Power counting in noncommutative Quantum Field Theory

• Eva-Maria Hekkelman (Max Planck Institute, Bonn)

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.

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

• Brian Street (University of Wisconsin)

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.

The Satake groupoid for Riemannian symmetric spaces and beyond

• Jacob Bradd (Université de Lorraine, Metz)

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.

Seiberg-Witten equations on Hermitian symmetric spaces, part II

• Rafailia Tsiavou (University of Thessaloniki)

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.

On the algebraic Mackey deformation

• Yvann Gaudillot-Estrada (Université de Lorraine, Metz)

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.

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

• Brian Street (University of Wisconsin)

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.

High-frequency analysis of PDEs and pseudo differential calculus

• Clotilde Fermanian (Université d'Angers)

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.

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

• Edward McDonald (Universität Bonn / Université Paris Cité)

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.

A regularity theorem for fully nonlinear maximally subelliptic PDEs

• Gautam Neelakantan Memana (University of Wisconsin)

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.

Trace and Index of Callias Operators on Hyperbolic Space

• Jilly Kevo (Universität Bonn)

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.

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

• Brian Street (University of Wisconsin)

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.

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

• Victor Nistor (Université de Lorraine, Metz)

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.

Family index on contact fibrations and families of Toeplitz operators

• Clément Cren (Universität Göttingen)

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.

Groupoid equivariant Roe algebras and coarse index map

• Kai Mao (Université de Lorraine, Metz)

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.

Localized equivariant index computations for manifold with boundary using groupoids

• Mathis Alleysson (Université de Toulouse)

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.