Introducing symplectic billiards

In this article we introduce a simple dynamical system called symplectic billiards. As opposed to usual/Birkhoff billiards, where length is the generating function, for symplectic billiards symplectic area is the generating function. We explore basic properties and exhibit several similarities, but also differences of symplectic billiards to Birkhoff billiards.Comment: 41 pages, 16 figure

Total Curvature of Graphs after Milnor and Euler

We define a new notion of total curvature, called net total curvature, for finite graphs embedded in Rn, and investigate its properties. Two guiding principles are given by Milnor's way of measuring the local crookedness of a Jordan curve via a Crofton-type formula, and by considering the double cover of a given graph as an Eulerian circuit. The strength of combining these ideas in defining the curvature functional is (1) it allows us to interpret the singular/non-eulidean behavior at the vertices of the graph as a superposition of vertices of a 1-dimensional manifold, and thus (2) one can compute the total curvature for a wide range of graphs by contrasting local and global properties of the graph utilizing the integral geometric representation of the curvature. A collection of results on upper/lower bounds of the total curvature on isotopy/homeomorphism classes of embeddings is presented, which in turn demonstrates the effectiveness of net total curvature as a new functional measuring complexity of spatial graphs in differential-geometric terms.Comment: Most of the results contained in "Total curvature and isotopy of graphs in $R^3$."(arXiv:0806.0406) have been incorporated into the current articl

Arbeitsgemeinschaft: Mathematical Billards

The workshop Mathematical Billiards, organised by Serge Tabachnikov (Penn State) and Serge Troubetzkoy (Marseille) was held April 4th–April 10th, 2010. This meeting was well attended by over 40 participants including a number of master and PhD students, with broad geographic representation. This workshop was a nice blend of researchers with various backgrounds who brought in their various point of views to cover the classics as well as recent advances in mathematical billiards and flat surfaces. The report consists in the abstracts for the 18 lectures, followed by the abstracts for the 4 short talks that took place in the evenings. During the workshop, there was also a demo of the mathematical software Sage