(Oxford University)

Standard twistor theory involves a complex projective
3-space PT which naturally divides into two halves PT+
and PT–, joined by their common 5-real-dimensional
boundary PN. However, this splitting has two quite
different basic physical interpretations, namely
positive/negative helicity and positive/negative
frequency, which ought not to be confused in the
formalism, and the notion of “bi-twistors” is introduced
to resolve this issue. It is found that quantized bi-
twistors have a previously unnoticed G2* structure,
which enables the split-octonion algebra to be directly
formulated in terms of quantized bi-twistors, once the
appropriate complex structure is incorporated.

We study the celestial description of the O(N) sigma model in the large N limit. Focusing on three dimensions, we analyze the implications of a UV complete, all-loop order 4-point amplitude of pions in terms of correlation functions defined on the celestial circle. We find these retain many key features from the previously studied tree-level case, such as their relation to Generalized Free Field theories and crossing-symmetry, but also incorporate new properties such as IR/UV softness and S-matrix metastable states. In particular, to understand unitarity, we propose a form of the optical theorem that controls the imaginary part of the correlator based solely on the presence of these resonances. We also explicitly analyze the conformal block expansions and factorization of four-point functions into three-point functions. We find that summing over resonances is key for these factorization properties to hold. This is a joint work with D. García-Sepúlveda, A. Guevara, J. Kulp.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome

Holography in asymptotically flat spaces is one of the most coveted goals of modern mathematical physics. In this talk, I will motivate a novel holographic description of self-dual SO(8) Yang-Mills + self-dual conformal gravity on a Euclidean signature, asymptotically flat background called Burns space. The holographic dual lives on a stack of D1-branes wrapping a CP^1 cycle in the twistor space of R^4 and is given by a gauged beta-gamma system with SO(8) flavor and a pair of defects at the north and south poles. It provides the first example of a stringy realization of (asymptotically) flat holography and is a Euclidean signature variant of celestial holography. This is based on ongoing work with Kevin Costello and Natalie Paquette.

Let G(R) be the real points of a complex reductive algebraic group G. There are many difficult questions about admissible representations of real reductive groups which have (relatively) easy answers in the case of complex groups. Thus, it is natural to look for a relationship between representations of G and representations of G(R). In this talk, I will introduce a functor from admissible representations of G to admissible representations of G(R). This functor interacts nicely with many natural invariants, including infinitesimal character, associated variety, and restriction to a maximal compact subgroup, and it takes unipotent representations of G to unipotent representations of G(R).

Motivated by the physical relevance of many Hodge Laplace-type PDEs from the finite element exterior calculus, we analyse the Hodge Laplacian boundary value problem arising from the strain space in the linear elasticity complex, an exact sequence of function spaces naturally arising in several areas of continuum mechanics. We propose a discretisation based on the adaptation of discontinuous Galerkin FEM for the incompatibility operator $\mathrm{inc} := \mathrm{rot}\circ\mathrm{rot}$, using the symmetric-tensor-valued Regge finite element to discretise  the strain field; via the 'Regge calculus', this element has already been successfully applied to discretise another metric tensor, namely that arising in general relativity. Of central interest is the characterisation of the associated Sobolev space $H(\mathrm{inc};\mathbb{R}^{d\times d}_{\mathrm{sym}})$. Building on the pioneering work of van Goethem and coauthors, we also discuss promising connections between functional analysis of the $\mathrm{inc}$ operator and Kröner's theory of intrinsic elasticity in the presence of defects.

This is based on ongoing work with Dr Kaibo Hu.

Regularization of linear least squares problems is necessary in a variety of contexts. However, the optimal regularization parameter is usually unknown a priori and is often to be determined in an ad hoc manner, which may involve solving the problem for multiple regularization parameters. In this talk, we will discuss three randomized algorithms, building on the sketch-and-precondition framework in randomized numerical linear algebra (RNLA), to efficiently solve this set of problems. In particular, we consider preconditioners for a set of Tikhonov regularization problems to be solved iteratively. The first algorithm is a Cholesky-based algorithm employing a single sketch for multiple parameters; the second algorithm is SVD-based and improves the computational complexity by requiring a single decomposition of the sketch for multiple parameters. Finally, we introduce an algorithm capable of exploiting low-rank structure (specifically, low statistical dimension), requiring a single sketch and a single decomposition to compute multiple preconditioners with low-rank structure. This algorithm avoids the Gram matrix, resulting in improved stability as compared to related work.