**Joan Bosa** (Glasgow University)

**Title:** Covering dimension of C*-algebras and 2-coloured classification.

**Abstract:** We introduce the concept of finitely coloured equivalence for *-homomorphisms of C*-algebras, for which unitary equivalence of unital *-homomorphisms is the 1-coloured case. We use this concept to classify *-homomorphisms from separable, unital, nuclear C*-algebras into ultrapowers of simple, unital, nuclear Z-stable C*-algebras with compact extreme tracial-state space up to 2-coloured equivalence by their behaviour on traces (Joint work with N. Brown, Y. Sato, A. Tikuisis, S. White and W. Winter).

**Martin Dindos** (The University of Edinburgh)

**Title:** The equivalence of BMO solvability of the Dirichlet problem for parabolic equation with A_{∞} condition for the parabolic measure.

**Abstract:** We define a class of admissible parabolic domains on which we consider the Dirichlet problem for variable coefficient scalar parabolic PDE. We establish that the (natural) parabolic measure associated with the PDE belongs to the A_{∞} class with respect to the surface measure if and only if the Dirichlet problem for this parabolic equation with BMO data is solvable.

The A_{∞} class is significant, since if a parabolic measure belongs to it then the Dirichlet problem for the associated PDE is solvable in L^{p} for some p>p_{0}.

The presented proof significantly simplifies a recent result of Kenig, Kircheim, Pipher and Toro for the elliptic PDEs that has improved an analogous equivalence result in the elliptic case of Kenig, Pipher and myself. Since our approach also applies to the elliptic case, we also re-prove the Kenig, Kircheim, Pipher and Toro result.

This is a joint work with J. Pipher and S. Petermichl.

**Heiko Gimperlein** (Heriot-Watt University, Edinburgh)

**Title:** Nonclassical spectral asymptotics of commutators.

**Abstract:** This talk considers the spectral properties of commutators [P,f] between a pseudodifferential operator P and a Holder continuous function f. The mapping properties of such commutators have been of interest in harmonic analysis since work by Calderon in the 1960s. Much less is known about their spectral theory, which relates to the classical theory of Hankel operators and is motivated by applications.

We discuss sharp upper estimates for the asymptotic behavior of the singular values (weak-Schatten class properties) as well as explicit formulas for their Dixmier traces on closed Riemannian or sub-Riemannian manifolds. The commutators exhibit a rich spectral asymptotics beyond classical Weyl laws, dominated by the points where $f$ is not differentiable.

A mix of ideas from harmonic analysis, spectral theory and operator algebras enters into the proofs. The results are applied to questions in noncommutative geometry and complex analysis of several variables. (joint work with M. Goffeng).

**José González Llorente** (Universitat Autònoma de Barcelona)

**Title:** Variations on the mean value property.

**Abstract:** The interplay between classical potential theory and probability relies on the well known mean value property for harmonic functions. In the last years some efforts have been made to clarify the probabilistic framework associated to some remarkable nonlinear differential operators(such as the p-laplacian or the ∞-laplacian) by means of appropriate (nonlinear) mean value properties. In the talk we will review some classical

facts about the converse mean value property and we will also introduce a nonlinear mean value property associated to the p-laplacian.

**Sukjung Hwang** (The University of Edinburgh)

**Title:** The Dirichlet boundary problem for second order parabolic operators satisfying Carleson condition.

**Abstract:** We consider boundary value problems for a parabolic equation in the form of $u_t-\mbox{div}(A\nabla u) – \boldsymbol{B}\cdot\nabla u =0$ on time-varying domains $\Omega$ with $L^{p}$ Dirichlet boundary data for $1 < p \leq \infty$. Many difficulties are caused by the fact that neither the operator nor the domain we consider are assumed to be smooth. When the coefficient matrices A=[a_{ij}] and **B** = [b_{i}] satisfy certain Carleson conditions with small norms, we establish the solvability of the Dirichlet boundary problems for second order parabolic equations by comparing L^{p} norm of the square and non-tangential maximal functions.

**Joaquim Martín** (Universitat Autònoma de Barcelona)

**Title:** Isoperimetric weights and generalized uncertainty inequalities in metric measure spaces.

**Abstract:** We extend the recent L1 uncertainty inequalities obtained by G. M. Dall’ara and D. Trevisan to the metric setting. For this purpose we introduce a new class of weights, named isoperimetric weights, for which the growth of the measure of their level sets $\mu(w\led r)$ can be controlled by rI(r); where I is the isoperimetric profile of the ambient metric space. We use isoperimetric weights, new localized Poincaré inequalities, and interpolation, to prove Lp-uncertainty inequalities on metric measure spaces.

**Jordi Pau** (Universitat de Barcelona)

**Title:** Schatten class Hankel operators on Bergman spaces.

**Abstract:** We completely characterize the simultaneous membership in the Schatten ideals S_{p}, 0<p<∞ of the Hankel operators H_{f} and H_{f} on the Bergman space, in terms of the behaviour of a local mean oscillation function, proving a conjecture of Kehe Zhu from 1991.