Séminaire – Mathématiques 19e-21e, histoire et philosophie

Jean-Jacques Szczeciniarz (Professeur émérite, Université Paris Cité-SPHERE)

About Penrose Transform the rôle played by the computation

Abstract: This presentation is all about showing you how to install a calculation in multiple variables. It’s all about embracing the power of plurality ! In particular, we’ll be exploring how matrices can be treated as variables, and how points can be represented by matrices.

Paul-Emmanuel Timotei (Doctorant, Université Paris Cité-SPHERE)

Georges-Henri Halphen’s approaches to the reduction of singularities

Abstract: In the second half of the 19th century, in Germany, England, Italy and France, mathematicians became interested in certain geometric invariants. These invariants enabled certain classifications to be made. The development and study of these invariants led mathematicians to study the singularities of geometric objects, insofar as they have an influence on the determination of these invariants. Once knowledge of the influence of « simple » singularities on invariants was established, mathematicians turned their attention to more complex singularities. One way of approaching this problem is to reduce these high singularities to simpler ones.

Georges-Henri Halphen (1844–1889) presented two methods for the reduction of singularities of plane algebraic curves. He developed his methods in reaction to the method that Max Noether (1844–1921) had published, which Halphen considered to be unclear with regard to the result.

In two later texts, Halphen qualified his methods in two different ways. He describes the first one as being more geometric than Noether’s method, while he comments on the second one, stating that it proceeds using of trulygeometric transformations. We notice that Halphen always compares his methods to Noether’s and he uses the term geometric to qualify them twice, but with different meanings.

What meaning does Halphen give to the word geometric in the context of the reduction of singularities?

To explore this issue, I will present Noether’s and Halphen’s methods of the reduction of singularities and their knowledge about this notion. Then I compare the three methods and I explore the different meanings Halphen gives to the word geometric.

Lieu : Salle 569 bâtiment Olympe de Gouges