Hodge Club

Here is the link for the last years’ talks: https://hodge.maths.ed.ac.uk/?page_id=902 .

Current Schedule of talks for 2025/26

19th September | Social

Arthur’s seat (4pm) & Dinner at ‘The Pakora Bar’ (7pm)

26th September | Julia Bierent | What is causal set theory?

Quantum mechanics is able to explain 3 out of the 4 fundamental forces: electromagnetic interaction, strong force and weak force. What about gravity? Causal Set Theory is an attempt of explaining it. When trying to theorise Quantum gravity, one needs to make choices, and this leads to different theories. I am aiming to explain these choices in the case of Causal Set Theory.

3rd October (5pm) | Isky Mathews | Sphere Packing & Fearful Symmetry

Packing spheres is a very old and seemingly very easy problem: every fruit seller can tell you intuitively how to pack oranges best in a crate. However, this apparent ease belies a staggeringly complex topic and we only know the densest arrangements in 1,2,3,8 & 24 dimensions. In this talk, we’ll discuss something about the 8 & 24 dimensional cases and the beautiful mathematics that is involved.

10th October (4pm) | Nikolai Perry | A grapple with BV algebras

Associated to any quiver are certain involutive Lie bialgebras whose operations cut and glue paths in specific ways. Verifying that these operations satisfy the required axioms, however, can be a very tedious affair. In this talk, we will see how viewing these structures as BV algebras within a representation-theoretic framework not only simplifies matters greatly, but also reveals their (arguable) conceptual origin. The broader aim of this story is to highlight two simple yet meaningful lessons: (1) naïve generalisations can be surprisingly fruitful, and (2) notation should be taken seriously. Time permitting, we will conclude with a few open questions.

17st October (4:30 pm) | Siddharth Setlur | A spooky sheaf theoretic tale involving quantum mechanics 

When Einstein criticized quantum mechanics for exhibiting spooky action at a distance, he was referring to quantum entanglement and the fact that it leads to quantum mechanics violating the principle of locality. Locality posits that particles can only be influenced by their immediate environment and that any interactions between particles cannot propagate faster than the speed of light. Accepting this principle necessitates the introduction of hidden variables of particles to explain certain quantum phenomena. Assuming the existence of hidden variables and locality, Bell showed that independent measurements performed on a separated pair of quantum entangled particles leads to a bound on how the two measurements are correlated (this bound is known as Bell’s inequality). Bell then shows that for certain entangled pairs (e.g., Bell states) and certain experimental setups, quantum physics predicts measurements that violate this bound, i.e. quantum systems are non-local. This is a special case of quantum contextuality, the fact that measurements are dependent on the context of other compatible measurements. This can be characterized as a failure to pass from local to global (i.e., obstructions to forming global sections). In this talk, we will see how we can frame important examples and theorems from quantum mechanics using sheaf theory and hopefully make them less spooky.

24th October (4:30 pm) | Tuan Anh Pham | Prime ideals of quantum algebras

Let A be a quantized coordinate ring of an affine algebraic variety V. A natural question is how the prime spectrum of A related to the prime spectrum of the classical coordinate ring O(V). In this talk, I will give a survey on the results and conjectures related to this questions, and discuss some examples including quantum affine space, quantum matrices and quantum nilpotent algebras.

31st October (4:30 pm) | Adrian Doña Mateo | What is an epimorphism of rings?

The category of rings is one of the first examples you meet where epimorphisms are not surjections: the unique homomorphism ℤ → ℚ is famously epic but not surjective. In the 60s, Silver characterised epimorphisms in Ring as those f : R → S such that the map of abelian groups S ⊗_R S → S is an isomorphism. These maps have are quite special in enriched category theory; they are the Cauchy dense Ab-functors between one-object Ab-categories. In this talk, I will present Silver’s characterisation and report on ongoing work with Isky Matthews trying to determine when the epimorphisms of monoids in a monoidal category 𝒱 are precisely the Cauchy dense 𝒱-functors.

7th November (4:30 pm) | Yan Yau Cheng | A Trace-Path Integral Formula over Function Fields

In a topological quantum field theory, path integrals can often be expressed instead as the trace of a monodromy action on a Hilbert space.

In this talk I will discuss an arithmetic analogue of this phenomena for function fields, where the phase space is replaced with the ℓ-torsion points of the Jacobian of a curve over a finite field, the path integral is replaced with a sum over the points of J[ℓ], and the monodromy is instead replaced with the Frobenius action. Time permitting, I will also briefly outline the proof of this arithmetic trace-path integral formula.

