Geometric and Algebraic Modeling

Organizers: Maria Alberich-Carramiñana and Edwin León-Cardenal

Date & Time: Tuesday, September 29, 2026 | 17:00 - 18:40
Place: Campus South, Mathematics Building 20.30 [Englerstraße 02], Room 2.059
Chairs: Edwin León-Cardenal

Abstract:

This mini-symposium will serve as a meeting point for researchers exploring geometry, algebra and its interdisciplinary applications. It aims to provide an overview of some recent progress in the field with views towards applied contexts in science and engineering.

Over the years, geometric and algebraic modeling has undergone significant expansion, evolving from its classical foundations to encompass a broad spectrum of modern topics. These include, among others, non-linear geometric and algebraic structures, computational geometry, algebraic geometry, approximation theory, computer-aided geometric design, and their interactions with other areas of mathematics and computation. Although traditionally rooted in pure and applied mathematics, geometric and algebraic tools have become a fundamental framework in diverse fields such as computer graphics, computer-aided design, robotics, biology, engineering, and data science.

By fostering interaction between theoretical and applied researchers, this mini-symposium aims to stimulate collaboration, generate fresh insights, and further advance the understanding of geometric and algebraic modeling both within mathematics and across disciplines.

Confirmed Speakers

Speaker: Àlex Garcia-Herranz (Universitat Politècnica de Catalunya)
Title: Lie-Poisson Reduction and Stability of Kirchhoff Rods Under Gravity.
Abstract: Modeling heavy continuous elastic rods under gravity typically requires computationally intensive discretizations or simplifying approximations. This work addresses these limitations by developing a continuous mathematical model and a low-dimensional representation of Kirchhoff rods suitable for robotic control. While gravity breaks the classical left-invariance symmetry of the Hamiltonian on , we recover a reducible geometric structure by embedding the system into the semidirect product group . Lie-Poisson reduction then projects the system from the cotangent bundle to a low-dimensional dual Lie algebra.
We validate this framework by recovering the classical balance laws for internal forces and moments. Crucially for robotic applications, we prove that the mapping from the initial base wrench to the corresponding equilibrium rod configuration is a global diffeomorphism, establishing the six-dimensional base wrench space as a global control atlas.. Combined with a proof of path-connectedness for segmented stable configurations, this geometric formulation provides a foundation for global motion planning and safe manipulation of soft continuum robots and cables.

Speaker: Werner Seiler (Universität Kassel)
Title: Geometric Modelling of Singular Initial and Boundary Value Problems: Theory and Numerics
Abstract: We consider quasilinear ordinary differential equations both theoretically and numerically. We call an initial or boundary value problem singular, if at an initial or boundary condition the coefficient of a derivative of highest order vanishes. In such a situation, the well-known existence and uniqueness theorems cannot be applied and also classical numerical integrators typically break down in the vicinity of such a singularity. Using a geometric view of a differential equation as a submanifold in a suitable jet bundle, we define an explicit dynamical system on this submanifold and relate questions of existence, (non)uniqueness and regularity of solutions to standard problems in dynamical systems theory. Furthermore, this point of view leads in a straightforward manner to a simple and robust numerical approach to singular initial and boundary value problems. As examples, we will consider classical equations like Lane-Emden or Thomas-Fermi and show how the for applications relevant initial and boundary value problems can be solved in a few lines of Maple code.

Speaker: Abhilash Nayak (Universitat Politècnica de Catalunya)
Title: Geometric Modeling of Kinematic Grasp Constraints for Robotic Cloth Folding
Abstract: Robotic cloth folding requires coupling the rigid motion of a gripper with the deformation of a flexible surface. We present a lightweight geometric interface in which the gripper is modeled by a pose, a jaw state, and an attached grasping volume, represented as a box or a square pyramid in the gripper frame. When the gripper closes, cloth vertices inside this volume are selected, stored in local coordinates, and transported by quaternion-based rigid transformations. The resulting constraints are integrated into an inextensible cloth simulator and tested on a Franka robot arm by performing a corner-folding motion, highlighting the role of grasp support and gripper orientation.