复变函数

We introduce irregular constructible sheaves, which are $\mathbb{C}$-constructible with coefficients in a finite version of Novikov ring $\Lambda$ and special gradings. We show that the bounded derived category of cohomologically irregular…

The so called Ahlfors function is the solution to an extremal problem. In this thesis, we study several properties of this function in a most general domain Omega. We consider the Ahlfors function in the context of analytic capacity, and…

This article contains a complete proof of Gabrielov's rank Theorem, a fundamental result in the study of analytic map germs. Inspired by the works of Gabrielov and Tougeron, we develop formal-geometric techniques which clarify the difficult…

We derive a formula for the number of pre-images under a non-degenerate harmonic mapping $f$, using the argument principle. This formula reveals a connection between the pre-images and the caustics. Our results allow to deduce the number of…

We find an explicit sequence of univariate polynomials of arbitrary degree with optimal condition number. This solves a problem posed by Michael Shub and Stephen Smale in 1993.

We study harmonic mappings of the form $f(z) = h(z) - \overline{z}$, where $h$ is an analytic function. In particular we are interested in the index (a generalized multiplicity) of the zeros of such functions. Outside the critical set of…

Let $X$ be a compact K\"ahler manifold. We study plurisupported currents on $X$, i.e. closed, positive $(1,1)$-currents which are supported on a pluripolar set. In particular, we are able present a technical generalization of…

We obtain new integral inequalities for the integrals of the difference of subharmonic functions in measure through their Nevanlinna characteristic and some functional characteristic of the measure. These results are new also for…

A finite Blaschke product, restricted to the unit circle, is a smooth covering map. The maximum and minimum values of the derivative of this map reflect the geometry of the Blaschke product. We identify two classes of extremal Blaschke…

This paper mainly uses the nonnegative continuous function $\{\zeta_n(r)\}_{n=0}^{\infty}$ to redefine the Bohr radius for the class of analytic functions satisfying $\real f(z)<1$ in the unit disk $|z|<1$ and redefine the Bohr radius of…

For a sum of squares domain of finite D'Angelo 1-type at the origin, we show that the polynomial model obtained from the computation of the Catlin multitype at the origin of such a domain is likewise a sum of squares domain. We also prove,…

For self maps of the disk, it can be shown that under the right conditions one can embed a discrete iteration of the map into a continuous semigroup. In this article we extend these results to two complex variables for maps of the unit ball…

In one complex variable, the cross ratio is a well-known quantity associated with four given points in the complex plane that remains invariant under linear fractional maps. In particular, if one knows where three points in the complex…

The Segal-Bargmann transform plays an essential role in signal processing, quantum physics, infinite-dimensional analysis, function theory and further topics. The connection to signal processing is the short-time Fourier transform, which…

We prove that if a compact set E in complex Euclidean space is contained in an arc J, then there is a choice of J whose polynomial hull is the union of J and the polynomial hull of E. This strengthens an earlier result of the author. We…

In his work on the Ramanujan theory of elliptic functions to alternative bases, Shen constructed two different elliptic functions in signature three; we determine the precise relationship between them, by precisely relating their coperiodic…

We prove that every polynomially convex arc is contained in a polynomially convex simple closed curve. We also establish results about polynomial hulls of arcs and curves that are locally rectifiable outside a polynomially convex subset.

Given pure-dimensional (generalized) cycles $\mu_1$ and $\mu_2$ on a complex manifold $Y$ we introduce a product $\mu_1\diamond_{Y} \mu_2$ that is a generalized cycle whose multiplicities at each point are the local intersection numbers at…

Let $\varphi=\{\varphi_k\}_{k=-\infty}^\infty$ denote the extended Takenaka--Malmquist system on unit circle $\mathbb T$ and let $\sigma_{n,\varphi}(f),$ $f\in L^1(\mathbb T)$, be the Fej\'er-type operator based on $\varphi$, introduced by…

We study Nevanlinna theory on complete K\"ahler manifolds. As a consequence of the main result, we prove a defect relation of holomorphic mappings from complete K\"ahler manifolds of non-positive sectional curvature into complex projective…