We present recent results on the connections existing between the facial

structure of the unit ball in a JB*-triple and the lattice of tripotents in its

bidual.

# Past Functional Analysis Seminar

For reasonable domains $\Omega\subseteq\mathbb{R}^{d+

1}$, and given some boundary data $f\in C(\partial\Omega)$, we can solve the Dirichlet problem and find a harmonic function $u_{f}$ that agrees with $f$ on $\partial\Omega$. For $x_{0}\in \Omega$, the association $f\rightarrow u_{f}(x_{0})$. is a linear functional, so the Riesz Representation gives us a measure $\omega_{\Omega}^{x_{0}}$ on $\partial\Omega$ called the harmonic measure with pole at $x_{0}$. One can also think of the harmonic measure of a set $E\subseteq \partial\Omega$ as the probability that a Brownian motion of starting at $x_{0}$ will first hit the boundary in $E$. In this talk, we will survey some very recent results about the relationship between the measure theoretic behavior of harmonic measure and the geometry of the boundary of its domain. In particular, we will study how absolute continuity of harmonic measure with respect to $d$-dimensional Hausdorff measure implies rectifiability of the boundary and vice versa.

We present some recent results on the study of Schatten-von Neumann properties for

operators on compact manifolds. We will explain the point of view of kernels and full symbols. In both cases

one relies on a suitable Discrete Fourier analysis depending on the domain.

We will also discuss about operators on $L^p$ spaces by using the notion of nuclear operator in the sense of

Grothendieck and deduce Grothendieck-Lidskii trace formulas in terms of the matrix-symbol. We present examples

for fractional powers of differential operators. (Joint work with Michael Ruzhansky)

In this presentation we study the asymptotic behaviour of infinite systems of coupled linear ordinary differential equations. Each subsystem has identical dynamics that are only dependent on the states of its immediate neigbours. Examples of such systems in particular include the infinite "robot rendezvous problem" and the "platoon system" that are used to approximate the dynamics of large configurations of vehicles. In the presentation introduce novel methods for studying the spectral properties and stability of infinite systems of differential equations. The latter question is particularly interesting due to the fact that the systems in our class are known to lack uniform exponential stability. As our main results, we introduce general conditions for strong stability and derive rational rates of convergence for the solutions using recent results in the theory of nonuniform stability of strongly continuous semigroups.

We study the Atiyah-Singer Dirac operator on smooth Riemannian Spin manifolds with smooth compact boundary. Under lower bounds on injectivity radius and bounds on the Ricci curvature and its first derivatives, we demonstrate that this operator is stable in the Riesz topology under bounded perturbations of local boundary conditions. Our work is motivated by the spectral flow and its connection to the Riesz topology. These results are obtained by obtaining similar results for a more wider class of elliptic first- order differential operators on vector bundles satisfying certain general curvature conditions. At the heart of our proofs lie methods from Calderón-Zygmund harmonic analysis coupled with the modern operator theory point of view developed in proof of the Kato square root conjecture.

Let $X$ be a compact Hausdorff space and $C(X)$ be the space of continuous real-valued functions on $X$ endowed with the topology of uniform convergence. Assume we are given a finite number of closed subalgebras $A_1, \dots A_k$ of $C(X)$. Our talk is devoted to the following problem. What conditions imposed on $A_1, \dots, A_k$ are necessary and/or sufficient for the representation $C(X) = A_1 +\dots + A_k$? For the case $k = 1$, the history of this problem goes back to 1937 and 1948 papers by M. Stone. A version of the corresponding famous result, known as the Stone-Weierstrass theorem, states that a closed subalgebra $A \subset C(X)$, which contains a nonzero constant function, coincides with the whole space $C(X)$ if and only if $A$ separates points of $X$.

A self-similar groupoid action (G,E) consists of a faithful action of a groupoid G on the path space of a graph which displays a notion of self-similarity. In this talk I will explain this concept and consider KMS states on associated Cuntz-Pimsner C*-algebras. This talk is based on joint work with Marcelo Laca, Iain Raeburn, and Jacqui Ramagge

The classical Banach-Stone theorem describes the structure of onto linear isometries of the Banach space $C(K)$ of all continuous functions on a compact Hausdorff space $K$. Namely, such an isometry is always a product of a composition operator with a homeomorphism symbol and a multiplication operator with a continuous symbol which has modulus 1.

Recently, similar results have been obtained in the setting of certain class of probability measures. In my talk first, I will give an overview of these results, and then I will present the main ideas of a recent work. Namely, I will provide a characterisation of all surjective isometries of the (non-linear) space of all Borel probability measures on an arbitrary separable Banach space with respect to the famous Levy-Prokhorov distance (which metrises the weak convergence). This is a recent joint work with Tamas Titkos (MTA Alfred Renyi Institute of Mathematics, Budapest, Hungary).

We study a spectral problem which is related to the propagation of electromagnetic waves in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic medium. The defect is infinitely extended and aligned with one of the coordinate axes. Under certain geometrical assumptions, the underlying Maxwell operator reduces to an elliptic operator and we study the effect of the perturbation by the waveguide on its spectrum. We show that the perturbation introduces guided mode spectrum inside the band gaps of the fully periodic, unperturbed spectral problem and use variational arguments to prove that guided mode spectrum can be created by arbitrarily small perturbations.