourier-Helgason analytic methods on symmetric spaces with Volterra-type constructions of heat kernels, adapted to the curved setting of hyperbolic space. In this talk, I present an explicit trace formula, which constitutes the first of its kind in a multi-dimensional, non-Euclidean setting.

Orateur: Eva-Maria Hekkelman (Max Planck Institute, Bonn)

10:30 Coffee

2 Revisiting Singular Integrals: Applications to Subelliptic PDEs (Part I)

Since work of Hilbert in 1908, singular integrals have played an important role in the study of PDEs. In the 1960s, Kohn and Nirenberg (building on work of Seeley, Unterberger, Bokobza) introduced pseudodifferential operators, which simplified many arguments that relied on singular integral operators. Since then, pseudodifferential operators have been used to great effect in the study of PDEs; especially elliptic PDEs. Because of their powerful algebraic structure, pseudodifferential operators have largely replaced singular integral operators in many applications.

However, there are modern settings where pseudodifferential operators do not apply and singular integrals once again become a central tool.
In this mini-course we:
• Define singular integral operators with motivating examples from Euclidean space.
• Describe function spaces from harmonic analysis and their connections with singular integral operators.
• Present maximally subelliptic PDEs as a case study where singular integrals give sharp results and standard pseudodifferential operators do not.
• Show how singular integrals allow us to not only study linear subelliptic PDEs, but also fully nonlinear subelliptic PDEs, subelliptic PDEs with rough coefficients, and subelliptic boundary value problems.

This course is designed for graduate students with some functional analysis (L^p spaces, Fourier transform). Some familiarity with distributions and pseudodifferential operators will be useful, but not necessary.

Orateur: Brian Street (University of Wisconsin)

3 The Satake groupoid for Riemannian symmetric spaces and beyond

With Nigel Higson and Robert Yuncken, we have constructed a groupoid for a Riemannian symmetric space, or $G/K$ where $G$ is a real reductive group with maximal compact subgroup $K$, based on a general construction due to Omar. Geometrically, it is formed as a smooth family of homogeneous spaces $G/H$, where $H$ ranges over all maximal compact subgroups (conjugates of $K$) and their "limit groups", which in particular contain the unipotent part of a parabolic subgroup (so that a limit homogeneous space looks roughly like $G/N$, the domain relevant to the principal series). This is related to the so-called "boundary degenerations" that appear in the study of real spherical varieties (related to Harish-Chandra's theory of the constant term). The key advantage is that this nice geometric object has a (Lie) groupoid structure, and so has a corresponding $C^*$ algebra. The construction is aimed to assist in applying $C^*$-algebraic techniques to representation theory. While studying a groupoid instead of a group may at first seem to make the problem harder, in fact the groupoid turns out to be very simple both geometrically and algebraically, and this groupoid gets at the geometric core of Harish-Chandra's "philosophy of cusp forms". Similar results about limit groups are known for more general non-Riemannian spaces, where $K$ is replaced by an open subgroup of the fixed points of an involution. This suggests possible generalizations to this case; we shall discuss progress made in this direction near the end of the talk.

Orateur: Jacob Bradd (Université de Lorraine, Metz)

16:00 Coffee

4 Seiberg-Witten equations on Hermitian symmetric spaces, part II

The Seiberg-Witten equations, originally formulated on a 4-dimensional manifold as the minima of a gauge-theoretic energy functional, have led to profound results regarding the differential structure of the manifold. In this second part, I will explain how representation theory can be used to formulate, generalise and solve these equations on Hermitian symmetric spaces and outline the idea behind the aforementioned results.

Orateur: Rafailia Tsiavou (University of Thessaloniki)

5 On the algebraic Mackey deformation

Let $G$ be a real reductive group with Lie algebra $\mathfrak{g}$ and let $K \subset G$ be a maximal compact subgroup.
The heart of the algebraic representation theory of $G$ consists in the study of $(\mathfrak{g},K)$-modules. The latter correspond to modules over a certain approximately unital ring $R$, known as the Hecke algebra of the pair $(\mathfrak{g},K)$. The contraction of $G$ onto its motion group $G_0$ naturally defines an algebraic family of rings $(R_t)_t$ over the affine line, with $R_t = R$ if $t\neq 0$ and $R_0$ being the Hecke algebra of the pair $(\mathfrak{g}_0,K)$ defined by the Cartan motion group. In this talk, we present a recent result showing that this algebraic family of rings is, in a sense, piecewise constant. We will then explain consequences for the continuity of the Mackey bijection.

