Evelyne Bachmann

Mathematisches Institut
Universität Heidelberg
Im Neuenheimer Feld 205
69120 Heidelberg, Germany


Opening of the Research Station

July 21 at IWH

Welcome to the Research Station Geometry + Dynamics! We will inaugurate the research station with a small celebration on July 21st at the International Academic Forum Heidelberg (IWH). For the scientific part three of our current international guests will give us short insights in their research, Michael Gekhtman (Notre Dame University), Sergei Tabachnikov (Penn State University) and Richard Schwartz (Brown University). Afterwards a reception will be held in the wonderful garden of the IWH.


Michael Gekhtman: Discrete Dynamics from Cluster Algebras

Cluster algebras were introduced by Fomin and Zelevinsky 20 years ago and have since found exciting applications in many areas including algebraic geometry, representation theory, integrable systems and theoretical physics. I will use examples to illustrate how combinatorial ingredients in the definition of a cluster algebra can be interpreted using Poisson geometry and Hamiltonian formalism and then sketch several applications of the theory, including Somos-5 recurrence, pentagram map and its generalizations and dilogarithm identities.

Sergei Tabachnikov: Cusps of caustics by reflection

The boundary of a planar oval is an ideal mirror, and one has a point source of light inside the oval. Consider the rays of light that have undergone N reflection in the mirror, where N=1,2,... The envelope of this system of rays is the Nth caustic by reflection. I shall explain that, for every N, this caustic has at least four cusps. Similar problems for convex surfaces were considered before: Caratheodory proved that the locus of points conjugated to a given point has at least four cusps, and Jacobi stated, in his "Lectures on dynamics", that this number is exactly four in the case of the ellipsoid (this is known as the "Last Geometric Statement of Jacobi"). Our problem is a billiard version of these problems of differential geometry of surfaces, and it belongs to an endless collections of results stemmed from and motivated by the classical 4-vertex theorem of Mukhopadhyaya.

Richard Schwartz: The farthest point map on platonic solids

Imagine two points, Alice and Bob, sitting on the surface of a regular dodecahedron (equipped with its path metric). Alice and Bob dislike each other, so Alice runs away to the point farthest away from Bob. Then Bob runs away to the point farthest away from the new location of Alice. And so on. Where do Alice and Bob go? In this talk I will explain the dynamics of this process, which is called "the farthest point map". A complete understanding of what happens involves a computer-assisted exploration of something called the cut locus of the regular dodecahedron.


14:00 - 14:30 Coffee Break
14:30 - 14:45 Welcome
14:45 - 15:15 Michael Gekhtman
15:15 - 15:45 Coffe Break
15:45 - 16:15 Sergei Tabachnikov
16:30 - 17:00 Richard Schwartz
17:00 Reception in the garden
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