ANALYSIS  SEMINAR (UTK)

Abstract:. We will explain how one may "weld" two Blaschke products to obtain a rational function, and address the question of whether the rational function one obtains by this procedure is unique. We will also explain how this question arose in the recent literature on applications of complex analysis to pattern recognition. Along the way we will talk about some relevant work of UTK professors Ken Stephenson and Carl Sundberg.   

Abstract: If $\mathcal H$ denotes a Hilbert space of analytic functions in the open unit disc $\mathbb D\subseteq \mathbb C$, then a function $f\in \mathcal H$ is called cyclic, if the polynomial multiples of $f$ are dense in $\mathcal H$. Cyclic functions cannot have any zeros in the open unit disc, and for many spaces $\mathcal H$ it is interesting to characterize cyclicity in terms of how small $|f(z)|$ can be near the boundary of $\mathbb D$. For example, if $\mathcal H=H^2$, the space of analytic functions in $\mathbb D$ whose Taylor coefficients are square summable, then $f$ is cyclic in $H^2$, if and only if it is a so-called outer function, i.e.  the harmonic function $\log|f(z)|$ equals the Poisson integral of an absolutely continuous measure. This excludes for example functions of the type $f(z)=e^{-\frac{1+z}{1-z}}$ where $|f|$ has exponential decline on the radius $\to 1$, and $\log |f(z)|$ equals the Poisson integral of a point mass. 

Let $D=\{f\in \mathrm{Hol}(\mathbb D): \int_{\mathbb D} |f'|^2dxdy<\infty\} $ denote the Dirichlet space of the unit disc $\mathbb D$. One checks that $D\subseteq H^2$ with $\|f\|_{H^2}\le \|f\|_D$. It follows that cyclic functions in $D$ must be outer, but deciding which outer functions  are cyclic in $D$ is an open problem. We show that an outer function  $f\in D$ is cyclic in $D$, whenever $\log f$ belongs to the Pick-Smirnov class $N^+(D)$. If $f$ is a bounded analytic function in $D$, then cyclicity can also be checked via iterated logarithms. For example, we show that an outer function $f$ is cyclic, whenever $\log(1+ \log(1/f))\in N^+(D)$. This condition can be checked by verifying that $\log(1+ \log(1/f))\in D$. If $f$ satisfies a mild extra condition, then the conditions also become necessary for cyclicity. The talk is based on joint work with Alexandru Aleman.

Abstract:. The study of the boundary behavior of quasiconformal mappings started with Beurling and Ahlfors. They proved that any quasiconformal mapping of the upper half plane extends to a quasisymmetric mapping of the real line and vice versa. Since then, quasisymmetric mappings have played a central role in Geometric Function Theory and have been used to characterize domains and mappings. In this talk, we will look at a finer version of such embeddings and use them to characterize certain domains in the plane. 

Abstract:. Cyclicity is a fundamental concept in the study of analytic function spaces, closely connected to invariant subspaces and the structure of multiplier algebras. A natural question is: under what conditions can a given analytic function generate the entire space? In this talk, I will discuss cyclicity in  Dirichlet-type spaces, with particular emphasis on spaces that do not possess the complete Pick property. Motivated by earlier results connecting cyclicity with logarithmic conditions, we introduce weak* sequential cyclicity in the multiplier algebra and an associated Smirnov class. We then investigate how the geometry of the range of a holomorphic logarithm influences cyclicity. In particular, we establish a criterion that replaces the classical bounded argument assumption with a more flexible sectorial condition, allowing the logarithm to have an unbounded imaginary part. To illustrate the significance of this extension, we present an explicit Hardy-space example of a cyclic function with unbounded logarithmic argument. We also discuss how the growth of the imaginary part of the logarithm affects cyclicity. The talk will be accessible to a general analysis audience, and the necessary background will be introduced along the way. 

Abstract:. Let \gamma=(x,y) be a smooth curve in the plane. Its signed area is defined by S(\gamma) := 1/2 \int_\gamma (x dy − y dx). When \gamma is a simple closed curve, this agrees, up to sign, with the area it encloses. By Young integration, S(\gamma) is well defined for every \alpha-Hölder curve with \alpha>1/2. In this talk, I will discuss a new sharp existence result for S(\gamma) at the critical exponent \alpha = 1/2, under an additional square-summability assumption. I will explain how ideas behind this result lead to a proof of the coarea formula for Lipschitz maps from H^1 to R^2, where H^1 denotes the first sub-Riemannian Heisenberg group. This gives a vector-valued coarea formula in the sub-Riemannian setting for merely Lipschitz maps, in a regime that was beyond the reach of previously available methods in geometric measure theory. This is joint work with Robert Young (NYU) 

Abstract:. A quasiconformal uniformization theorem of Rickman \cite{Rickman} asserts that every bounded turning planar arc is the image of the unit segment under a quasiconformal self-map of the plane. We prove an analogous uniformization theorem for planar trees by showing that every planar quasiconformal tree with uniform branch separation is quasiconformally equivalent to a quasiconvex planar tree with uniform branch separation and Hausdorff dimension arbitrarily close to 1. Along the way, we show that every quasisymmetric (resp. bi-Lipschitz) homeomorphism between planar quasiconformal trees that respects the counterclockwise orientation of the branches at every branch point can be extended quasisymmetrically (resp. bi-Lipschitz) to the whole plane. As a corollary of the extension theorem we construct a class of quadrivalent quasiconformal trees, which includes the Vicsek fractal, so that each tree in this class is the Julia set of a uniformly quasiregular map of $\mathbb{R}^2$.