Abstract. This talk concerns analytic tent spaces induced by radial weights admitting certain doubling properties. Several technical tools such as maximal theorems, Littlewood-Paley formulas, reproducing kernel estimates, fractional derivative estimates, tent space embedding theorems and duality characterizations are established. Within this framework, we study the Toeplitz operators induced by positive measures.

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Abstract. In 2021 Boc Thaler showed that any bounded regular open set without holes can be realised as the wandering domain of some entire function. This result was later refined by Martí-Pete, Rempe and Waterman. The natural question is whether we can obtain a similar result for unbounded sets. Here we run into some technical issues, namely that by just using approximation techniques it is difficult to ensure a function remains injective on an unbounded set. If we instead only care about the approximate shape of a wandering domain, this turns out to not be such a big issue. We show that any closed set is in some sense arbitrarily close to a wandering domain. In particular this shows there exists an entire function with a wandering domain with complement of arbitrary small area.

Nevertheless, prescribing the shape of an unbounded wandering domain is not completely hopeless. We show that a suitable open set contained inside a strip can in fact be realised as the wandering domain of some entire function. Thus this is a generalization of Boc Thalers result to certain unbounded sets.

Abstract. The classical uniqueness theorem for the (bilateral) Laplace transform states that if an analytic function has two Laplace representations \(F=\mathcal{L}(f_1)\) and \(F=\mathcal{L}(f_2)\) on overlapping strips, then \(f_1=f_2\) almost everywhere. This may fail if the strips are disjoint, even if \(F\) extends analytically between them. Using an analytic continuation method developed by Harper, we relate the problem of uniqueness of Laplace representation on separate strips to the uniqueness of solutions of the Cauchy problem for the heat equation on the infinite rod. We then adapt methods of holomorphic approximation theory to construct examples of entire functions satisfying \(F=\mathcal{L}(f_1)\) on \(\Re(z)>0\) and \(F=\mathcal{L}(f_2)\) on \(\Re(z)<0\), for which \(f_1-f_2\) is nontrivial on a set of positive Lebesgue measure. In this way, we recover well-known examples of non-uniqueness solutions for the heat equation.

Abstract. We present the notion of a vanishing zone for holomorphic map-germs \(f:(X,0)\rightarrow(\mathbb{C},0)\), where \(X\) is a complex analytic set with an isolated singularity at \(0\). We then establish criteria for the homological equivalence between the boundary of the Milnor fiber of \(f\) and its link, in the case where the singular locus of \(f\) is one-dimensional. Finally, we present hypotheses under which these criteria can be extended to functions defined on analytic sets with non-isolated singularities.

Abstract. In this talk, we define the notion of a Reinhardt domain in a toric variety \(X_{\sigma}\) and provide some examples. We restrict our attention to toric varieties \(X_{\sigma}\) constructed from an affine simplicial strictly convex rational polyhedral cone \(\sigma\). We introduce the main result of our recent article by stating that for every Reinhardt domain \(D\subset X_{\sigma}\), there is a Stein Reinhardt domain \(\widehat{D}\subset X_{\sigma}\) that is a holomorphic extension of \(D\). We use that there exist a finite subgroup \(\Gamma\) of \(GL_n(\mathbb{C})\), a morphism \(\pi:\mathbb{C}^n\rightarrow X_{\sigma}\), and an isomorphism \(\varphi:\mathbb{C}^n/\Gamma\rightarrow X_{\sigma}\) such that \(\pi=\varphi \circ \pi_{\Gamma}\), where \(\pi_\Gamma:\mathbb{C}^n\rightarrow\mathbb{C}^n/\Gamma\) is the quotient mapping. Using this structure, we conclude that for every Reinhardt domain \(D\subset X_{\sigma}\), the preimage \(\pi^{-1}(D)\) is a Reinhardt domain in \(\mathbb{C}^n\). Finally, we state some open questions in the literature.

