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
 

Suppose 𝐺_𝕜 is a connected reductive algebraic 𝕜-group where 𝕜 is an algebraically closed field. If 𝑉_𝕜 is a 𝐺_𝕜-module then, using geometric invariant theory, Kempf has defined the nullcone 𝒩(𝑉_𝕜) of 𝑉_𝕜. For the Lie algebra 𝔤_𝕜 = Lie(𝐺_𝕜), viewed as a 𝐺_𝕜-module via the adjoint action, we have 𝒩(𝔤_𝕜) is precisely the set of nilpotent elements.

We may assume that our group 𝐺_𝕜 = 𝐺 ×_ℤ 𝕜 is obtained by base-change from a suitable ℤ-form 𝐺. Suppose 𝑉 is 𝔤 = Lie(G) or its dual 𝔤* = Hom(𝔤, ℤ) which are both modules for 𝐺, that are free of finite rank as ℤ-modules. Then 𝑉 ⨂_ℤ 𝕜, as a module for 𝐺_𝕜, is 𝔤_𝕜 or 𝔤_𝕜* respectively.

It is known that each 𝐺_ℂ -orbit 𝒪 ⊆ 𝒩(𝑉_ℂ) contains a representative ξ ∈ 𝑉 in the ℤ-form. Reducing ξ one gets an element ξ_𝕜 ∈ 𝑉_𝕜 for any algebraically closed 𝕜. In this talk, we will explain two ways in which we might want ξ to have “good reduction” and how one can find elements with these properties. We will also discuss the relationship to Lusztig’s special orbits.

This is on-going joint work with Adam Thomas (Warwick).

For any smooth complex variety Y with an action of a finite group W, Etingof defines the global Cherednik algebra H_c and its spherical subalgebra B_c as certain sheaves of algebras over Y/W. When Y is an n-dimensional abelian variety, the algebra of global sections of B_c is a polynomial algebra on n generators, as shown by Etingof, Felder, Ma, and Veselov. This defines an integrable system on Y. In the case of Y being a product of n copies of an elliptic curve E and W=S_n, this reproduces the usual elliptic Calogero­­--Moser system. Recently, together with P. Argyres and Y. Lu, we proposed that many of these integrable systems at the classical level can be interpreted as Seiberg­­--Witten integrable systems of certain super­symmetric quantum field theories. I will describe our progress in understanding this connection for groups W=G(m, 1, n), corresponding to the case Y=E^n where E is an elliptic curves with Z_m symmetry, m=2,3,4,6.