Abstract:
The purpose of the present article is threefold. First of all, we rebuild the whole theory of cosimplicial models of mapping spaces by using systematically Kan adjunction techniques. Secondly, given two topological spaces X and Y, we construct a cochain algebra which is quasi-isomorphic (as an algebra) to the singular cochain algebra of the corresponding mapping space from X to Y. Here, X has to be homotopy equivalent to the geometric realization a finite simplicial set and of dimension less or equal to the connectivity of Y. At last, we apply these results to the study of finite group actions on mapping spaces that are induced by an action on the source.

Abstract:
We will study torus actions on Cuntz--Krieger algebras trivially acting on its canonical maximal abelian $C^*$-subalgebras from the view points of continuous orbit equivalence of one-sided topological Markov shifts and flow equivalence of two-sided topological Markov shifts.

Abstract:
This paper develops a duality theory for connected cochain DG algebras, with particular emphasis on the non-commutative aspects. One of the main items is a dualizing DG module which induces a duality between the derived categories of DG left-modules and DG right-modules with finitely generated cohomology. As an application, it is proved that if the canonical module $A / A^{\geq 1}$ has a semi-free resolution where the cohomological degree of the generators is bounded above, then the same is true for each DG module with finitely generated cohomology.

Abstract:
A Poisson analog of the Dixmier-Moeglin equivalence is established for any affine Poisson algebra $R$ on which an algebraic torus $H$ acts rationally, by Poisson automorphisms, such that $R$ has only finitely many prime Poisson $H$-stable ideals. In this setting, an additional characterization of the Poisson primitive ideals of $R$ is obtained -- they are precisely the prime Poisson ideals maximal in their $H$-strata (where two prime Poisson ideals are in the same $H$-stratum if the intersections of their $H$-orbits coincide). Further, the Zariski topology on the space of Poisson primitive ideals of $R$ agrees with the quotient topology induced by the natural surjection from the maximal ideal space of $R$ onto the Poisson primitive ideal space. These theorems apply to many Poisson algebras arising from quantum groups. The full structure of a Poisson algebra is not necessary for the results of this paper, which are developed in the setting of a commutative algebra equipped with a set of derivations.

Abstract:
Consider a local chain Differential Graded algebra, such as the singular chain complex of a pathwise connected topological group. In two previous papers, a number of homological results were proved for such an algebra: An Amplitude Inequality, an Auslander-Buchsbaum Equality, and a Gap Theorem. These were inspired by homological ring theory. By the so-called looking glass principle, one would expect that analogous results exist for simply connected cochain Differential Graded algebras, such as the singular cochain complex of a simply connected topological space. Indeed, this paper establishes such analogous results.

Abstract:
We consider torus actions on Mori dream spaces and ask whether the associated Chow quotient is again a Mori dream space and, if so, what does its Cox ring look like. We provide general tools for the study of these problems and give solutions for k*-actions on smooth quadrics.

Abstract:
We study the equivariant cobordism theory of schemes for torus actions. We give the explicit relation between the equivariant and the ordinary cobordism of schemes with torus action. We deduce analogous results for action of arbitrary connected linear algebraic groups. We prove some structure theorems for the equivariant and ordinary cobordism of schemes with torus action and derive important consequences. We establish the localization theorems in this setting. These are used to describe the structure of the ordinary cobordism ring of certain smooth projective varieties.

Abstract:
Let $f$ be a germ of biholomorphism of $\C^n$, fixing the origin. We show that if the germ commutes with a torus action, then we get information on the germs that can be conjugated to $f$, and furthermore on the existence of a holomorphic linearization or of a holomorphic normalization of $f$. We find out in a complete and computable manner what kind of structure a torus action must have in order to get a Poincar\'e-Dulac holomorphic normalization, studying the possible torsion phenomena. In particular, we link the eigenvalues of $df_O$ to the weight matrix of the action. The link and the structure we found are more complicated than what one would expect; a detailed study was needed to completely understand the relations between torus actions, holomorphic Poincar\'e-Dulac normalizations, and torsion phenomena. We end the article giving an example of techniques that can be used to construct torus actions.

Abstract:
This is a survey on natural local torus actions which arise in integrable dynamical systems, and their relations with other subjects, including: reduced integrability, local normal forms, affine structures, monodromy, global invariants, integrable surgery, convexity properties of momentum maps, localization formulas, integrable PDEs.

Abstract:
For any affine variety equipped with coordinates, there is a surjective, continuous map from its Berkovich space to its tropicalisation. Exploiting torus actions, we develop techniques for finding an explicit, continuous section of this map. In particular, we prove that such a section exists for linear spaces, Grassmannians of planes (reproving a result due to Cueto, H\"abich, and Werner), matrix varieties defined by the vanishing of 3 times 3 minors, and for the hypersurface defined by Cayley's hyperdeterminant.