Title: Inception of Diffeology

By:    Patrick Iglesias-Zemmour

         (Aix-Marseille Université, & the Hebrew University)

Abstract:

I will summarise the beginning of diffeology when a diffeological space was still called an “espace différentiel”. Then I’ll describe the category {Diffeology} and its main properties. We shall review in details a few examples which emphasises the very special skills of diffeology, compared with other approaches.

Title: Moduli space of Chen’s connections and characteristic classes

By:   Takahiro Matsuyuki

        (Tokyo Institute of Technology)

Abstract:

For a manifold, we can regard the set of homotopy classes of Chen's connections as diffeological space through the diffeology of the de Rham complex. According to Chen's theorem, the quotient space of this space by the structure group plays a role as a classifying space of a smooth fiber bundle, and we can get characteristic classes of a fiber bundle through the cohomology of the space.

Title: Homological properties of decomposition spaces

By:   Akira Koyama

        (Waseda Unversity)

Abstract:

Let $X$ be a continuum in $¥R^n$. It is known that  the homotopy of the decomposition space $¥R^n/X$  depends on only the shape type of $X$. In this talk we introduce an inverse sequence of compact ANR-sets whose limit has the same shape of $¥R^n/X$. Using the inverse sequence, we construct a shape equivalence between the suspension $¥Sigma(X)$ of $X$ and $¥R^n/X$. Then we investigate homologically local properties of the decomposition space $¥R^n/X$ by certain continua such as the Case-Chamberlin curve $C$ and the solenoids. As its consequence, we give an alternative fact that $¥R^3/C$ has the trivial shape but $¥R^3/¥Sigma$ does not have.

Title: Generalized functions and Diffeology

　　 （一般関数とディフェオロジー）

By:   Kazuhisa Shimakawa

        (Okayama University)

Abstract:

We discuss a possibility of extending the definition of morphisms in the category of diffeological spaces. In particular, we aim at constructing a category of smooth spaces in which real functions on the Euclidean n-spaces are those generalized functions (including Schwartz distributions).

（微分空間のカテゴリーにおける射の定義の拡張について論じる。とくに，n 次元ユークリッド空間上の実数値関数の全体が（シュワルツ超関数を含む）一般関数の集合となるようなカテゴリーの構成を目指す）

Abstracts (please find schedule.pdf and abstract.pdf)