14th November | SKIPPED SESSION |

21st November (4:30 pm) | Emmanouil Sfinarolakis | Hypersets: Taming Self-Reference

Self-reference has traditionally been viewed as the bête noire of set theory. Triggered by Russell’s Paradox, the mathematical community erected the Axiom of Foundation as a barrier, effectively banishing circular membership structures (like x = {x}) from the universe of “safe” sets. However, excluding these structures limits our ability to model naturally circular phenomena found in computer science, linguistics and game theory. This talk introduces hypersets: a rigorous extension of the set-theoretic universe that tames self-reference while avoiding contradictions. We will explore how replacing the Foundation Axiom with Aczel’s Anti-foundation Axiom (AFA) opens the door to a uniform treatment of circularity. Join us to see how the “vicious circle” is not a paradox to be avoided, but a rich mathematical structure to be understood.

28st November | (CANCELLED) |

5th December (4:30 pm) | Emanuel Roth | Normality of the stack of (parahoric) Higgs bundles

We’d like to prove that a space parametrizing objects of interest has as many nice properties as possible! I care about spaces parametrizing bundles over a complex projective curve, often with added structure (Higgs fields, parabolic structures, symplectic forms etc.). Unfortunately, not all these spaces are smooth, as I will explain through deformation theory. Some however, are at least normal, so their singular locus is of codimension ≥2. I will partly recall Simpson’s proof that the moduli space of semistable Higgs bundles is normal, and explain an approach to showing that the moduli stack of parahoric Higgs bundles is normal, using the Serre criterion for normality.

12th December (4:30 pm) | Loïc Bramley | Line operators and representation theory

Line operators in 3D topological QFTs assemble into braided monoidal categories. These often admit characterisations as categories of modules for certain quasitriangular Hopf algebras. I will outline different ways of extracting such algebras in the perturbative and non-perturbative setting. The former will provide motivation for the latter, in which categorical reconstruction and (relative) Drinfeld centres/doubles take on a clear physical meaning. The main example throughout will be 3D Chern-Simons theory for a Lie algebra g wherein I will show how U_q(g) arises in two different ways.

16th January (5 pm) | Social activity

23rd January (4:30 pm) | Lucy Spouncer | Tangent Bundles for Categories and Quillen Cohomology 

The tangent bundle encodes infinitesimal information about smooth manifolds, providing the foundation for differential geometry and deformation theory. But what happens when we venture beyond manifolds? How can we capture infinitesimal deformations of objects in arbitrary categories—algebras, operads, or even categories themselves?

In this talk, I’ll explain how the geometric intuition of tangent spaces extends to a powerful categorical framework through the abstract cotangent complex. The key insight is that “linearisation” naturally leads to stable categories and spectra: just as tangent spaces are vector spaces (the universal linear objects), tangent categories to ∞-categories are stable ∞-categories. This perspective unifies and generalizes classical cohomology theories—from ordinary cohomology of spaces to Hochschild cohomology of algebras—within a single formalism known as (spectral) Quillen cohomology.

The talk will be based on recent work by Harpaz, Nuiten, and Prasma, with an emphasis on examples and geometric intuition over technical details. Time permitting, I’ll discuss connections to my current work on deformation of operads.

30th January (4:30 pm) | Tudor-Ioan Caba |  A story about 3D TFT feat. Rozansky-Witten theory

Jones discovered his famous polynomial invariant of knots in 1984. A bunch of questions emerged – most notably, where is it coming from? Witten answered this question (in arguably his most well-known paper) by providing a physical interpretation of the Jones polynomial by quantizing Chern-Simons theory – a 3-dimensional topological quantum field theory (TQFT) – kickstarting the field of quantum topology. This sparked an immense effort by mathematicians to understand, define and formalize what a TQFT is. In our current understanding, a (three-dimensional) TQFT is a gadget which assigns information to 0, 1, 2 and 3-dimensional manifolds, in a way which is compatible with operations such as gluing, giving rise to a fantastic amount of topological information such as 3-manifold invariants and representations of mapping class groups of surfaces.

I will describe how this story unfolded historically and the stark contrast between what mathematicians and physicists mean by TQFT. I will end by describing my recent results on formalizing Rozansky-Witten theory, a 3D TQFT introduced in 1996 which takes as input a holomorphic symplectic variety and sits at the intersection of a lot of fascinating (and poorly understood) mathematics. The talk will be informal, and I will focus on painting the broad picture rather than technical details.