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• Question. Can Euclidean space be two-coloured, in every dimension, with no red copy of three equally spaced points on a line and no blue copy of some longer such configuration?
• Result. Yes: for some natural number m there is a red/blue colouring of Euclidean space in every dimension with no red copy of l_3 and no blue copy of l_m, answering a question of Conlon and Fox, and independently of Arman and Tsaturian.

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• Problem. The link between residual networks and their continuous-depth analogue, neural ODEs, lacked a mathematical foundation.
• Approach. Study gradient flow on deep residual networks initialized as discretizations of a neural ODE.
• Result. The discretization structure is preserved throughout training, for finite time and, under a Polyak–Łojasiewicz condition, as time goes to infinity, with convergence to a global minimum.

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• Problem. The exact minimizer of denoising score matching memorizes the training data, yet trained models memorize only mildly.
• Approach. Analyze gradient descent with large steps, which can only converge to minima whose sharpness is bounded by the inverse learning rate.
• Result. The memorizing solution is too sharp to be reached; large learning rates prevent memorization.

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• Problem. The last steps of latent diffusion can degrade samples, which does not happen in pixel-space diffusion.
• Approach. A Gaussian model with linear autoencoders that links latent dimension, the constraints of score matching and the stopping time.
• Result. Low-dimensional latents call for earlier stopping, high-dimensional ones for later; stopping time is a key hyperparameter of latent diffusion.

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• Problem. FID needs tens of thousands of samples and can be fooled by matching moments.
• Approach. A sliced Wasserstein distance between Inception features, computed by sorting one-dimensional projections, with no high-dimensional means or covariances to estimate.
• Result. An order of magnitude more sample-efficient, two orders of magnitude faster, more robust; 5k samples match FID at 50k.

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• Problem. In diffusion models memorization and generalization are incompatible, so the supervised-learning theory of generalization does not transfer.
• Approach. Survey the three families of explanations and run controlled CIFAR-10 sweeps over dataset size, model size, batch size and learning rate.
• Position. Why models do not memorize is largely settled by early stopping and the linear scaling of memorization time; what is learned before memorization is the open question.

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• Problem. Autoregressive LLMs decode one token at a time, a sequential bottleneck for generation speed.
• Approach. Fine-tune Gemma 4 into a block-diffusion model that refines 256 tokens in parallel: supervised denoising, then reinforcement learning with sampler distillation, using under 10% of the base model’s training tokens.
• Result. About 1,500 output tokens per second on a single H100, a new speed-capability Pareto frontier, while keeping thinking mode, multimodal inputs and long context.

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• Problem. Autoregressive ML emulators of PDEs accumulate error over long horizons and miss the stochasticity of physical systems.
• Approach. Adapt a deterministic physics foundation model into a generative surrogate with two-stage inference, mean-prediction regularization and spatial gradient matching.
• Result. 42.9% average reduction in forecasting error over the Walrus fine-tuning baseline on The Well, better on 8 of 10 datasets.

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