Sergei Tabachnikov

Arnold’s Problem

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Centers of Mass of Poncelet Polygons, 200 Years After

The locus of the centers of mass of the family of Poncelet polygons, inscribed into a conic

Hyperbolic Caratheodory Conjecture

\Gamma

Vladimir Igorevich Arnold. 12 June 1937—3 June 2010

and circumscribed about a conic

Billiard transformations of parallel flows: A periscope theorem

\gamma

Osculating Curves: Around the Tait-Kneser Theorem

, is a conic homothetic to

Quasiperiodic motion for the pentagram map

\Gamma

Dragon Curves Revisited

.

Tire Tracks and Integrable Curve Evolution

A quadratic point on a surface in ℝP3 is a point at which the surface can be approximated by a quadric abnormally well (up to order 3). We conjecture that the least number of quadratic points on a generic compact nondegenerate hyperbolic surface is 8; the relation between this and the classic Carathéodory conjecture is similar to the relation between the six-vertex and the four-vertex theorems on plane curves. Examples of quartic perturbations of the standard hyperboloid confirm our conjecture. Our main result is a linearization and reformulation of the problem in the framework of the 2-dimensional Sturm theory; we also define a signature of a quadratic point and calculate local normal forms recovering and generalizing the Tresse-Wilczynski theorem.

A Singular Mathematical Promenade by Étienne Ghys

Vladimir Arnold was a pre-eminent mathematician of the second half of the twentieth and early twenty-first century. Kolmogorov–Arnold–Moser (KAM) theory, Arnold diffusion, Arnold tongues in bifurcation theory, Liouville–Arnold theorem in completely integrable systems, Arnold conjectures in symplectic topology—this is a very incomplete list of notions and results named after him. Arnold was a charismatic leader of a mathematical school, a prolific writer, a flamboyant speaker and a tremendously erudite person. Our biographical sketch describes his extraordinary personality and his major contributions to mathematics.