Du 11 au 12 juin 2015
IRMA
The conference "Geometry in History" will take place at Institut de Recherche Mathématique Avancée (University of Strasbourg and CNRS) on June 11-12, 2015.
Organizers: S.G. Dani (Bombay) and A. Papadopoulos (Strasbourg)
Each talk will present a major geometrical idea in a historical context together with its modern ramifications. The talks are given by mathematicians.
The speakers will include:
- N. A'Campo (Basel)
- J.-P Bourguignon (IHES)
- C. Frances (Strasbourg)
- L. Ji (Ann Arbor)
- K. Ohshika (Osaka)
- V. Poenaru (Orsay)
- N. Schappacher (Strasbourg)
- A. Sossinsky (Moscow)
- T. Sunada (Tokyo)
- J. Vargas (PST Associates, LLC, USA)
The talks will be in English and they are intended for a general audience.
_ Graduate students and young mathematicians are welcome. Registration is required (and free of charge) at this link. Hotel booking can be asked for through the registration link.
_ For practical matters and other questions please contact the organizers :
- Shrikrishna Dani : shrigodani@gmail.com
- Athanase Papadopoulos : athanase.papadopoulos@math.unistra.fr
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Jeudi 11 juin 2015
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09:00
Ken'ichi Ohshika, Osaka
The origin of the notion of manifold
Abstract.--- Starting from Kant’s philosophy and Riemann’s Habilitationsvorstag, I will review how the notion of manifold came into world, and developed into the one as it is today. I shall also talk about philosophical aspects of this development. -
10:00
Coffee Break
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10:30
Alexey Sossinsky, Independent University of Moscow
On the approval of non-Euclidean geometries by the mathematical community
Abstract.--- The question that I try to answer in this talk is why it took the mathematical community so long to approve non-Euclidean geometry. I treat this question from a viewpoint that is more philosophical and psychological than historical -- essentially the answer is related to the fundamental question "what is mathematics about?" and to the inertia of the human mind.
I begin with a short account of the 80 year chronology of events related to non-Euclidean geometries, from 1817, when Gauss understood what he had discovered, until the turn of the 19th century, when, following the publication of Hilbert's "Grundtlagen der Geometrie" and Poincar\'e's "Science et
Hypoth\`ese", it became clear that non-Euclidean geometries were an acceptable and important part of mathematics.
This will be followed by an analysis of 20th century criticism of Euclid's "Elements": this criticism is based, in my opinion, on a total misunderstanding of what the "Elements" are about, and I will try to explain what the correct understanding of Euclid's approach should be, and how it is related to the real world and to Plato's world of ideal forms.
I will then explain why Gauss, Bolyai, and Lobachesky, blinded as they were by their naive Platonicism and Kant's understanding of geometry as a typical item in the category of synthetic apriori, were unable to prove the equiconsistency of hyperbolic and Euclidean geometry, although they had all the necessary ingredients at their fingertips.
Finally, I give my answer to the main question of this talk. The answer is related to what may be called the "Copernican revolution" of the role of mathematics and to different answers to the philosophical question: What is mathematics about? I distinguish three main philosophical trends in answering the latter question: formalism (Aristotle, Leibnitz, Hilbert, Bourbaki), realism (Euclid, Newton, Kolmogorov, Arnold), pragmatism (Poincar\'e, Carnap, Wittgenstein) and analyze their understanding of the role the non-Euclidean geometries in contemporary science.
As an afterthought, I will remark on the strange workings of the human mind whenever faced with any fundamentally new idea. -
11:30
Charles Frances, Strasbourg
Cartan geometries: some classical and recent aspects
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14:00
Lizhen Ji, Ann Arbor
A short history of moduli and Teichmueller spaces
Abstrac.--- Though moduli spaces of Riemann surfaces and Teichmuller spaces are some of the central objects in mathematics and have been intensively and extensively studied by many people, their history seems to be complicated and confusing to various people. In this talk, I will describe some milestones in their history and applications. This talk is based on joint work with Norbert A'Campo and Athanase Papadopoulos. -
15:00
Coffee Break
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15:30
Valentin Poenaru, (Orsay)
A fast glimpse into the big problems of four dimensions
Abstract.--- I will try to explain the special place of 4d ,inside the topology of manifolds. A fast survey of Casson Handles, Yang-Mills and exotic R4’s will be presented.Some open pronblems will also be explained. -
19:30
Conference Dinner
Restaurant "Au Petit Bois vert", quartier Petite France. Everybody is invited. We shall leave IRMA at 19:00
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Vendredi 12 juin 2015
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09:00
Jean-Pierre Bourguignon, IHES et Bruxelles
Parallel Transport and Holonomy, between Mathematics and Physics
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10:00
Coffee Break
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10:30
Jose G. Vargas, PST Associates, LLC (USA)
From Grassmann to Kaehler, through Clifford and Elie Cartan
Abstract.--- The Ausdehnungslehre of Grassmann (1844) marks the belated beginning of two other major developments in mathematics, namely Cli fford algebra and Cartan's exterior calculus. Kaehler later produced a calculus which is to Cli fford algebra what Cartan's is to exterior algebra.
