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Extensions of the notions of polynomially and rationally convex hulls are introduced. Using these notions, a generalization of a result of Duval and Levenberg on polynomially convex hulls containing no analytic discs is presented. As a…

Several results concerning pairs of polynomially convex sets whose union is not even rationally convex are given. It is shown that there is no restriction on how two spaces can be embedded in some $\C^N$ so as to be polynomially convex but…

In this paper it is shown that every compact two-dimensional manifold $S$, with or without boundary, can be embedded in $\mathbb C^3$ as a smooth submanifold $\Sigma$ in such a way that the polynomially convex hull of $\Sigma$, though…

It is shown that if $A$ is a unital commutative Banach algebra with a dense set of invertible elements, then the maximal ideal space of $A$ contains no compact, locally connected, simply coconnected subspace of topological dimension $\geq…

The paper is concerned with the boundary behaviour of polynomially and rationally convex hulls in pseudoconvex domains in $\mathbb{C}^n$. As an application, it is shown that every connected polynomially or rationally convex compact set with…

The existence of a nontrivial polynomially convex hull with every point a one-point Gleason part and with no nonzero bounded point derivations is established. This strengthens the \hbox{celebrated} result of Stolzenberg that there exists a…

It is shown that there exist arcs and simple closed curves in ${\mathbb C}^3$ with nontrivial polynomial hulls that contain no analytic discs. It is also shown that in any bounded Runge domain of holomorphy in ${\mathbb C}^N$ ($N \geq 2$)…

A generalization of a result of Wermer concerning the existence of polynomial hulls without analytic discs is presented. As a consequence it is shown that there exists a Cantor set $X$ in ${\mathbb C}^3$ whose polynomial hull is strictly…

In this note, a general result for determining the rational hulls of fibered sets in $\mathbb{C}^2$ is established. We use this to compute the rational hull of Rudin's Klein bottle, the first explicit example of a totally real nonorientable…

Let T be the unit circle in the complex plane C. This paper proves the existence of analytic structure in a compact subset K of T X C^n, where K has so-called "lineally convex" or "hypoconvex" fibers over T. It also addresses a related…

We discuss the relation between pluripolar hulls and fine analytic structure. Our main result is the following. For each non polar subset $S$ of the complex plane $\mathbb C$ we prove that there exists a pluripolar set $E \subset (S \times…

A rational polytope is the convex hull of a finite set of points in $\R^d$ with rational coordinates. Given a rational polytope $P \subseteq \R^d$, Ehrhart proved that, for $t\in\Z_{\ge 0}$, the function $#(tP \cap \Z^d)$ agrees with a…

It is a conjecture of Koll\'ar that a variety $X$ with rational singularities in some open subvariety $U$ has a rationalification; that is, a proper, birational morphism $f: Y \rightarrow X$ such that $Y$ has rational singularities, and…

Let $L$ be a simply-connected simple connected algebraic group over a number field $F$, and $H$ be a semisimple absolutely maximal connected $F$-subgroup of $L$. Under a cohomological condition, we prove an asymptotic formula for the number…

For a Euclidean building $X$ of type $A_{2}$, we classify the 0-dimensional subbuildings $A$ of $\partial_{T}X$ that occur as the asymptotic boundary of closed convex subsets. In particular, we show that triviality of the holonomy of a…

The main result of this paper is a characterization of the minimal surface hull of a compact set $K$ in $\mathbb R^3$ by sequences of conformal minimal discs whose boundaries converge to $K$ in the measure theoretic sense, and also by…

A well known conjecture asserts that a cubic fourfold X is rational if it has a cohomologically associated K3 surface. G.Ouchi proved that if X admits a finite group G of symplectic automorphisms, whose order is different from 2, then X has…

We show that a rational holonomy representation of any flat manifold except torus must have at least two non-equivalent irreducible subrepresentations. As an application we show that if a K\"ahler flat manifold is not a torus then its…

We prove for a tropical rational map that if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical…

In arXiv:1008.3825, Totaro gave examples of a K3 surface such that its automorphism group is not commensurable with an arithmetic group, answering a question of Mazur. We give examples of rational surfaces with the same property. Our…