Orateur: Yvann Gaudillot-Estrada (Université de Lorraine, Metz)

mer. 27 mai

6 Revisiting Singular Integrals: Applications to Subelliptic PDEs (Part II)

Since work of Hilbert in 1908, singular integrals have played an important role in the study of PDEs. In the 1960s, Kohn and Nirenberg (building on work of Seeley, Unterberger, Bokobza) introduced pseudodifferential operators, which simplified many arguments that relied on singular integral operators. Since then, pseudodifferential operators have been used to great effect in the study of PDEs; especially elliptic PDEs. Because of their powerful algebraic structure, pseudodifferential operators have largely replaced singular integral operators in many applications.

However, there are modern settings where pseudodifferential operators do not apply and singular integrals once again become a central tool.
In this mini-course we:
• Define singular integral operators with motivating examples from Euclidean space.
• Describe function spaces from harmonic analysis and their connections with singular integral operators.
• Present maximally subelliptic PDEs as a case study where singular integrals give sharp results and standard pseudodifferential operators do not.
• Show how singular integrals allow us to not only study linear subelliptic PDEs, but also fully nonlinear subelliptic PDEs, subelliptic PDEs with rough coefficients, and subelliptic boundary value problems.

This course is designed for graduate students with some functional analysis (L^p spaces, Fourier transform). Some familiarity with distributions and pseudodifferential operators will be useful, but not necessary.

Orateur: Brian Street (University of Wisconsin)

10:30 Coffee

7 High-frequency analysis of PDEs and pseudo differential calculus

In this talk, we will discuss some traditional questions from PDEs on manifolds and explain the role of pseudo differential calculus in the manner they are solved. We will explain how to extend this approach on filtered manifolds and give example on contact manifolds and nilmanifolds.

Orateur: Clotilde Fermanian (Université d'Angers)

8 Log-polyhomogeneous pseudodifferential operators on filtered manifolds and their spectral asymptotics

A classical theorem of Euler is that a function $f$ on $\mathbb{R}^d$ is homogeneous of degree $m$ if and only if $Ef=mf,$ where $E$ is the Euler vector field. If we relax this to $Ef=mf+g,$ where $g$ is Schwartz-class, we say that $f$ is approximately homogeneous. Classical pseudodifferential operators have a symbol function $\sigma(x,\xi)$ which admits an asymptotic expansion into a series $\sum_{n=0}^{\infty} \sigma_{n}(x,\xi),$ where $\sigma_n(x,\xi)$ is homogeneous in its second argument. Van Erp and Yuncken observed that polyhomogeneous functions can be characterised by their having extension to an approximately homogeneous function on a higher-dimensional space. Similarly the Schwartz kernels of classical pseudodifferential operators can be characterised by their having approximately homogeneous extension to the tangent groupoid. If we relax the condition $(E-m)f = 0$ to $(E-m)^k f=0$ for some $k\geq 1,$ we get the class of log-homogeneous functions. There is a corresponding theory of log-polyhomogeneous pseudodifferential operators and via the tangent groupoid we can say some things about their spectral theory.

Orateur: Edward McDonald (Universität Bonn / Université Paris Cité)

16:00 Coffee

9 A regularity theorem for fully nonlinear maximally subelliptic PDEs

Maximally subelliptic PDEs are a far-reaching generalization elliptic PDEs, and they enjoy many desirable properties analogous to elliptic PDEs. In this talk, we will discuss a sharp regularity theorem for a general fully nonlinear maximally subelliptic PDE in Sobolev spaces and recovering the regularity theorem for fully nonlinear elliptic PDEs as a special case.

Orateur: Gautam Neelakantan Memana (University of Wisconsin)

10 Trace and Index of Callias Operators on Hyperbolic Space

I consider Callias operators (that is, Dirac-Schrödinger operators perturbed by an operator-valued potential $A$) on odd, $d$-dimensional hyperbolic space. Without imposing uniform invertibility of the potential outside a compact region, the operators are no longer Fredholm, so that the Witten index, a type of regularized index defined via an iterated trace limit, becomes the natural substitute for the classical notion of index.

An explicit trace formula, which leads to a new index formula, has already been found in the Euclidean case, in dimension $d=1$ by Pushnitski, Gesztesy et al. and others, and recently, by Fürst, for $d \geq 3$.

