Lecture Proposal 2nd Congress of Graduate Students in Mathematics from IME-USP
• Author’s name: Jean Cerqueira Berni.
• Institution of origin: IME-USP.
• Title of the work: An overview of the Synthetic Differential Geometry.
• 2010 Mathematics Subject Classification: 51K10: Synthetic Differential Geometry; 18axx: General theory of categories and functors.
In this lecture we shall present an overview of Synthetic Differential Geometry, by introducing the famous Kock-Lawvere axiom, which states that every map is smooth, in a sense we shall make precise along the presen- tation. We will see that this axiom captures the essence of ”infinitesimality”and streamlines some proofs of the usual derivations, such as the Chain Rule, the ordinary rules of derivation of sums and products of maps and the expansion of any map in a unique Taylor series - which will give us a stronger statement of the axiom.
It is well known that Synthetic Differential Geometry gives us a much more intuitive approach of the concepts of Differential Geometry, by providing us with infinitesimal objects that fit more in our perception of the geometric theory of curves, surfaces etc. The price paid for this more natural approach is paid by slackening the underlying Logic to this theory, using a constructivist logic, where the principle of the excluded middle is no longer available.
We intend to show in our lecture that this slackening pays off very well, since we can ”connect”several concepts such as a flow of a vector field on a manifold, tangent bundles, vector fields and regard the relationship between Lie algebras and Lie groups using a categorical viewpoint, namely the adjointness property.
We shall be concerned with the categorical approach, by generalizing some objects of Differential Geometry through functors that ”perceive”diagrams that are not necessarily limits or colimits as if they were, the so-called
”microlinear”objects. These objects will broaden the category of objects we shall be concerned with, composing the category of Smooth Spaces, that is more manageable than the category of manifolds for having more categorical constructions, such as products, which allows us to deal in a more simple way with tangent bundles and vector fields, since these are, as we shall present in our lecture, adjoint to each other in a Cartesian closed category.
The stated goal of this research is to analyze whether there is a synthetic version of De Rham’s theorem and, in this case, to find the classifying topos for the geometric theories on which this theorem holds. So far we have studied the fundamentals of Synthetic Differential Geometry and of the Theory of Categories. In the present stage, we are studying Moerdjik’s book [7] in order to understand the essence of the models of the Synthetic Differential Geometry, and then try to elaborate such classifying topos.
References
1 Bell, John L.Two approaches to modelling the Universe: Synthetic Differential Geometry and Frame Valued Sets.
2 Schulman, Michael. Synthetic Differential Geometry. Maio, 2006.
3 Kock, Anders. Synthetic Differential Geometry, http://home.imf.au.dk/kock/sdg99.pdf
4 Kostecki, Ryszard Pawel.Differential Geometry on Toposes. http://www.fuw.edu.pl/ kostecki/sdg.pdf.
5 Lavendhome, Ren´e. Basic Concepts of Synthetic Differential Geometry, Springer-Verlag, 1996.
6 www.wikipedia.org
7 Moerdjik, I. Reyes, G.Models for Smooth Infintesimal Analysis, Springer Verlag, 1991
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