Abstract:
Let $\xi\geq 1$ be a countable ordinal. We study the Borel subsets of the plane that can be made ${\bf\Pi}^0_\xi$ by refining the Polish topology on the real line. These sets are called potentially ${\bf\Pi}^0_\xi$. We give a Hurewicz-like test to recognize potentially ${\bf\Pi}^0_\xi$ sets.

Abstract:
We prove that a nonempty closed and geodesically convex subset of the $l_{\infty}$ plane $\mathbb{R}^2_{\infty}$ is hyperconvex and we characterize the tight spans of arbitrary subsets of $\mathbb{R}^2_{\infty}$ via this property: Given any nonempty $X\subseteq\mathbb{R}^2_{\infty}$, a closed, geodesically convex and minimal subset $Y\subseteq\mathbb{R}^2_{\infty}$ containing $X$ is isometric to the tight span $T(X)$ of $X$.

Abstract:
Let xi be a non-null countable ordinal. We study the Borel subsets of the plane that can be made $\bormxi$ by refining the Polish topology on the real line. These sets are called potentially $\bormxi$. We give a Hurewicz-like test to recognize potentially $\bormxi$ sets.

Abstract:
Let $X$ be a Banach space and $Conv_H(X)$ be the space of non-empty closed convex subsets of $X$, endowed with the Hausdorff metric $d_H$. We prove that each connected component of the space $Conv_H(X)$ is homeomorphic to one of the spaces: a singleton, the real line, a closed half-plane, the Hilbert cube multiplied by the half-line, the separable Hilbert space, or a Hilbert space of density not less than continuum.

Abstract:
In the paper we study relations of rigidity, equicontinuity and pointwise recurrence between a t.d.s. $(X,T)$ and the t.d.s. $(K(X),T_K)$ induced on the hyperspace $K(X)$ of all compact subsets of $X$, and provide some characterizations. Among other examples, we construct a minimal, non-equicontinuous, distal and uniformly rigid t.d.s. and a t.d.s. which has dense small periodic sets but does not have dense distal points, solving that way open questions existing in the literature.

Abstract:
For an arbitrary simple Lie algebra $\g$ and an arbitrary root of unity $q,$ the closed subsets of the Weyl alcove of the quantum group $U_q(\g)$ are classified. Here a closed subset is a set such that if any two weights in the Weyl alcove are in the set, so is any weight in the Weyl alcove which corresponds to an irreducible summand of the tensor product of a pair of representations with highest weights the two original weights. The ribbon category associated to each closed subset admits a ``quotient'' by a trivial subcategory as described by Brugui\`eres and M\"uger, to give a modular category and a framed three-manifold invariant or a spin modular category and a spin three-manifold invariant. Most of these theories are equivalent to theories defined in previous work of the author, but several exceptional cases represent the first nontrivial examples to the author's knowledge of theories which contain noninvertible trivial objects, making the theory much richer and more complex.

Abstract:
We prove the elementary but surprising fact that the Hofer distance between two closed subsets of a symplectic manifold can be expressed in terms of the restrictions of Hamiltonians to one of the subsets; this helps explain certain energy-capacity inequalities that appeared recently in work of Borman-McLean and Humiliere-Leclercq-Seyfaddini. We also build on arXiv:1201.2926 to obtain new vanishing results for the Hofer distance between subsets, applicable for instance to singular analytic subvarieties of Kahler manifolds.

Abstract:
Some properties of minimal closed sets and maximal closed sets are obtained, which are dual concepts of maximal open sets and minimal open sets, respectively. Common properties of minimal closed sets and minimal open sets are clarified; similarly, common properties of maximal closed sets and maximal open sets are obtained. Moreover, interrelations of these four concepts are studied.

Abstract:
The causal structure of space-time offers a natural notion of an opposite or orthogonal in the logical sense, where the opposite of a set is formed by all points non time-like related with it. We show that for a general space-time the algebra of subsets that arises from this negation operation is a complete orthomodular lattice, and thus has several of the properties characterizing the algebra physical propositions in quantum mechanics. We think this fact could be used to investigate causal structure in an algebraic context. As a first step in this direction we show that the causal lattice is in addition atomic, find its atoms, and give necesary and sufficient conditions for ireducibility.

Abstract:
Considering the space of closed subsets of $\mathbb{R}^d$, endowed with the Chabauty-Fell topology, and the affine action of $SL_d(\mathbb{R})\ltimes\mathbb{R}^d$, we prove that the only minimal subsystems are the fixed points $\{\varnothing\}$ and $\{\mathbb{R}^d\}$. As a consequence we resolve a question of Gowers concerning the existence of certain Danzer sets: there is no set $Y \subset \mathbb{R}^d$ such that for every convex set $\mathcal{C} \subset \mathbb{R}^d$ of volume one, the cardinality of $\mathcal{C} \cap Y$ is bounded above and below by nonzero contants independent of $\mathcal{C}$. We also provide a short independent proof of this fact and deduce a quantitative consequence: for every $\varepsilon$-net $N$ for convex sets in $[0,1]^d$ there is a convex set of volume $\varepsilon$ containing at least $\Omega(\log\log(1/\varepsilon))$ points of $N$.