Benchmarks in Leipzig

Between April 1 and May 15, 2026, a group of 49 mathematicians compiled a dataset of research-level mathematics questions with known answers. Most of the work was done during the 3-day workshop *Benchmarks in Leipzig* with 35 participants at the Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany. We present the resulting collection of 100 questions. We evaluated these questions in three stages: a single attempt by five state-of-the-art LLMs, followed by a 20-runs-per-model evaluation with three of these models, and finally a 3-run attempt with two heavy-thinking models. After Stage 1, 41 questions remained completely unsolved; after Stage 2, this count dropped to 16; and we concluded Stage 3 with only 2 unsolved questions. This demonstrates that the mathematical reasoning capabilities of LLMs are becoming impressive.

On the Universal Representation Property of Spiking Neural Networks

Inspired by biology, spiking neural networks (SNNs) process information via discrete spikes over time, offering an energy-efficient alternative to the classical computing paradigm and classical artificial neural networks (ANNs). In this work, we analyze the representational power of SNNs by viewing them as sequence-to-sequence processors of spikes, i.e., systems that transform a stream of input spikes into a stream of output spikes. We establish the universal representation property for a natural class of spike train functions. Our results are fully quantitative, constructive, and near-optimal in the number of required weights and neurons. The analysis reveals that SNNs are particularly well-suited to represent functions with few inputs, low temporal complexity, or compositions of such functions. The latter is of particular interest, as it indicates that deep SNNs can efficiently capture composite functions via a modular design. As an application of our results, we discuss spike train classification. Overall, these results contribute to a rigorous foundation for understanding the capabilities and limitations of spike-based neuromorphic systems.

A Continual and Incremental Learning Approach for TinyML On-device Training Using Dataset Distillation and Model Size Adaption

A new algorithm for incremental learning in the context of Tiny Machine learning (TinyML) is presented, which is optimized for low-performance and energy efficient embedded devices. TinyML is an emerging field that deploys machine learning models on resource-constrained devices such as microcontrollers, enabling intelligent applications like voice recognition, anomaly detection, predictive maintenance, and sensor data processing in environments where traditional machine learning models are not feasible. The algorithm solve the challenge of catastrophic forgetting through the use of knowledge distillation to create a small, distilled dataset. The novelty of the method is that the size of the model can be adjusted dynamically, so that the complexity of the model can be adapted to the requirements of the task. This offers a solution for incremental learning in resource-constrained environments, where both model size and computational efficiency are critical factors. Results show that the proposed algorithm offers a promising approach for TinyML incremental learning on embedded devices. The algorithm was tested on five datasets including: CIFAR10, MNIST, CORE50, HAR, Speech Commands. The findings indicated that, despite using only 43% of Floating Point Operations (FLOPs) compared to a larger fixed model, the algorithm experienced a negligible accuracy loss of just 1%. In addition, the presented method is memory efficient. While state-of-the-art incremental learning is usually very memory intensive, the method requires only 1% of the original data set.