Related papers: Extremal holomorphic maps and the symmetrised bidi…

Consider a fixed connected, finite graph $\Gamma$ and equip its vertices with weights $p_i$ which are non-negative integers. We show that there is a finite number of possibilities for the coefficients of the canonical cycle of a numerically…

Let $\Gamma \subset \mathbb C$ be a curve of class $C(2,\alpha)$. For $z_{0}$ in the unbounded component of ${\mathbb C}\setminus \Gamma$, and for $n=1,2,...$, let $\nu_n$ be a probability measure with supp$(\nu_{n})\subset \Gamma$ which…

The Baum-Connes assembly map with coefficients $e_{\ast}$ and the Mishchenko-Kasparov assembly map with coefficients $\mu_{\ast}$ are two homomorphisms from the equivariant $K$-homology of classifying spaces of groups to the $K$-theory of…

We apply a notion of geodesics of plurisubharmonic functions to interpolation of compact subsets of $C^n$. Namely, two non-pluripolar, polynomially closed, compact subsets of $C^n$ are interpolated as level sets $L_t=\{z: u_t(z)=-1\}$ for…

Let $X$ be a locally symmetric space $\Gamma\backslash G/K$ where $G$ is a connected non-compact semisimple real Lie group with trivial centre, $K$ is a maximal compact subgroup of $G$, and $\Gamma\subset G$ is a torsion-free irreducible…

This article proposes a bivariate polynomial problem for finite-order real matrices that endows a \textit{`sufficient condition'} for a map from the standard vector spaces of finite-order real matrices to the same dimensional bivariate…

We prove two disc formulas for the Siciak-Zahariuta extremal function of an arbitrary open subset of complex affine space. We use these formulas to characterize the polynomial hull of an arbitrary compact subset of complex affine space in…

We prove some estimates for elementary symmetric polynomials on $\mathbb D^n.$ We show that these estimates are sharp which allow us to study the properties of closed symmetrized polydisc $\Gamma_n.$ Furthermore, we show the existence and…

We establish rigidity results for holomorphic mappings and plurisubharmonic functions in complex geometry. First, under mild conditions, we show that the gradient of a $\operatorname{U}(1)$-invariant strictly plurisubharmonic function in…

We introduce a coarse analog of the classical Lusternik-Schnirelmann category which we denote by $\text{c-cat}$, defined for metric spaces in the coarse homotopy category. This provides a new tool for studying large-scale topological…

Let $X$ be a smooth complex quasi-projective variety and $\Gamma=\pi_1(X)$. Let $\chi \colon \Gamma \to \mathbb{R}$ be an additive character. We prove that the ray $[\chi]$ does not belong to the BNS set $\Sigma(\Gamma)$ if and only if it…

A graph is $n$-existentially closed ($n$-e.c.) if for any disjoint subsets $A$, $B$ of vertices with $|{A \cup B}|=n$, there is a vertex $z \notin A \cup B$ adjacent to every vertex of $A$ and no vertex of $B$. For a block design with block…

We define the class of high dimensional graph manifolds. These are compact smooth manifolds supporting a decomposition into finitely many pieces, each of which is diffeomorphic to the product of a torus with a finite volume hyperbolic…

We give conditions under which a germ of a holomorphic mapping in $\Bbb C^N$, mapping an irreducible real algebraic set into another of the same dimension, is actually algebraic. Let $A\subset \bC^N$ be an irreducible real algebraic set.…

We study the problem of embedding bipartite graphs in Ahlfors-David regular sets of large dimension using results from extremal graph theory. Our main theorem states that any graph satisfying a power-improving bound on the extremal number…

We consider the convex set $\Gamma_{m,n}$ of $m\times n$ stochastic matrices and the convex set $\Gamma_{m,n}^\pi\subset \Gamma_{m,n}$ of $m\times n$ centrosymmetric stochastic matrices (stochastic matrices that are symmetric under rotation…

Let $\Gamma$ be the fundamental group of a closed orientable surface of genus at least two. Consider the composition of a uniformly random element of $\mathrm{Hom}(\Gamma,S_n)$ with the $(n-1)$-dimensional irreducible representation of…

Let $M$ be a Hadamard manifold with curvature bounded above by a negative constant $-\alpha$, satisfying the "strict convexity condition", and assume that $M$ admits a "helicoidal" one-parameter subgroup $G$ of isometries of $M$. Then,…

We consider extremal problems for subgraphs of pseudorandom graphs. For graphs $F$ and $\Gamma$ the generalized Tur\'an density $\pi_F(\Gamma)$ denotes the density of a maximum subgraph of $\Gamma$, which contains no copy of~$F$. Extending…

We study $ \mathcal{H}ol(\Gamma\cup\text{Int}(\Gamma)) $, the normed algebra of all holomorphic functions defined on some simply connected neighbourhood of a simple closed curve $\Gamma$ in $\mathbb{C} $, equipped with the supremum norm on…