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Many classically used function space structures (including the topology of pointwise convergence, the compact-open topology, the Isbell topology and the continuous convergence) are induced by a hyperspace structure counterpart. This scheme is used to study local properties of function space structures on $C(X,\mathbb R)$, such as character, tighntess, fan-tightness, strong fan-tightness, the Fr{\'e}chet property and some of its variants. Under mild conditions, local properties of $C(X,\mathbb R)$ at the zero function correspond to the same property of the associated hyperspace structure at $X$. The latter is often easy to characterize in terms of covering properties of $X$. This way, many classical results are recovered or refined, and new results are obtained. In particular, it is shown that tightness and character coincide for the continuous convergence on $C(X,\mathbb R)$ and is equal to the Lindel{\"o}f degree of $X$. As a consequence, if $X$ is consonant, the tightness of $C(X,\mathbb R)$ for the compact-open topology is equal to the Lindel{\"o}f degree of $X$.

We survey research on the homotopy theory of the space map(X, Y) consisting of all continuous functions between two topological spaces. We summarize progress on various classification problems for the homotopy types represented by the path-components of map(X, Y). We also discuss work on the homotopy theory of the monoid of self-equivalences aut(X) and of the free loop space LX. We consider these topics in both ordinary homotopy theory as well as after localization. In the latter case, we discuss algebraic models for the localization of function spaces and their applications.

We explain how to calculate the correlation function for SYM N=2 SO(3)-gauge QFT with 4 flavors in terms of top Chern classes of the universal bundles over the moduli spaces of rank 2 stable torsion free coherent sheaves with det=-1 on the complex projective plane. We give a direct geometrical description of these moduli spaces, study a 2-dimensional torus action on them, and calculate several initial coefficients of the correlation function via Bott residue formula.

Coorbit space theory is an abstract approach to function spaces and their atomic decompositions. The original theory developed by Feichtinger and Gr{\"o}chenig in the late 1980ies heavily uses integrable representations of locally compact groups. Their theory covers, in particular, homogeneous Besov-Lizorkin-Triebel spaces, modulation spaces, Bergman spaces, and the recent shearlet spaces. However, inhomogeneous Besov-Lizorkin-Triebel spaces cannot be covered by their group theoretical approach. Later it was recognized by Fornasier and the first named author that one may replace coherent states related to the group representation by more general abstract continuous frames. In the first part of the present paper we significantly extend this abstract generalized coorbit space theory to treat a wider variety of coorbit spaces. A unified approach towards atomic decompositions and Banach frames with new results for general coorbit spaces is presented. In the second part we apply the abstract setting to a specific framework and study coorbits of what we call Peetre spaces. They allow to recover inhomogeneous Besov-Lizorkin-Triebel spaces of various types of interest as coorbits. We obtain several old and new wavelet characterizations based on precise smoothness, decay, and vanishing moment assumptions of the respective wavelet. As main examples we obtain results for weighted spaces (Muckenhoupt, doubling), general 2-microlocal spaces, Besov-Lizorkin-Triebel-Morrey spaces, spaces of dominating mixed smoothness, and even mixtures of the mentioned ones. Due to the generality of our approach, there are many more examples of interest where the abstract coorbit space theory is applicable.

Given a map $f: X\rightarrow Y$ of simply connected spaces of finite type such. The space of based loops at $f$ of the space of maps between $X$ and $Y$ is denoted by $\Omega_{f} Map(X,Y)$. For $n> 0$, we give a model categorical interpretation of the existence (in functorial way) of an injective map of $\mathbb{Q}$-vector spaces $\pi_{n} \Omega_{f}Map(X,Y_{\mathbb{Q}}) \rightarrow HH^{-n}(C^{\ast}(Y),C^{\ast}(X)_{f})$, where $HH^{\ast}$ is the (negative) Hochschild cohomology and $C^{\ast}(X)_{f}$ is the rational cochain complex associated to $X$ equipped with a structure of $C^{\ast}(Y)$-differential graded bimodule via the induced map of differential graded algebras $f^{\ast}: C^{\ast}(Y)\rightarrow C^{\ast}(X)$. Moreover, we identifiy the image in presice way by using the Hodge filtration on Hochschild cohomology. In particular, when $X=Y$, we describe the fundamental group of the identity component of the monoid of self equivalence of a (rationalization of) space $X$ i.e., $\pi_{1} Aut(X_{\mathbb{Q}})_{id}$.

Given a map $f: X\rightarrow Y$ of simply connected spaces of finite type such. The space of based loops at $f$ of the space of maps between $X$ and $Y$ is denoted by $\Omega_{f} Map(X,Y)$. For $n> 0$, we give a model categorical interpretation of the existence (in functorial way) of an injective map of $\mathbb{Q}$-vector spaces $\pi_{n} \Omega_{f}Map(X,Y_{\mathbb{Q}}) \rightarrow HH^{-n}(C^{\ast}(Y),C^{\ast}(X)_{f})$, where $HH^{\ast}$ is the (negative) Hochschild cohomology and $C^{\ast}(X)_{f}$ is the rational cochain complex associated to $X$ equipped with a structure of $C^{\ast}(Y)$-differential graded bimodule via the induced map of differential graded algebras $f^{\ast}: C^{\ast}(Y)\rightarrow C^{\ast}(X)$. Moreover, we identifiy the image in presice way by using the Hodge filtration on Hochschild cohomology. In particular, when $X=Y$, we describe the fundamental group of the identity component of the monoid of self equivalence of a (rationalization of) space $X$ i.e., $\pi_{1} Aut(X_{\mathbb{Q}})_{id}$.