The SFB/TRR 109 "Discretization in Geometry and Dynamics" has been funded by the Deutsche Forschungsgemeinschaft e.V. (DFG) since 2012.
The project is a collaboration between:
- the Technische Universität Berlin as lead university,
- the Technische Universität München as partner university,
- and individual scientists from
The central goal of the SFB/Transregio is to pursue research on the discretization of differential geometry and dynamics. In both fields of mathematics, the objects under investigation are usually governed by differential equations. Generally, the term "discretization" refers to any procedure that turns a differential equation into difference equations involving only finitely many variables, whose solutions approximate those of the differential equation.
The common idea of our research in geometry and dynamics is to find and investigate discrete models that exhibit properties and structures characteristic of the corresponding smooth geometric objects and dynamical processes. If we refine the discrete models by decreasing the mesh size they will of course converge in the limit to the conventional description via differential equations. But in addition, the important characteristic qualitative features should be captured even at the discrete level, independent of the continuous limit. The resulting discretizations constitutes a fundamental mathematical theory, which incorporates the classical analog in the continuous limit.
The SFB/Transregio brings together scientists from the fields of geometry and dynamics, to join forces in tackling the numerous problems raised by the challenge of discretizing their respective disciplines.
New film featuring the work of the SFB
- 31.05.2016, 14:15 - 16:30
14:15 - 15:15
Tire track geometry and the filament equation: results and conjectures,
Sergei Tabachnikov (Pennsylvania State University)
The simplest model of a bicycle is a segment of fixed length that can move, in n-dimensional Euclidean space, so that the velocity of the rear end is always aligned with the segment (the rear wheel is fixed on the frame). The rear wheel track and a choice of direction uniquely determine the front wheel track; changing the direction to the opposite, yields another front track. The two track are related by the bicycle (Darboux) transformation which defines a discrete time dynamical system on the space of curves. I shall discuss the symplectic, and in dimension 3, bi-symplectic, nature of this transformation and, in dimension 3, its relation with the filament equation.
An interesting problem is to describe the curves that are in the bicycle correspondence with themselves (in this case, given the front and rear tracks, one cannot tell which way the bicycle went). In dimension two, such curves yield solutions to Ulam's problem: is the round ball the only body that floats in equilibrium in all positions? I shall discuss F. Wegner's results on this problem and relate them with the planar filament equation.
Open problems and conjectures will be emphasized.
15:30 - 16:30
Dynamic isoperimetry and Lagrangian coherent structures,
Gary Froyland (UNSW, Australia)
- The study of transport and mixing processes in dynamical systems is important for the analysis of mathematical models of physical systems. I will describe a novel, direct geometric method to identify subsets of phase space that remain strongly coherent over a finite time duration. The method is based on a dynamic extension of classical (static) isoperimetric problems; the latter are concerned with identifying submanifolds with the smallest boundary size relative to their volume. I will introduce dynamic isoperimetric problems; the study of sets with small boundary size relative to volume as they are evolved by a general dynamical system. I will state dynamic versions of the fundamental (static) isoperimetric (in)equalities; a dynamic Federer-Fleming theorem and a dynamic Cheeger inequality. I will also introduce a dynamic Laplace operator and describe a computational method to identify coherent sets based on eigenfunctions of the dynamic Laplacian. Our results include formal mathematical statements concerning geometric properties of finite-time coherent sets, whose boundaries can be regarded as Lagrangian coherent structures. The computational advantages of this approach are a well-separated spectrum for the dynamic Laplacian, and flexibility in appropriate numerical approximation methods. Finally, we demonstrate that the dynamic Laplace operator can be realised as a zero-diffusion limit of a recent probabilistic transfer operator method for finding coherent sets, based on small diffusion.
- 01.06.2016, 14:15 - 15:15
14:15 - 15:15
Ricci flow, part 2,
- After the introduction to the smooth and discrete Ricci flow of a few weeks ago, I will look deeper into the properties of the first of the discrete Ricci flows based on a weighted triangulation. I will discuss some parts of the proof of convergence of this Ricci flow to a metric of constant curvature and the existence and uniqueness of such metric.
- 07.06.2016, 14:15 - 15:15
14:15 - 15:15
Smooth polyhedral surfaces,
Felix Günther (Max-Planck-Institut für Mathematik, Bonn)
We study the geometry of polyhedral surfaces, which are fundamental
objects in architectural geometry. The aim of this talk is to discuss
suitable assessments of smoothness of polyhedral surfaces. A smooth
reference surface which the polyhedral surface should approximate is not
needed. To describe such properties, we analyze the Gaussian image
of vertex stars and derive restrictions on its shape. By investigating the
discrete Dupin indicatrix we will show that star-shapedness of the
Gaussian images is a good indicator for smoothness in a region of
non-vanishing discrete Gaussian curvature.
(Joint work with Helmut Pottmann.)
- Closing Date: 20.04.2016
- Location: TU Berlin
Current Guests and Visitors
- Prof. Dr. Bernd Sturmfels as Einstein Visiting Fellow at TU Berlin (01.05.2015 - 30.04.2018)
- Prof. Dr. Francisco Santos as Einstein Visiting Fellow at FU Berlin (01.04.2016 - 31.03.2019)
- Albert Chern as Visitor at TU Berlin (07.04.2016 - 30.09.2016)
- Prof. Dr. Peter Schröder as Guest Professor at TU Berlin (09.05.2016 - 26.08.2016)
Forthcoming Guests and Visitors
- Prof. Dr. Sergei Tabachnikov as Guest Professor at TU Berlin (29.05.2016 - 08.06.2016)