Algebraists are familiar with the expression "The tragedy of Grassmann", meaning in particular the neglect and/or misinterpretation of Grassmann's ideas, which continue to this day. Given the tone in the preface of
their related work, some algebraists of the last half century have anointed themselves experts on the tragedy, but they disagree.
Disagreements sometimes arise because they overlook that their considerations pertain to algebraic/geometric invariant theory, but not to
the theory of di fferential/integral invariants. In the first of those theories, one is interested in the interaction of elements of diff erent, arbitrary grades; the main ideas, like union and intersection, originate in set theory.
In the second of those theories, one is primarily interested in the action of a di fferential operator of grade one on an arbitrary di fferential form; this operator does not have its origin in structures simpler than di fferentiable
manifolds.
I invite judgement of those disagreeing interpretations on the basis of the following:
a) Cartan's exposition of Grassmann's progressive-regressive system,
b) Cartan's occasional use of this system in the physics part of his papers on the theory of affine and Euclidean connections,
c) Cartan's view of di fferential geometry as pertaining to the product of two exterior algebras, its elements being exterior-valued exterior diff erential forms, and
d) Kaehler's innovative calculus of tensor-valued (a step backwards) Cli fford-based (a major step forward) di fferential forms.
In the author's opinion, a more Cartanian view of the Kaehler calculus is obtained through replacement of tensor-valuedness algebra with Cli fford valuedness algebra. This results
in a structure suitable for the two types of invariant theory to which we referred above. For the algebraic invariant theory, one simply chooses
the available products that best fit each speci fic application. For the differential/integral invariant theory, the Kaehler calculus has no competitors. -
11:30
Toshikazu Sunada, Meiji University, Tokyo
Riemann sums and near quasicrystals
Abstract.--- Counting things is a great favorite of children, and
mathematicians as well, whatever the things are. The primary aim of this expository talk is, interspersing with historical background, to explain how the idea of Riemann sum is linked to other branches of mathematics, especially various counting problems. The materials we treat are ones available to the ``mathematician in the streets" except for a few. However one may still see interesting inter-connection and cohesiveness in mathematics. -
14:30
Norbert Schappacher, Strasbourg
Critical moments from the History of Intersection Theory, 1890 - 1990
Abstract.--- Since it is impossible to present the history of intersection theory in algebraic geometry
in one talk, this talk will follow one concrete example through the century indicated in the
title, from Kronecker to Fulton. The debates that this example trigerred naturally reflect the
evolution of algebraic geometry, but they also show that this evolution is more complicated
than we usually tend to think ….. -
15:30
Coffee Break
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16:00
Tom Archibald, (Simon Fraser University, Vancouver, Canada)
Counterexamples in Weierstrass’ Work (This talk is part of the IRMA colloquium series talks)
Abstract.--- Weierstrass is well known for having produced a variety of counterexamples that show certain mathematical distinctions to be critical or that demonstrate certain assumptions unwarranted. In this paper we look at three of these involving the Dirichlet principle, the existence of everywhere continuous nowhere differentiable functions, and natural boundaries for the domain of analytic functions. All these are very well-known, but we draw attention to the different force that they had for Weierstrass and his readers and hearers than they would have today. The reason for this is allied to changes in the view of mathematics that emerged over the course of the nineteenth century. In the early decades of the century, most writers saw mathematics as more akin to a natural science than to a body of knowledge created by human effort independent of nature. In this pre-axiomatic era (before Hilbert and Russell, for example) the force of a definition was descriptive and somewhat looser than in the strict semantic role that we now give it. Weierstrass’ work is thus a participant in and a driving force for this transition, and it is this issue we will try to illustrate. The examples likewise show an interesting side of the competitiveness Weierstrass felt with respect to the achievements of Riemann.