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.
Film featuring the work of the SFB
- 23.04.2019, 14:15 - 15:15
14:15 - 15:15
Commuting Hamiltonian Flows of Space Curves,
Albert Chern (TU Berlin)
- Starting from the vortex filament flow introduced in 1906 by Da Rios, there is a hierarchy of commuting geometric flows on space curves. The traditional approach relates those flows to the nonlinear Schrödinger hierarchy satisfied by the complex curvature function of the space curve. Rather than working with this infinitesimal invariant, we describe the flows directly as vector fields on manifolds of space curves, which carries a canonical symplectic form introduced by Marsden and Weinstein. The flows are precisely the symplectic gradients of a natural hierarchy of invariants, beginning with length, total torsion, and elastic energy. There are a number of advantages of our geometric approach. For instance, the spectral curve is geometrically realized as the motion of the monodromy axis when varying total torsion. This insight provides a new explicit formula for the hierarchy of Hamiltonians. We also complete the hierarchy of Hamiltonians by adding area and volume. These allow for the characterization of elastic curves as solutions to an isoperimetric problem: elastica are the critical points of length while fixing area and volume.
- 07.05.2019, 14:15 - 16:30
14:15 - 15:15
Perfectly matched layers with discrete complex analysis,
Albert Chern (Technische Universität Berlin)
- Seeking reflectionless boundary conditions for wave equations has been one of the most classical non-trivial problems in numerical PDEs. The method of perfectly matched layer (PML) is regarded as the state-of-the-art approach. A PML is a wave-absorbing layer attached to the boundary designed so ideally that the interface produces no reflection wave. Mathematically, PML equations are viewed as alternate realizations of the analytic continuation of the wave equation. However, it had been believed that numerical reflections are inevitable in discretized PML equations due to discretization error. In this work we adopt the linear discrete complex analysis and replicate the PML theory in the discrete setup. The discovered discrete PML becomes the first truly reflectionless boundary treatment for the discrete wave equation.
- 10.05.2019, 12:00 - 13:00
12:00 - 13:00
Ideal triangulations on hyperbolic surfaces,
Current Guests and Visitors
- Prof. Dr. Bernd Sturmfels as Einstein Visiting Fellow at TU Berlin (01.05.2015 - 31.07.2020)
- Prof. Dr. Peter Schröder as Einstein Visiting Fellow at TU Berlin (01.03.2018 - 28.02.2021)
- Prof. Dr. Francisco Santos as Einstein Visiting Fellow at FU Berlin (01.04.2019 - 31.03.2021)