[geometry-ml:05162] 信州トポロジーセミナー (23-1)

[Keiichi Sakai]

皆様

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信州大学理学部 数学科（松本キャンパス）では，不定期で信州トポロジーセミナーを開催しています．
下記のように、本年度第1回の信州トポロジーセミナー（対面）が開催されます．
（過去の記録につきましては、下記URLをご覧ください）

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■ 2023年6月7日（水）16:30～18:00 ■
題目：On nonstandard and asymptotic extensions of smooth maps between
diffeological spaces
講演者：島川 和久（岡山大学）
会場：理学部 A 棟 4 階 数理・自然情報合同研究室 (A-401)
概要：
In this talk I introduce certain generalizations of smooth maps between
diffeological spaces by utilizing the method of Colombeau's algebra of
generalized functions combined with the idea of nonstandard analysis
originated by A. Robinson. Main topics include the following:
1. On arbitrary diffeological spaces, we can construct two types of
extensions of smooth functions called nonstandard functions and asymptotic
functions, respectively. It is shown that nonstandard (resp. asymptotic)
functions form a smooth algebra over the non-Archimedean field of
Robinson's nonstandard (resp. asymptotic) numbers.
2. If the base space is a Euclidean open set then the algebra of asymptotic
functions comes equipped with partial differential operators and there is a
continuous natural transformation of differential algebras from the
Colombeau algebra of generalized functions to the algebra of asymptotic
functions, which is injective on the linear subspace consisting of Schwartz
distributions and restricts to the identity on the common subspace
consisting of smooth functions.
3. Similarly, nonstandard functions on a Euclidean open set forms a smooth
differential algebra containing the space of Schwartz distributions as a
(topological) linear subspace. But, unlike the case of asymptotic
functions, this inclusion does not extend to a natural transformation over
the Colombeau algebra and restricts to the embedding of smooth functions
only up to negligible difference.
4. Nonstandard functions can be generalized to morphisms between arbitrary
diffeological spaces in such a way that the resulting category is enriched
over Diff (the category of diffeological spaces and smooth maps) and
inherits such noble properties as completeness and cocompleteness.
5. Unfortunately, we cannot extend asymptotic functions to morphisms of a
supercategory of Diff due to the fact that our asymptoticity is not
compatible with composition. Still, there is a partial extension of
asymptotic functions to certain morphisms, called asymptotic maps, in the
case target spaces are subsets of Euclidean spaces. We present several
examples showing that the use of such asymptotic maps provides a flexible
tool to study homotopy theory of diffeological spaces.

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Date: 7 June 2023, 16:30 -- 18:00
Title:  On nonstandard and asymptotic extensions of smooth maps between
Speaker: Kazuhisa Shimakawa (Okayama University)
Room: A-401, Faculty of Science, Shinshu University
Abstract: as above

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--
境 圭一
sakaikeiichi ＠ gmail.com

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