Abstract. Bellman algorithm is a recently developed algorithm by me and my supervisor. It solves a fully non-linear elliptic Monge-Ampere equation with a Dirichlet boundary condition by approximating the solution with solutions to linear elliptic PDEs with the same boundary condition. For the construction of the aforementioned linear elliptic PDEs, Bellman principle was utilized as it relates non-linear determinant of a matrix to its linear trace.

In this talk, I will briefly introduce the algorithm and numerical results in order to mainly focus on presenting the proof of convergence which explains the numerical observations and illuminates other interesting aspects of behaviour of the method.

Abstract. In this talk, we will discuss the rigidity problem of holomorphicity of Kobayashi isometry. Given an isometry map between two domains in complex Euclidean space with respect to their Kobayashi distance/metric, it is an interesting problem to know when this isometry is holomorphic. We will see through a few examples that Kobayashi isometry need not be holomorphic and mention some important results in this context. In the end, we will show that for the domain diamond \(\triangle=\{|z_1|+|z_2|<1\}\subset \mathbb{C}^2\) and special Carathéodory sets of Tridisc \(D_{a,b}=\{(z,w)\in \mathbb D^2 : |az_1+bz_2-z_1z_2|<|\overline{a}z_2+\overline{b}z_1-1|\}\) for \(\{|a|,|b|,1\}\) forms sides of a triangle, the Kobayashi isometry is indeed holomorphic.

Abstract. This talk focuses on the optimal domains for bounded Volterra integration operators \(T_g\) between distinct Hardy spaces \(H^p\) and \(H^q\) of the unit ball. It is shown that the intersection of the optimal domains is equal to \(H^p\) if \(p> q\), whereas if \(p<q\), we show that this intersection is genuinely larger.

Abstract. Spheres are central objects in Cauchy-Riemann (CR) geometry as model hypersurfaces with rich symmetry groups. In particular, the embeddability of real hypersurfaces into spheres in higher-dimensional complex spaces has been extensively studied. In this talk, after a brief overview of known results on the existence and uniqueness of embeddings into spheres, we focus on CR embeddings of strictly pseudoconvex connected real hypersurfaces in \(\mathbb{C}^2\) into the sphere in \(\mathbb{C}^3\) and present rigidity results for certain classes of such hypersurfaces with non-umbilical points.

Abstract. For a radial weight \(\omega\), let \(B_{\omega}\) denote the Berezin transform induced by the given weight. We show an interesting connection between the Carleson measures of the weighted Bergman space \(A^p_{\omega}\) and the boundedness of \(B_{\omega}\) in the case of when \(\omega\) is doubling. We also discuss how our results are related to the connection between the Carleson measures of the Hardy space \(H^p\) and the boundedness of the Poisson transform.

Abstract. In this talk, we discuss a class of complex Schrödinger equation with a $q$-difference term: \begin{align}\tag{†}\label{dagger} f'(z) = a(z)f(qz) + R(z, f(z)), \quad R(z, f(z)) = \frac{P(z, f(z))}{Q(z, f(z))}, \end{align} where $a(z) \not\equiv 0$ is a small meromorphic function with respect to $f(z)$, and all the coefficient functions of $R(z, f(z))$ are also small meromorphic functions with respect to $f(z)$. We assume that $q\in\mathbb{C}\setminus \left \{ 0,-1,1 \right \} $ and that $R(z, f(z))$ is an irreducible rational function in both $f(z)$ and $z$. We obtain some necessary conditions for \eqref{dagger} to have meromorphic solutions of zero order and non-constant entire solutions, respectively. In particular, we prove the existence of entire solutions in many cases, study their number, and further investigate the local and global meromorphic solutions to \eqref{dagger}. Additionally, we consider the possible forms of the meromorphic solutions to \eqref{dagger} in certain conditions and examine exponential polynomials as possible solutions of \eqref{dagger}.

Abstract. I will discuss the construction of bounds for the Bergman kernel form on Riemann surfaces with bounded Ricci curvature. The main part of the talk will be concerned with the construction of Ohsawa-Takegoshi estimates which will be equivalent to the lower bound in this situation. I will also discuss how one may construct a global upper bound with the use of the Hele-Shaw exponential introduced by Hedenmalm-Shimorin.