Thu, 26 Jun 2025
13:30
L5

Generalised symmetries and scattering amplitudes

Lea Bottini
Abstract

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.

Mon, 16 Jun 2025

15:30 - 16:30
L5

A unitary three-functor formalism for commutative Von Neumann algebras

Thomas Wasserman
((Oxford University))
Abstract

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.

Fri, 20 Jun 2025
13:00
L5

Latent Space Topology Evolution in Multilayer Perceptrons

Eduardo Paluzo Hidalgo
(University of Seville)
Abstract

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.

 

[1] Paluzo-Hidalgo, E. (2025). Latent Space Topology Evolution in Multilayer Perceptrons arXiv:2506.01569 
Fri, 13 Jun 2025
13:00
L5

The Likelihood Correspondence

Hal Schenck
(Auburn University)
Abstract

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

Fri, 06 Jun 2025
13:00
L5

Topologically good cover from gradient descent

Uzu Lim
(Queen Mary University London)

Note: we would recommend to join the meeting using the Teams client for best user experience.

Abstract

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.

Mon, 09 Jun 2025
15:30
L5

Planar loops and the homology of Temperley-Lieb algebras

Guy Boyde
(Universiteit Utrecht)
Abstract

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.

Mon, 02 Jun 2025
15:30
L5

Some geometry around torsion homology

Cameron Gates Rudd
(Oxford University )
Abstract

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.

Thu, 05 Jun 2025
13:30
L5

Seiberg-Witten theory

Harshal Kulkarni
Abstract
Seiberg-Witten theory is a powerful framework for understanding the exact non-perturbative dynamics of 4d $\mathcal{N} = 2$ supersymmetric QFTs. On the Coulomb branch of the moduli space, the low-energy physics is described by an abelian gauge theory with a holomorphic structure constrained by supersymmetry and duality. In this talk, I will explain the emergence of $PSL(2,\mathbb{Z})$ invariance in this effective field theory and how this naturally leads to a fibration of elliptic curves over the Coulomb branch. Focusing on the simplest case of $\mathcal{N} = 2$ SU(2) gauge theory without flavors, I will discuss the singularity structure of the Coulomb branch and the physical significance of these special points. I will conclude by briefly commenting on the central role that the singular structure of the moduli space plays in the classification of 4d $\mathcal{N}=2$ SCFTs.
 

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.

Fri, 30 May 2025
13:00
L5

A unified theory of topological and classical integral transforms

Vadim Lebovici

Note: we would recommend to join the meeting using the Teams client for best user experience.

Abstract

Alesker's theory of generalized valuations unifies smooth measures and constructible functions on real analytic manifolds, extending classical operations on measures. Therefore, operations on generalized valuations can be used to define integral transforms that unify both classical Radon transforms and their topological analogues based on the Euler characteristic, which have been successfully used in shape analysis. However, this unification is proven under rather restrictive assumptions in Alesker's original paper, leaving key aspects conjectural. In this talk, I will present a recent result obtained with A. Bernig that significantly closes this gap by proving that the two approaches indeed coincide on constructible functions under mild transversality assumptions. Our proof relies on a comparison between these operations and operations on characteristic cycles.

Fri, 30 May 2025
14:30
L5

Minimal tension holography from a String theory in twistor space

Nathan McStay
(Cambridge )
Abstract

Explicit examples of the AdS/CFT correspondence where both bulk and boundary theories are tractable are hard to come by, but the minimal tension string on AdS_3 x S^3 x T^4  is one notable example. In this paper, we discuss how one can construct sigma models on twistor space, with a particular focus on applying these techniques to the aforementioned string theory. We derive novel incidence relations, which allow us to understand to what extent the minimal tension string encodes information about the bulk. We identify vertex operators in terms of bulk twistor variables and a map from twistor space to spacetime is presented. We also demonstrate the presence of a partially broken global supersymmetry algebra in the minimal tension string and we argue that this implies that there exists an N=2 formulation of the theory. The implications of this are studied and we demonstrate the presence of an additional constraint on physical states. This is based on work with Ron Reid-edwards https://arxiv.org/abs/2411.08836.

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