News & Events

We analyze the qualitative behavior of stochastic partial differential equations (SPDEs) with a particular focus on bifurcations. To this aim we investigate a change of sign in the finite-time Lyapunov exponents (FTLEs) of the SPDE in a small noise regime and close to a phase transition. Under suitable assumptions, the FTLEs are positive and thus indicate a change of stability. These results are applied to the stochastic Allen-Cahn and Burgers equation with non-Markovian noise and to singular SPDEs. Moreover, we also discuss properties of FTLEs, in particular large deviations type results.

We study uniform computability properties of PAC learning using Weihrauch complexity. We focus on closed concept classes, which are either represented by positive, by negative or by full information. Among other results, we prove that proper PAC learning from positive information is equivalent to the limit operation on Baire space, whereas improper PAC learning from positive information is closely related to Weak König's Lemma and even equivalent to it, when we have some negative information about the admissible hypotheses. If arbitrary hypotheses are allowed, then improper PAC learning from positive information is still in a finitary DNC range, which implies that it is non-deterministically computable, but does not allow for probabilistic algorithms. These results can also be seen as a classification of the degree of constructivity of the Fundamental Theorem of Statistical Learning. All the aforementioned results hold if an upper bound of the VC dimension is provided as an additional input information. We also study the question of how these results are affected if the VC dimension is not given, but only promised to be finite or if concept classes are represented by negative or full information. Finally, we also classify the complexity of the VC dimension operation itself, which is a problem that is of independent interest. For positive or full information it turns out to be equivalent to the binary sorting problem, for negative information it is equivalent to the jump of sorting. This classification allows also conclusions regarding the Borel complexity of PAC learnability. (joint work with Guillaume Chirache, École polytechnique, France) More information: https://theory.cca-net.de/seminar.php (prior registration requrired)

Neural networks have achieved striking empirical success, yet their theoretical foundations remain only partially understood. When equipped with the widely used ReLU activation function, such networks compute continuous piecewise linear (CPWL) functions, which makes it possible to study them using tools from polyhedral geometry, combinatorics, and computational complexity. In this talk, I will outline results and open questions concerning three aspects of ReLU networks: expressivity, verification, and parameterization. I will discuss how architectural choices constrain the class of CPWL functions that can be represented, complexity-theoretic challenges in verifying basic properties of the functions they compute, and how different parameter choices may correspond to the same function.

Creating physician rosters is a challenging task due to varying shift structures, qualifications, and department- or hospital-specific regulations. These variations mean that department-specific tools often fail to generalize across hospital settings. To address this, we developed a flexible mixed-integer programming (MIP) model capable of representing different roster structures, and we embedded it into an adaptable web application with an advanced graphical user interface (GUI), allowing physicians to specify preferences and hospital staff to configure the MIP model to their roster requirements without any mathematical or technical background. The practical implementation of such a system is essential for ensuring long-term acceptance in clinical environments. A sustainable solution must be easy to use, accessible, and well integrated into existing IT workflows. This talk presents the implementation process of our physician rostering framework and highlights the key design decisions that support its usability in practice. Particular emphasis is placed on interface elements that must be operated frequently during the rostering process, as well as on features that allow departments to tailor the system to their specific requirements. We conclude by discussing challenges during integration and showing how the application has been successfully deployed in a hospital department.