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 2024/25 academic year is organised by **Chun-Yu Bai **,** Julia Bierent**,** Adrian Cook **and** Matthias Vancraeynest.**

Current Schedule of talks for 2024/25

**6th December** | Teo Gil Moreno de Mora Sarda | Complete 3-manifolds of positive scalar curvature?

TBD

**29th November** | Arun Soor | TBD

TBD

**22th November** | Joseph Malbon | TBD

TBD

**15th November** | Diogo Freire De Andrade | A combinatorial approach to Morita categories.

The study of higher Morita categories of En-algebras plays a central role in the understanding of topological quantum field theories (TQFTs) and higher algebra. In this talk, we’ll explore a combinatorial approach to constructing these categories, as formulated by Rune Haugseng. This approach offers a framework for working with higher categorical structures, which are fundamental to modern mathematical physics and category theory.

A key part of our discussion will involve Lurie-Dunn Additivity, a result we will unpack during the talk, so no prior knowledge of it is required. To fully grasp this framework, we will also introduce non-symmetric infinity operads—powerful algebraic tools that generalize classical operads and provide a unified language for both monoidal and double infinity-categories.

The talk is intended for a broad audience, and I will make an effort to keep it self-contained and approachable for those unfamiliar with these ideas.

**8th November** | Jeff Hicks | TBD

TBD

**1th November** | Chun-Yu Bai | Factorization homology of Enriched (infty,2)-categories, String-Nets and fully extended TQFT.

Factorization homology can be view as a new kind of calculas, which can be used to compute global topological observables for a QFT. The global topological observables getting from the ordinary version (or so-called alpha version) with an E_n algebras of low dimensional defects as an input, however, are not genuine ones but only those global topological observables that are determined by low dimensional defects. What we really need is so-called Beta-version whose input is a higher category of all dimensional defects, rather than E_n algebra. For physical interests, given a symmetric monoidal infty-category V, I will consider such a higher category to be V-enriched cases, especially when V=Vect or Chain. In this talk, I will introduce a developing theory of factorization homology of enriched (infty,2)-categories, build the relation between factorization homology of Vect-enriched (infty,2)-categories and linear string-nets theory, and also construct a collection of fully extended 2D TQFTs by Beta-factorization homology. This is an ongoing work joint with Diogo Freire De Andrade.

**25th October** | Willow Bevington | What is Tannakian duality, anyway?

Pontrjagin duality gives a notion of duality to locally compact abelian groups, interchanging the role of discrete and compact groups. For instance, the Pontrjagin dual of ℤ (discrete) is S¹ (compact). Generalising this gives rise to Tannakian duality, which allows us to reconstruct a scheme from its category of quasi-coherent sheaves. This talk is entirely introductory and aims to give an introduction to the Tannakian formalism via Grothendieck’s description of the fundamental group via Galois theory.

**18th October** | Matthias Vancraeynest | Quantisation of character varieties

The term character variety refers to the moduli space of G-local systems on an oriented surface S for a reductive group G. This can be equivalently described as the space of homomorphisms from the fundamental group of the surface to G, quotiented by the conjugation action. It carries a Poisson structure, introduced by Attiyah, Bott and Goldman, which can be quantised. Different quantisation procedures are known and in my talk I will give a rough overview of two techniques, one using cluster varieties and another using factorisation homology. Lastly, I will talk about the bridge between these two procedures and extending what is known for SL2 to higher rank groups.

**11th October** | Daniel Bernal | C-instantons

Instanton bundles appeared initially as an analogue of finding solutions of a differential equation in physics via the so-called Atiyah-Ward correspondence. This converted the problem into an algebraic setting, and thus the study of instanton bundles started. More particularly, people were interested in studying possible deformations of such objects, which gave rise to the definition of instanton sheaves on projective spaces, and recently they have been adopted to more general smooth varieties. However, when we want to study the Bridgeland moduli space of such objects we encounter some strange “deformations” of instantons, called C-instantons. Our objective is to motivate this definition, and to give some properties that show that they behave nicely and in fact generalise all the previous definitions.

