One of the many meaningful equivalent norms on BMO uses a Carleson-measure condition on the gradient of the Poisson extension. This is closely related to the Dirichlet problem for the Laplacian in the upper half-space with boundary data in BMO. The Poisson semigroup provides the unique solution in appropriate classes, and it is bounded on BMO, that is, it propagates the space boundary space in the transversal direction. If the tangential Laplacian is replaced by a general elliptic operator in divergence form, boundedness of the Poisson semigroup on BMO can fail in any dimension n ≥ 3. Somewhat unexpectedly, its gradient persists to give rise to a Carleson measure with norm equivalent to the BMO-norm at the boundary in dimensions n = 3, 4 and hence a unique solution to the corresponding Dirichlet problem. In my talk, I will try to explain the broader context behind this phenomenon and why we still do not know if the result is sharp.
Based on joint work with (of course) Pascal. It is Chapter 18 of our book but you will not have to read the seventeen preceding chapters to follow.[-]

For an open set $\Omega \subset \mathbb{R}^{d}$ with an Ahlfors regular boundary, solvability of the Dirichlet problem for Laplaces equation, with boundary data in $L^{p}$ for some $p<\infty$, is equivalent to quantitative, scale invariant absolute continuity (more precisely, the weak- $A_{\infty}$ property) of harmonic measure with respect to surface measure on $\partial \Omega$. A similar statement is true in the caloric setting. Thus, it is of interest to find geometric criteria which characterize the open sets for which such absolute continuity (hence also solvability) holds. Recently, this has been done in the harmonic case. In this talk, we shall discuss recent progress in the caloric setting, in which we show that quantitative absolute continuity of caloric measure, with respect to surface measure on the parabolic Ahlfors regular (lateral) boundary $\Sigma$, implies parabolic uniform rectifiability of $\Sigma$. We observe that this result may be viewed as the solution of a certain 1-phase free boundary problem. This is joint work with S. Bortz, J. M. Martell and K. Nyström.[-]

The $T(1)$ theorem of David and Journé is one of the most remarkable theorems in harmonic analysis. The theorem reduces the study of $L^{p}$ boundedness of a singular integral operator, $T$ to testing a 'testing condition', that is, verifying $T(1)$ is in the space $B M O$. A simplistic view of these theorems is that they shift the task of verifying boundedness for all functions (globally) to that of verifying a condition on all cubes. More general testing conditions, e.g. 'local $T(b)$' conditions, allow one to adapt the testing function to the cube and/or weaken conditions on the operator. These 'local $T(b)$ theorems' are an important ingredient to the initial solution to the Kato problem.
The project will introduce the concepts of $T(1) / T(b)$ theory for singular integrals, Littlewood-Paley theory, Carleson measures and stopping time arguments. The goal is to present the 'original' proof of the Kato problem and, possibly, look at more recent developments.[-]

Nigel Kalton played a prominent role in the development of a holomorphic functional calculus for unbounded sectorial operators. He showed, in particular, that such a calculus is highly unstable under perturbation: given an operator $D$ with a bounded functional calculus, fairly stringent conditions have to be imposed on a perturbation $B$ for $DB$ to also have a bounded functional calculus. Nigel, however, often mentioned that, while these results give a fairly complete picture of what is true at a pure operator theoretic level, more should be true for special classes of differential operators. In this talk, I will briefly review Nigel's general results before focusing on differential operators with perturbed coefficients acting on $L_p(\mathbb{R}^{n})$. I will present, in particular, recent joint work with $D$. Frey and A. McIntosh that demonstrates how stable the functional calculus is in this case. The emphasis will be on trying, as suggested by Nigel, to understand what makes differential operators so special from an operator theoretic point of view.[-]

In this lecture we shall present some recent results in collaboration with B. Geshkovski (MIT) on the design of optimal sensors and actuators for control systems. We shall mainly focus in the finite-dimensional case, using the Brunovsky normal form. This allows to reformulate the problem in a purely matricial context, which permits rewriting the problem as a minimization problem of the norm of the inverse of a change of basis matrix, and allows us to stipulate the existence of minimizers, as well as non-uniqueness, due to an invariance of the cost with respect to orthogonal transformations. We will present several numerical experiments to both visualize these artifacts and also point out towards further directions and open problems, in particular in the context of PDE infinite-dimensional models.[-]