L5

I will present a pair of off-shell functionals in position space, localized on the self-dual and the anti-self-dual planes which naturally give the Parke-Taylor denominator. These can therefore be used: 
i) to compute scattering amplitudes of particles with different spins and helicities; and 
ii) develop a Lagrangian description. 
Using Witten's half-Fourier transform, I will express these functionals in twistor space and present the kernels in a closed compact form. For even multiplicities, I will show how to obtain this form geometrically which than then be “folded” to get the one-less odd-multiplicity result. 
 

I will review some recent developments in effective field theory of  composite higher-spin particles, namely, Zinoviev's massive gauge symmetry and 
the new chiral-field approach. The latter approach was inspired by a simple spinor-helicity structure first singled out by Arkani-Hamed, Huang and Huang, which encodes the higher-spin information of two massive particles. It turned out to be persistent in tree-level amplitudes with any number of additional identical-helicity gluons or gravitons, leading to the discovery of the chiral-field approach. I will mention the applications of massive higher-spin scattering amplitudes to classical gravitational dynamics of rotating black  holes. 
 

In this talk we review some recent applications of generalised symmetries to scattering amplitudes. We start in 4d by describing the connection between spontaneously broken higher-form symmetries and soft theorems for scattering amplitudes of the associated Nambu-Golstone bosons, and show a new soft theorem for theories with a so-called 2-group symmetry. Then, we switch gears and consider non-invertible symmetries in 2d theories. We show that the standard form of the S-matrix is incompatible with the non-invertible symmetry, and derive new S-matrices satisfying a modified crossing symmetry.

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.

Six-functor formalisms are ubiquitous in mathematics, and I will start this talk by giving a quick introduction to them. A three-functor formalism is, as the name suggests, (the better) half of a six-functor formalism. I will discuss what it means for such a three-functor formalism to be unitary, and why commutative Von Neumann algebras (and hence, by the Gelfand-Naimark theorem, measure spaces) admit a unitary three-functor formalism that can be viewed as mixing sheaf theory with functional analysis. Based on joint work with André Henriques.

In this talk, we present a topological framework for interpreting the latent representations of Multilayer Perceptrons (MLPs) [1] using tools from Topological Data Analysis. Our approach constructs a simplicial tower, a sequence of simplicial complexes linked by simplicial maps, to capture how the topology of data evolves across network layers. This construction is based on the pullback of a cover tower on the output layer and is inspired by the Multiscale Mapper algorithm. The resulting commutative diagram enables a dual analysis: layer persistence, which tracks topological features within individual layers, and MLP persistence, which monitors how these features transform across layers. Through experiments on both synthetic and real-world medical datasets, we demonstrate how this method reveals critical topological transitions, identifies redundant layers, and provides interpretable insights into the internal organization of neural networks.

An arrangement of hypersurfaces in projective space is strict normal crossing if and only if its Euler discriminant is nonzero. We study the critical loci of all Laurent monomials in the equations of the smooth hypersurfaces. These loci form an irreducible variety in the product of two projective spaces, known in algebraic statistics as the likelihood correspondence and in particle physics as the scattering correspondence. We establish an explicit determinantal representation for the bihomogeneous prime ideal of this variety.

Joint work with T. Kahle, B. Sturmfels, M. Wiesmann

The cover of a dataset is a fundamental concept in computational geometry and topology. In TDA (topological data analysis), it is especially used in computing persistent homology and data visualisation using Mapper. However only rudimentary methods have been used to compute a cover. In this talk, we formulate the cover computation problem as a general optimisation problem with a well-defined loss function, and use gradient descent to solve it. The resulting algorithm, ShapeDiscover, substantially improves quality of topological inference and data visualisation. We also show some preliminary applications in scRNA-seq transcriptomics and the topology of grid cells in the rats' brain. This is a joint work with Luis Scoccola and Heather Harrington.

Temperley-Lieb algebras are certain finite-dimensional algebras coming originally from statistical physics and knot theory. Around 2019, they became one of the first examples of homological stability for algebras (homology is here taken to be certain Tor-groups), when Boyd and Hepworth showed that in low dimensions the homology vanishes. We're now able to give complete calculations of their homology, which has a surprisingly rich structure (and in particular is very far from vanishing). This is joint work in progress with Rachael Boyd, Oscar Randal-Williams, and Robin Sroka. Prerequisites will be minimal: it will be enough to know what Tor is.

Given a space with some kind of geometry, one can ask how the geometry of the space relates to its homology. This talk will survey some comparisons of geometric notions of complexity with homological notions of complexity. We will then focus on hyperbolic 3-manifolds and the main result will replace a spectral gap problem related to torsion in homology with a geometric version involving geodesic length and stable commutator length. As an application, we provide "bad" examples of hyperbolic 3-manifolds with bounded geometry but extremely small (1-form) spectral gaps.

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.