In the Segal-Bargmann space (also Fock space) $A^2(\mathbb C^n, e^{-|z|^2})$ of entire functions the differentiation operators $a_j(f)= \frac{\partial f}{\partial z_j}$ (annihilation) and the multiplication operators $a^*_j(f)= z_j f$ (creation) are unbounded, densely defined adjoint operators with $\|f\|^2 \le \|a_j( f) \|^2 + \| a^*_j(f)\|^2$ for $f\in {\text{dom}}(a_j),$ which corresponds to the uncertainty principle. We study certain densely defined unbounded operators on the Segal-Bargmann space, related to the annihilation and creation operators of quantum mechanics. We consider the corresponding $D$-complex and study properties of the complex Laplacian $\tilde \Box_D = D D^* + D^* D,$ where $D$ is a differential operator of polynomial type, in particular we discuss the corresponding basic estimates, where we express a commutator term as a sum of squared norms. The basic estimates can be seen as a generalization of the uncertainty principle represented in the Segal-Bargmann space. In addition we show that K\"ahler manifolds $(M,h)$ admitting a real holomorphic vector field, i.e. for which there exists a smooth real-valued function $\psi: M \longrightarrow \mathbb R$ such that $h^{j \bar k}\frac{\partial \psi}{\partial \bar z^k} \frac{\partial}{\partial z^j}$ is a holomorphic vector field, have the property that the weighted Bergman space $A^2(M,h, e^{-\psi})$ exhibits the same duality between differentiation and multiplication as in the Segal-Bargmann space.

We investigate the question of existence of plurisubharmonic defining functions for smoothly bounded, pseudoconvex domains in $\mathbb{C}^2$. In particular, we construct a family of simple counterexamples to the existence of plurisubharmonic smooth local defining functions. Moreover, we give general criteria equivalent to the existence of plurisubharmonic smooth defining functions on or near the boundary of the domain. These equivalent characterizations are then explored for some classes of domains. This is joint work with A.-K. Gallagher.

The Diederich-Fornæss worm domain is a central object of study in Several Complex Variables. Originally constructed as the first example of a smoothly bounded, pseudoconvex domain without a Stein neighborhood basis, it also serves as a counterexample to several interesting questions in the field of Several Complex Variables. In this talk, after reviewing properties of the worm domain, I will present some results on the group of its biholomorphic automorphisms, its dimension and its connection to the unit sphere.

We present a selection theorem for domains in $\mathbb{C}^n$, $n\geq 1$, which states that any tamed sequence of pointed connected open subsets admits a subsequence convergent to its own kernel in the sense of Carath\'eodory. Not only is this analogous to the well-known Blaschke selection theorem for compact convex sets, but it fits better in the study of normal families of biholomorphic maps with varying domains and ranges. The research on this topic is joint-work with Kang-Tae Kim.

The Oka-Weil theorem on a Stein manifold is quite stable: small changes in a holomorphic function typically lead to small changes in the global holomorphic function approximating it. But what happens when we perturb the underlying complex structure? We show that the Oka-Weil theorem remains stable under deformations of Stein structures, provided these deformations satisfy mild regularity conditions. This is joint work with Franc Forstneric.

An intriguing phenomenon regarding Levi-degenerate hypersurfaces is the existence of nontrivial infinitesimal symmetries with vanishing 2-jets at a point. In this talk we consider polynomial models of Levi-degenerate real hypersurfaces in $\mathbb{C}^3$. of finite Catlin multitype. Exploiting the structure of the corresponding Lie algebra, we characterize completely models without 2-jet determination, including an explicit description of their symmetry algebras. The talk is based on joint work with Petr Liczman and Francine Meylan.

Finding non-trivial CR structures on a given manifold is an important topic in CR geometry. During my research on infinitesimal CR automorphisms, I discovered a method for defining a "general CR structure" on an arbitrary complex vector bundle, which I applied to solve the finite jet determination property for infinitesimal CR automorphisms. In this talk, I will introduce the concept of general CR structures, discuss the differences and similarities between them and state some useful propositions.

The aim of this talk is to present some of my results, as well as joint work with Chin-Yu Hsiao, on embedding theorems. The main tool in the proof is the asymptotic expansion of the Szegő kernel in different settings: $(0,q)$-forms, the equivariant case, and, if time permits, the orbifold setting.