Abstract:
We prove that if the outer billiard map around a plane oval is algebraically integrable in a certain non-degenerate sense then the oval is an ellipse.

Abstract:
The classical Sturm-Hurwitz-Kellogg theorem asserts that a function, orthogonal to an n-dimensional Chebyshev system on a circle, has at least n+1 sign changes. We prove the converse: given an n-dimensional Chebyshev system on a circle and a function with at least n+1 sign changes, there exists an orientation preserving diffeomorphism of the circle that takes this function to a function, orthogonal to the Chebyshev system. We also prove that if a function on the real projective line has at least four sign changes then there exists an orientation preserving diffeomorphism of the projective line that takes this function to the Schwarzian derivative of some function. These results extend the converse four vertex theorem of H. Gluck and B. Dahlberg: a function on a circle with at least two local maxima and two local minima is the curvature of a closed plane curve.

Abstract:
Given a plane oval, is it possible to go around it so that, at all times, the two tangent segments to the oval from the moving point have unequal lengths? In this note we construct an example of such an oval.

Abstract:
A manifold is T-embedded into an affine space if its tangent spaces at distinct points are disjoint. We prove that an n-dimensional disc cannot be T-embedded into 2n-dimensional space.

Abstract:
We deduce a recent theorem by R. Schwartz on the structure of the so-called Poncelet grid from complete integrability of the billiard in an ellipse

Abstract:
Many classical facts in Riemannian geometry have their pseudo-Riemannian analogs. For instance, the spaces of space-like and time-like geodesics on a pseudo-Riemannian manifold have natural symplectic structures (just like in the Riemannian case), while the space of light-like geodesics has a natural contact structure. We discuss the geometry of these structures in detail, as well as introduce and study pseudo-Euclidean billiards. In particular, we prove pseudo-Euclidean analogs of the Jacobi-Chasles theorems and show the integrability of the billiard in the ellipsoid and the geodesic flow on the ellipsoid in a pseudo-Euclidean space.

Abstract:
Following a recent paper by Baryshnikov and Zharnitskii, we consider outer billiards in the plane possessing invariant curves consisting of periodic orbits. We prove the existence and abundance of such tables using tools from sub-Riemannian geometry. We also prove that the set of 3-periodic outer billiard orbits has zero measure.

Abstract:
The model of a bicycle is a unit segment AB that can move in the plane so that it remains tangent to the trajectory of point A (the rear wheel is fixed on the bicycle frame); the same model describes the hatchet planimeter. The trajectory of the front wheel and the initial position of the bicycle uniquely determine its motion and its terminal position; the monodromy map sending the initial position to the terminal one arises. According to R. Foote's theorem, this mapping of a circle to a circle is a Moebius transformation. We extend this result to multi-dimensional setting. Moebius transformations belong to one of the three types: elliptic, parabolic and hyperbolic. We prove a 100 years old Menzin's conjecture: if the front wheel track is an oval with area at least pi then the respective monodromy is hyperbolic. We also study bicycle motions introduced by D. Finn in which the rear wheel follows the track of the front wheel. Such a ''unicycle" track becomes more and more oscillatory in forward direction. We prove that it cannot be infinitely extended backward and relate the problem to the geometry of the space of forward semi-infinite equilateral linkages.

Abstract:
An affine Hopf fibration is a fibration of n-dimensional real affine space by p-dimensional pairwise skew affine subspaces. An example is a fibration of 3-space by pairwise skew lines, the result of the central projection of the classical Hopf fibration of 3-sphere. In this expository article, we describe the solution of the following problem: for which values of n and p does an affine Hopf fibration exist? The answer is given in terms of the Hurwitz-Radon function.

Abstract:
We obtain bounds on the least dimension of an affine space that can contain an $n$-dimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points. This problem is closely related to the generalized vector field problem, non-singular bilinear maps, and the immersion problem for real projective spaces.