The problem of whether the pullback of a singular analytic germ via a proper holomorphic self-map germ of (Cn,0) stays singular dates back to 2007 and has not been solved yet, apart from a number of special cases. In this paper we start with a brief overview of the main advances made so far. Then, we observe that analogous results can be proved globally, for algebraic sets and a proper polynomial map. Secondly, we study the case of proper map germs Cn→Cm, i.e. with m≥n. We also present some additional results, the first of which is an application of a slightly generalised version of the Giraldo–Roeder reduction. Another one involves Lipschitz geometry and Lipschitz normally embedded sets.

We consider a finite-time stochastic drift control problem under the assumption that the control is bounded and the system is controlled until the state process leaves the half-line. Assuming general conditions, it is proved that the resulting parabolic Hamilton–Jacobi–Bellman equation has a classical solution. In fact, we consider an even more general family of semilinear equations, which might be helpful in solving other control or game problems. Not only is the existence result proved, but also a recursive procedure for finding a solution resulting from a fixed-point argument is provided.

The article presents the main scientific achievements of Arkadiusz Płoski (1947–2024).

We solve an open problem posed by Thamrongthanyalak on the definable Banach fixed point property. Lipschitz curve selection is a key to our solution. In addition, we show a definable version of the Caristi fixed point theorem.

We generalize the Bernstein–Walsh–Siciak theorem on polynomial approximation in Cn to the case where the polynomial ring P(Cn) is replaced by a subring PS(Cn) consisting of all polynomials with exponents restricted to sets mS, where S is a compact convex subset of Rn+ with 0∈S and m=0,1,2,…, and uniform estimates of error in the approximation are replaced by weighted uniform estimates with respect to an admissible weight function.

We investigate a class of fractional Schrödinger–Poisson systems with critical exponents and steep potential well. By using the constraint variational method and a quantitative deformation lemma, the existence of positive ground state solutions and sign-changing ground state solutions is obtained. It is shown that the energy of the sign-changing ground state solution is strictly larger than twice the energy of the positive ground state solution. Moreover, we also study the asymptotic behavior of the sign-changing ground state

We investigate the relationship between the valuations of a germ of a singular foliation $\mathcal F$ on the complex plane and those of a balanced equation of separatrices for $\mathcal F$, extending a theorem by Mattei–Salem. Under certain conditions, we

We prove that every polynomial submersion (f, g) : R 2 R 2 such that deg f 5 and deg g 6 is injective.

We establish a generalized defect relation for holomorphic maps into algebraic varieties intersecting arbitrary families of linearly equivalent divisors. Based on this, we also give an improvement of many previous second main theorems for holomorphic maps

Let $X, Y$ be locally compact Hausdorff spaces, and let $C_0(X)$, $C_0(Y)$ be the Banach spaces of all real continuous functions on $X,Y$, respectively, vanishing at infinity endowed with the usual sup-norm. Suppose that $A, B$ are subspaces of $C_0(X), C