## Prochainement

Lundi 8 mars 14:00-15:00 Thierry De Pauw (East China Normal University, Shangai)
Sur les ensembles Lebesgue négligeables, les ensemble de Nikodym et un problème de Zygmund

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Résumé : On considère des entiers $0 < m < n$ et $R^n \to G(n,m) : x \mapsto W(x)$ un champ de m-plans tel que $x \in W(x)$. Si $2 = n = m+1$ il existe un ensemble conégligeable $A \subset [0,1] \times [0,1]$ (appelé ensemble de Nikodym) et il existe $W$ continu tels que pour tout $x \in A$ on a $A \cap W(x) = x$. Nous démontrons, en toutes dimensions et codimensions, que si $W$ est lipschitzien et $A$ est borélien alors $A$ est négligeable si et seulement si $H^m(A \cap W(x)) = 0$ pour presque tout $x \in A$, où $H^m$ désigne la mesure de Hausdorff de dimension $m$. On obtient en fait un résultat quantitatif plus fort : Pour preque tout $x \in A$ on a $\limsup_r \to 0 \fracH^m(A \cap W(x) \cap B(x,r))r^m \geq c(n,m)$.

## Passés

Lundi 1er mars 14:00-15:00 Slawomir Dinew (Jagiellonian University, Krakow, Poland)
Plurisubharmonic functions with subharmonic singularities

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Résumé : We shall discuss extension of plurisubharmonic functions past small sets.
We shall prove that for any closed set E of measure zero subharmonic
functions which are plurisubharmonic off E extends plurisubharmonically

Lundi 8 février 14:00-15:00 Alexander Strohmaier (Leeds Univ. )
A trace formula in obstacle scattering

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Résumé : I will give a brief introduction about the spectral properties of the Laplace operator in obstacle scattering. I will explain the relation to Zeta functions that appear in local quantum field theory. I will then show that an appropriately defined stress energy tensor is the integral kernel of a trace-class operator whose trace can be interpreted as the vacuum energy. I will introduce a more abstract and more generally applicable version of this statement and an associated trace-formula.
This also proves the equivalence and convergence of several methods used in the physics literature to compute Casimir interactions. The trace-formula is somewhat different from the Birman-Krein formula and is interesting in its own right.
Knowledge of quantum field theory will not be assumed. All quantities will be defined in terms of functional calculus of the Laplace operator. The talk summarises joint work with F. Hanisch, Yan-Long Fang, Alden Waters, Timo Betcke, and Xiaoshu Sun.

Lundi 1er février 14:00-15:00 Siarhei Finski (Université Grenoble Alpes,)
On characteristic forms of positive vector bundles, local Kempf-Laksov formula and mixed discriminants.

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Notes de dernières minutes : By a theorem of Kodaira, for a line bundle over a compact Kähler manifold, the positivity of the first Chern class is equivalent to its ampleness. For vector bundles of higher rank, there are several widely used notions of positivity, and the precise relation between them and ampleness is still only conjectural. In this talk we will discuss a certain relation between those notions of positivity for a vector bundle and the positivity of the associated characteristic forms. In particular, we will establish a differential-geometric version of the result of Fulton-Lazarfeld about the description of positive characteristic classes for ample vector bundles through Schur polynomials.

Lundi 25 janvier 14:00-15:00 Francisco Torres de Lizaur  (University of Toronto)
A characterization of steady Euler flows

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Résumé : I will show how to characterize those non-singular volume-preserving
vector fields on a closed manifold that are steady solutions to the
Euler equation for some Riemannian metric. Given a vector field, the
existence of such a metric depends on the existence of a limit to the
precision with which the asymptotic cycles can be approximated by
certain classes of loops. This is joint work with Daniel Peralta-Salas
and Ana Rechtman.

 Département de Mathématiques Bâtiment 307 Faculté des Sciences d'Orsay Université Paris-Saclay F-91405 Orsay Cedex Tél. : +33 (0) 1-69-15-79-56 Département Fermeture du département Laboratoire Formation