Abstract. Brück's conjecture is as follows:

Conjecture. Let $f$ be a non-constant entire function with hyper-order $\rho_{2}(f) \not\in \mathbb{N}\cup \{\infty\}$. If $f$ and $f'$ share a finite value $\alpha$ CM, resp. a suitable meromorphic function $\alpha$, then there exists $ c \in \mathbb{C}\setminus\{0\}$ such that $$ \frac{f'-\alpha}{f-\alpha}=c. $$

The conjecture holds in a number of special situations as described in a survey paper by Lahiri. In this talk, we show a fairly general situation, where this conjecture fails.

Joint work with A. El Farissi, R. Dida, M. A. Zemirni.

Abstract. The \(n\)-th Chebyshev polynomial of a compact set in the complex plane minimizes the supremum norm on that set. While it is a classical result that the \(n\)-th root of this minimum norm converges to the set's logarithmic capacity, finer asymptotic behavior is captured by "Widom factors"—the ratio of the Chebyshev norm to the \(n\)-th power of the capacity.

This talk explores the interplay between the geometry of a complex continuum and the behavior of these factors. Moving beyond the well-developed theory of real subsets, we will first establish a natural baseline: which complex sets share the asymptotic behavior of the unit disk, where Widom factors converge to 1?

We then turn to the extremes. While Widom factors remain uniformly bounded for sets with smooth boundaries, it is a major open problem whether there exists any complex continuum where they become unbounded. We will discuss the ongoing search for such a set, examining theoretical lower bounds and the specific fractal geometries that might finally force indefinite growth.

Abstract. Given an ideal \(\mathcal J_x\subset \mathcal O_x\) generated by a tuple \(f\) of holomorphic functions at \(x\in \mathbb{C}^n\) with common zero set \(\{x\}\), the classical King's formula asserts that the Lelong number at \(x\) of the Monge-Ampère product \((dd^c \log |f|^2)^n\) is the Hilbert-Samuel multiplicity of \(\mathcal J_x\).

I will discuss a joint work in progress with Mats Andersson, Richard Lärkäng, and Rahim Nkunzimana where we generalize this result to modules. Given a submodule \(\mathcal K_x\subset \mathcal O_x^s\) such that \(\mathcal O_x^s/\mathcal K_x\) has support at \(x\), we prove that the so-called Buchsbaum-Rim multiplicity of \(\mathcal K_x\) can be represented as the Lelong number of a current constructed in terms of Monge-Ampère products of generators of \(\mathcal K_x\).

Abstract. Properties of images and preimages of a polynomial selfmap $P:K^n\rightarrow K^n$, here $K$ is $\mathbb{R}$ or $\mathbb{C}$, are useful for many questions in analysis and algebra. In this talk I will present 2 main results:

- Result 1: For any algebraic subvariety $Z\subset \mathbb{C}^N$ of codimension at least $2$, there is a subvariety $W\subset \mathbb{C}^N$ birational to $Z$ and a surjective algebraic map $F:\mathbb{C}^N\rightarrow \mathbb{C}^N\backslash W$. This makes Chevalley's theorem more explicit. This is joint work with Viktor Balch Barth.

- Result 2: There are linear algebra criteria to check if a polynomial map $P$ is proper/non-proper.