My approach combines Fourier-Helgason analytic methods on symmetric spaces with Volterra-type constructions of heat kernels, adapted to the curved setting of hyperbolic space. In this talk, I present an explicit trace formula, which constitutes the first of its kind in a multi-dimensional, non-Euclidean setting.

Orateur: Jilly Kevo (Universität Bonn)

19:00 Dinner (Le Square)

jeu. 28 mai

11 Revisiting Singular Integrals: Applications to Subelliptic PDEs (Part III)

Since work of Hilbert in 1908, singular integrals have played an important role in the study of PDEs. In the 1960s, Kohn and Nirenberg (building on work of Seeley, Unterberger, Bokobza) introduced pseudodifferential operators, which simplified many arguments that relied on singular integral operators. Since then, pseudodifferential operators have been used to great effect in the study of PDEs; especially elliptic PDEs. Because of their powerful algebraic structure, pseudodifferential operators have largely replaced singular integral operators in many applications.

However, there are modern settings where pseudodifferential operators do not apply and singular integrals once again become a central tool.
In this mini-course we:
• Define singular integral operators with motivating examples from Euclidean space.
• Describe function spaces from harmonic analysis and their connections with singular integral operators.
• Present maximally subelliptic PDEs as a case study where singular integrals give sharp results and standard pseudodifferential operators do not.
• Show how singular integrals allow us to not only study linear subelliptic PDEs, but also fully nonlinear subelliptic PDEs, subelliptic PDEs with rough coefficients, and subelliptic boundary value problems.

This course is designed for graduate students with some functional analysis (L^p spaces, Fourier transform). Some familiarity with distributions and pseudodifferential operators will be useful, but not necessary.

Orateur: Brian Street (University of Wisconsin)

10:30 Coffee

12 Applications of operator algebras and operator theory to boundary value problems and layer potentials

One of the main and oldest approaches to boundary value problems is through layer potentials. The study of layer potentials has lead, more than one hundred years ago, to the development of operator theory (including compact and Fredholm operators). The layer potential operators associated to a suitable boundary value problem on a smooth, bounded domain are pseudodifferential operators and, in order to establish the solvability of the given boundary value problem, the basic theory of compact and Fredholm operators suffices. By contrast, on a non-smooth polyhedral domains, the relevant operator algebras have a more complicated structure and the basic theory of compact and Fredholm operators does not suffice anymore. In my talk, I will present some ideas from operator algebras that are useful in order to treat boundary value problems on domains with cylindrical ends (equivalently, on domains with conical points).

The main results of my talk are from joint works with Marius Mitrea and Mirela Kohr.

Orateur: Victor Nistor (Université de Lorraine, Metz)

13:00 Free afternoon

ven. 29 mai

13 Family index on contact fibrations and families of Toeplitz operators

At the end of the 80's, Boutet de Monvel gave an index formula for Toeplitz operators on certain complex domains. This formula was later re-interpreted by Epstein and Melrose using Heisenberg calculus on contact manifolds. In a joint work with Erfan Rezaei, we extend this formula to the case of families. More generally, we extend the index theorem of Baum and van Erp to families of contact manifolds. I will present some aspects of the proof, and if time permits, a potential obstruction to generalizations.

Orateur: Clément Cren (Universität Göttingen)

10:30 Coffee

14 Groupoid equivariant Roe algebras and coarse index map

In this talk we start with an introduction to Roe algebras and the coarse assembly map. Some difficulties appear when one is trying to define the groupoid equivariant analogue. We will share some techniques to formulate a framework to define groupoid equivariant Roe algebras and the coarse index map, and show that the coarse index map is compatible with the Baum-Connes assembly map for groupoids.

Orateur: Kai Mao (Université de Lorraine, Metz)

15 Localized equivariant index computations for manifold with boundary using groupoids

In the second paper of the series "The Index of Elliptic Operators", M. Atiyah and G. Segal study the G-equivariant topological index associated with the action of a compact Lie group G on a manifold. After localization at a conjugacy class, this index can be expressed in terms of fixed-point data, leading to Lefschetz-type fixed point formulas. In this talk, we study this localized computation in the case of manifold with boundary, using Monthubert's groupoid and pushforward in K-theory.

Orateur: Mathis Alleysson (Université de Toulouse)