Research in IRTG 3132

Program Overview

Our International Research Training Group (IRTG) represents a unique collaboration between the University of Freiburg, the University of Pisa, and the Scuola Normale Superiore in Pisa. This partnership creates an exceptional research environment where doctoral and postdoctoral researchers pursue cutting-edge mathematics while benefiting from complementary strengths of each participating institution. The program centers on three interconnected research themes that demonstrate how modern mathematical analysis focuses on both fundamental theoretical questions and real-world challenges.

Students in the IRTG benefit from regular exchanges, joint supervision, and access to resources and complementary expertise unavailable at a single institution. Regular workshops, conferences, and summer schools provide opportunities for presentation, learning, and network building. Connections to major research initiatives and international collaborations expose students to the broader mathematical community and current trends.

Research Themes

Theme A: Conformal Problems and Curvature Energies

This theme explores how shapes and surfaces behave when minimizing their bending energy, leading to mathematical structures that appear throughout nature and technology. Central to this research is the Willmore energy, which measures how much a surface deviates from a perfect sphere. This concept emerges in contexts ranging from soap bubbles and cell membranes to thin elastic materials, with mathematical challenges involving how surfaces evolve over time and form singularities.

The research also encompasses conformal Dirac equations, which provide frameworks for understanding spinorial geometry with applications in modern physics, particularly in modeling electron transport in materials like graphene. Additionally, Liouville problems and Toda systems arise in areas from Chern-Simons theory to complex geometry, sharing the common feature of conformal invariance that brings both mathematical elegance and analytical challenges.

Theme B: Shape Optimization for Spectral and Non-local Problems

Shape optimization addresses the fundamental question: what is the best possible shape for a given purpose? This field has revolutionized engineering design, enabling the creation of lightweight structures that achieve maximum strength with minimal material usage, a capability particularly important for 3D printing and additive manufacturing.

The applications extend far beyond engineering. In physics, shape optimization helps to understand atomic nuclei structure through liquid drop models balancing attractive and repulsive forces. Similar frameworks describe animal flocking behavior and enable computer vision applications like object recognition and image segmentation. Our research emphasizes non-local and spectral problems, where behavior at one point depends on the entire system configuration, requiring advanced mathematical tools and often exhibiting surprising behaviors.

A computer simulation shows the formation of droplet-like structures through the interplay of attractive and repulsive forces.

Theme C: Nonlinear and Stochastic Aspects of Fluids and Solids

Continuum mechanics provides the mathematical foundation for understanding material behavior, from airflow over aircraft to biological tissue deformation. A central challenge is dimension reduction, where three-dimensional phenomena are modeled using structures with smaller dimensions like thin shells or rods, requiring sophisticated techniques to maintain accuracy while reducing computational complexity.

Fluid turbulence presents fundamental unsolved problems affecting weather prediction and industrial design. Our research addresses both mathematical foundations and practical computational methods, including large eddy simulation techniques that capture essential turbulence features while filtering computationally expensive scales. We also study non-Newtonian fluids common in biological and industrial systems, and stochastic systems where randomness plays fundamental roles. Development of efficient numerical and machine learning based methods for these potentially high-dimensional problems is a further important aspect of the theme.