分野別セミナー

(1) The history of Polar Varieties starts with Blaise Pascal (1623-1662) and his work on conics. Then Jean-Victor Poncelet (1788-1867) introduced the notion of duality by poles and polars, or polar transformation. Examples of polar transformation in Euclidean space $R^3$ give the idea of polar variety. The generalisation by Francesco Severi (1879-1961) and John Arthur Todd (1908-1994) led to the relationship between polar varieties and characteristic classes of smooth manifolds. I will end with more recent results by Lê Dung Trang and Bernard Teissier. They define polar varieties for singular varieties and provide the relation with the characteristic classes of singular varieties, as defined by Marie-Hélène Schwartz and Robert MacPherson.


(2) The Poincaré-Hopf theorem says that, for a compact, manifold without boundary, the Euler-Poincaré characteristic is equal to the sum of the indices of a tangent vector field with isolated singularities. It is well known that the Poincaré-Hopf theorem is no longer true in the case of singular varieties. Counterexamples are not difficult to produce (elementary examples will be given during the lecture). Marie-Hélène Schwartz showed how the result can be retrieved using special vector fields which she called radials and using a Whitney stratification of the singular variety. However, within the framework of Whitney stratifications, the proof is complicated and delicate. We show that using the Lipschitz framework, Marie-Hélène Schwartz's idea of radial vector fields makes sense with nice and easier proof.
This is a joint work with Tadeusz Mostowski and Thuy Nguyen Thi Bich.

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