Related papers: Abstract Swiss Cheese Space and the Classicalisati…
Swiss cheese sets have been used in the literature as useful examples in the study of rational approximation and uniform algebras. In this paper, we give a survey of Swiss cheese constructions and related results. We describe some notable…
In this paper we consider the compact plane sets known as Swiss cheese sets, which are a useful source of examples in the theory of uniform algebras and rational approximation. We introduce a notion of 'allocation map' connected with Swiss…
A theory of allocation maps has been developed by J. F. Feinstein and M. J. Heath in order to prove a theorem, using Zorn's lemma, concerning the compact plane sets known as Swiss cheese sets. These sets are important since, as domains,…
We introduce a new operad, which we call the Swiss-cheese operad. It mixes naturally the little disks and the little intervals operads. The Swiss-cheese operad is related to the configuration spaces of points on the upper half-plane and…
It is shown that there exists a compact set $X$ in ${\bf C}^N$ ($N\geq 2$) such that $\widehat X\setminus X$ is nonempty and the uniform algebra $P(X)$ has a dense set of invertible elements, a large Gleason part, and an abundance of…
It is shown that there exists a compact planar set K such that the uniform algebra R(K) is nontrivial and strongly regular. This settles an issue raised by Donald Wilken 55 years ago. It is shown that the set K can be chosen such that, in…
We prove a conjecture of Kontsevich which states that if $A$ is an $E_{d-1}$ algebra then the Hochschild cohomology object of $A$ is the universal $E_d$ algebra acting on $A$. The notion of an $E_d$ algebra acting on an $E_{d-1}$ algebra…
We study bicolored configurations of points in the Euclidean $n$-space that are constrained to remain either inside or outside a fixed Euclidean $m$-subspace, with $n - m \ge 2$. We define a higher-codimensional variant of the Swiss-Cheese…
The notion of soft sets is introduced as a general mathematical tool for dealing with uncertainty. In this paper, we consider the concepts of soft compactness, countably soft compactness and obtain some results. We study some soft…
We build a model in groupoids for the Swiss-Cheese operad, based on parenthesized permutations and braids, and we relate algebras over this model to the classical description of algebras over the homology of the Swiss-Cheese operad. We…
We prove that for a large family of product graphs, and for Kneser graphs $K(n,\alpha n)$ with fixed $\alpha <1/2$, the following holds. Any set of vertices that spans a small proportion of the edges in the graph can be made independent by…
By considering the quantum Oppenheimer-Snyder model in loop quantum cosmology, a new quantum black hole model whose metric tensor is a suitably deformed Schwarzschild one is derived. The quantum effects imply a lower bound on the mass of…
In this note we construct Swiss cheeses X such that R(X) is non-regular but such that R(X) has no non-trivial Jensen measures. We also construct a non-regular uniform algebra with compact, metrizable character space such that every point of…
For an associative algebra A we consider the pair "the Hochschild cochain complex C*(A,A) and the algebra A". There is a natural 2-colored operad which acts on this pair. We show that this operad is quasi-isomorphic to the singular chain…
From a coloured operad $\mathcal{P}$ and a $\mathcal{P}$-algebra $A$, we construct a new operad $\mathrm{SC}(\mathcal{P})$ and a Hochschild object $\mathrm{Hoch}(A)$ together with an $\mathrm{SC}(\mathcal{P})$-action on the pair…
In this article, we study the Lipschitz Geometry at infinity of complex analytic sets and we obtain results on algebraicity of analytic sets and on Bernstein's problem. Moser's Bernstein Theorem says that a minimal hypersurface which is a…
In the present work, we extract pairs of topological spaces from maps between coloured operads. We prove that those pairs are weakly equivalent to explicit algebras over the one dimensional Swiss-Cheese operad SC_{1}. Thereafter, we show…
Among other things, we show that the ideal sheaf of a complex Hilbert submanifold of a pseudoconvex open subset of Hilbert space is acyclic over the ambient pseudoconvex open set. We also prove a vanishing theorem for a fairly general class…
In view of interpreting the cosmological observations precisely, especially when they involve narrow light beams, it is crucial to understand how light propagates in our statistically homogeneous, clumpy, Universe. Among the various…
We give a brief review of our previous works: [1,2]. We discuss two sets of issues. The first has to do with the possibility of getting a non-supersymmetric dS minimum without the addition of anti-D3 branes as in KKLT, and axionic slow-roll…