Euclidean Ramsey Theory is a natural multidimensional version of Ramsey Theory. A subset of Euclidean space is called Ramsey if, for any $k$, whenever we partition Euclidean space of sufficiently high dimension into $k$ classes, one class much contain a congruent copy of our subset. It is still unknown which sets are Ramsey. We will discuss background on this and then proceed to some recent results.

Recently, the Newman-Janis shift has been revisited from the angle of scattering amplitudes in terms of the so-called "massive spinor-helicity variables," tracing back to Penrose and Perjés in the 70s. However, well-established results are limited in the same-helicity (self-dual) sector, while a puzzle of spurious poles arises in mixed-helicity sectors. This talk will outline how massive twistor theory can reproduce the same-helicity results while offering a possible solution to the spurious pole puzzle. Firstly, the Newman-Janis shift in the same-helicity sector is derived from a complexified version of the equivalence principle. Secondly, the massive twistor particle is coupled to background fields from bottom-up and top-down perspectives. The former is based on perturbations of symplectic structures in massive twistor space. The latter provides a generalization of Newman-Janis shift in generic backgrounds, which also leads to "curved massive twistor space" and its deformed massive incidence relation. Lastly, the Feynman rules of the first-quantized massive twistor particle and their physical interpretation are briefly discussed. Overall, a significant emphasis is put on the Kähler geometry ("zig-z̄ag structure") of massive twistor space, which eventually connects to a worldsheet structure of the Kerr solution.

High-dimensional data is often assumed to be distributed near a smooth manifold. But should we really believe that? In this talk I will introduce HADES, an algorithm that quickly detects singularities where the data distribution fails to be a manifold.

By using hypothesis testing, rather than persistent homology, HADES achieves great speed and a strong statistical foundation. We also have a precise mathematical theorem for correctness, proven using optimal transport theory and differential geometry. In computational experiments, HADES recovers singularities in synthetic data, road networks, molecular conformation space, and images.

Paper link: https://arxiv.org/abs/2311.04171
Github link: https://github.com/uzulim/hades