The Hodge Club is the seminar for Hodge Institute graduate students and postdocs. That means we’re interested in Algebra, Geometry, Topology, Number Theory, and all possible combinations and derivations of the four. Before the 2016/17 academic year, the Hodge Club was known as the Geometry Club.
We meet every Friday at 16:00, where we take it in turns to present a topic of interest to the rest of the group. We hope to run this as a hybrid seminar, so in particular we will have both an in person and virtual audience. If you are attending in person, we will normally meet at Bayes 5.46 (exceptions announced by e-mail). If you are attending virtually, you can join using the following Zoom link. If you do not receive the weekly emails and would like to be added to the mailing list, please get in touch with one of the organisers.
Talks tend to be fairly informal and provide excellent practice for conference talks in front of a friendly audience. You can find our current schedule and a historical list of talks below.
The Hodge Club for the 2025/26 academic year is organised by João Camarneiro, Alexandra Ciotau and Isky Mathews.
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 (4:30 pm) | TBD | Title&Abstract TBD
5th December (4:30 pm) | TBD | Title&Abstract TBD
12th December (4:30 pm) | Loïc Bramley | Title&Abstract TBD