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## Multi angle  Some projective invariants of convex domains coming from differential geometry Loftin, John C. (Auteur de la Conférence) | CIRM (Editeur )

I will discuss some projective differential geometric invariants of properly convex domains arising from affine dfferential geometry. Consider a properly convex domain $\Omega$ in $R^n\subset RP^n$, and the cone $C$ over $\Omega$ in $R^{n+1}$. Then Cheng-Yau have shown that there is a unique hyperbolic affine sphere which is contained in $C$ and asymptotic to the boundary $\partial C$. The hyperbolic affine sphere is invariant under special linear automorphisms of $C$ , and carries an invariant complete Riemannian metric of negative Ricci curvature, the Blaschke metric. The Blaschke metric descends to a projective-invariantmetric on $\Omega$.
I will also address the relationship between the Blaschke metric and Hilbert metric, which is recent and is due to Benoist-Hulin. At the end, I will discuss applications to the geometry of real projective structures on surfaces.
I will discuss some projective differential geometric invariants of properly convex domains arising from affine dfferential geometry. Consider a properly convex domain $\Omega$ in $R^n\subset RP^n$, and the cone $C$ over $\Omega$ in $R^{n+1}$. Then Cheng-Yau have shown that there is a unique hyperbolic affine sphere which is contained in $C$ and asymptotic to the boundary $\partial C$. The hyperbolic affine sphere is invariant under special ...

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## Multi angle  Metric geometry in homogeneous spaces of the unitary group of a $C^*$-algebra. Part 2: minimal curves Recht, Lazaro (Auteur de la Conférence) | CIRM (Editeur )

Let $P$ be of the unitary group $U_A$ of a $C^*$-algebra $A$. The main result: in the von Neumann algebra context (i.e. if the isotropy sub-algebra is a von Neumann algebra), for each unit tangent vector $X$ at a point, there is a geodesic $\delta (t)$, wich is obtained by the action on $P$ of a $1$-parameter group in $U_A$. This geodesic is minimizing up to length $\pi /2.$

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## Multi angle  Metric geometry in homogeneous spaces of the unitary group of a $C^*$-algebra. Part 1: the quotient metric Andruchow, Esteban (Auteur de la Conférence) | CIRM (Editeur )

définition of the quotient norm - basic properties - existence of minimal liftings: von Neumann algebras - finite dimensional cases - non-uniqueness results - counter-examples: the unitary Fredholm group

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