Successful Dissertation on Geometric Analysis and Convex Integration

Sandra Ried has successfully defended her dissertation on the method of convex integration in problems of geometric nature and applications in fluid mechanics. Convex integration can be used to test the limits of well-posedness in mathematical problems i.e., the existence, uniqueness, and continuous dependence of solutions on the data. 

Her dissertation, titled “Geometric Analysis and Convex Integration,” was supervised by László Székelyhidi. In her research, Sandra explores structural properties of nonlinear partial differential equations (PDEs), focusing on questions of (non-)uniqueness and (ir)regularity. She describes her research goal as follows: “The main tool in this analysis so far has been the technique of convex integration. This method has been extended to various applications in the last two decades. However, it often relies on a specific structure of the PDE and its non-linearity. In the future, I want to work towards better understanding the full scope and flexibility of convex integration schemes.”

In non-linear PDEs, solutions of low regularity tend to be non-unique whereas higher regularity can ensure uniqueness. A central question is identifying the regularity threshold separating these regimes. Convex integration is a method for constructing solutions in low regularity settings, where classical compactness tools fail. It leverages the flexibility arising from the equation’s nonlinearity.

In her thesis, Sandra investigated the flexibility of the convex integration method itself on the basis of two examples.

1. Hall-Magnetohydrodynamics (Hall-MHD):
   This system describes the motion of a fluid influenced by a magnetic field, and can be used to model liquid metals and plasmas. For this system, it is possible to construct solutions with very little regularity to any given energy profile, and thus strongly violating well-posedness. Studying the stationary Hall equation, she could show that the Hall effect induces enough rigidity to prevent the application of one of the basic convex integration frameworks.
2. Isometric Embeddings in Contact Geometry:
   In order to construct distance-preserving embeddings between contact manifolds, Sandra demonstrated that it is possible to implement the constraint arising from the contact setting into the quantitative Nash scheme for Riemannian manifolds. This implies that such embeddings are non-unique as long as their differentials are Hölder continuous below a certain threshold.

Academic Path

Sandra Ried received her Bachelor's and Master's degrees in Mathematics from the Karlsruhe Institute of Technology. She pursued her PhD at Leipzig University under the supervision of László Székelyhidi, and later continued her doctoral work at the Max Planck Institute for Mathematics in the Sciences, following her advisor’s appointment as director.

In August 2024, she moved to Atlanta, where she taught as a part-time lecturer in the School of Mathematics at Georgia Tech during the spring term of 2025. Starting in Fall 2025, she will continue there as a Visiting Assistant Professor under the mentorship of Michael Loss.

We warmly congratulate Sandra and wish her all the best in both her personal life and her academic career at the Georgia Institute of Technology!