**4th October** | Xiang Li | Introduction of Chabauty-Kim Methods on S-unit Equations

Abstract: The Chabauty-Kim method is a p-adic method to find rational or integral points on algebraic curves effectively. Kim conjectured that his machinery will give the solution sets eventually. In the talk, I will first introduce the Chabauty-Kim methods, and some known results on verifying Kim’s conjecture on S-unit equations where S is a set of primes of size 2. Later, I will discuss about the case when S is of size 3 (which is not known) to give an upper bound of the number of solutions using Chabauty-Kim methods and weight filtrations, and also discuss about the Chabuaty-Kim methods over number fields if time permitted.

**27th September** | Karim Rega | What is/are moduli spaces?

Abstract: The main goal in a lot of mathematical problems is to classify some type of mathematical object. In this talk we will clarify what this means within the field of algebraic geometry with the aid of moduli functors. This will naturally lead to the definition of a fine moduli space, which is the ideal solution to such a classification problem. After showing that this ideal situation rarely occurs, we will delve into two other candidates for a new framework, Geometric Invariant Theory and/or the theory of stacks.

**20th September** | Social Meeting

### Talks for 2023/24

**21st ****June** | Malthe Sporring | (Co)ends in infinity-categories

Abstract: (Co)ends are certain (co)limits associated to bifunctors of the form F: C^op x C -> D. As an example, let R be a ring viewed as an one-object Ab-enriched category so an R-bimodule M is just a functor M:R^op x R -> Ab. Then the end of this functor is the centre of M and the coend is the space of coinvariants. (Co)ends are both abundant and admit a sort of calculus, making them useful for a large class of categorical arguments. In this talk I will give an overview of (co)ends, and explain how the definition can be extended to the world of infinity-categories.

**14th ****June** | Catrin Mair | Condensed Mathematics & Homotopy Theory

Abstract: The theory of condensed mathematics offers a new approach to the notion of topological spaces that is better behaved algebraically. Classically, topological spaces also play an important role in homotopy theory, where they find a replacement in the world of ∞-categories by the notion of anima (also called spaces or ∞-groupoids). These two directions can be combined by the ∞-category Cond(Ani) of condensed anima. In my talk, I will explain the role of Cond(Ani) in relation to the homotopy theory of topological spaces and schemes. I aim to give a gentle introduction to condensed mathematics and assume no familiarity with ∞-categories, in the hope that the talk will be accessible to anyone with a mild interest in homotopy theory or condensed mathematics.

**31st ****May** | Antoine Pinardin | G-solid Rational Surfaces

Abstract: A rational surface is a surface S such that there exists a birational map between S and the projective plane. Given a rational surface S and a finite subgroup G of Aut(S), we are interested in determining whether or not there exists a G-equivariant birational map between S and a G-conic bundle. If not, we say that S is G-solid. The Minimal Model Program for surfaces implies that it is enough to consider the case where S is a smooth Del Pezzo surface. After introducing this formalism, we will present the full classification of pairs (G,S) such that the surface S is G-solid. This classification is motivated by the long lasting problem of classifying the conjugacy classes of finite subgroups of the group of birational self maps of the projective space in dimensions 2 and 3.

**24th May** | Karim Rega | Anti-invariant bundles and moduli

Abstract: Anti-invariant bundles are vector bundles that are isomorphic to their dual when pulled back under an involution. These can be interpreted as torsors for some parahoric group scheme. After showing this interpretation, we will discuss what the correct notion of Higgs bundles in this case is and show the existence of a moduli space for these objects.

**17th May** | Social activity

###### Semester 2

**26th April** | Danil Kozevnikov | Homological Mirror Symmetry for C*

Abstract: Homological mirror symmetry (HMS) is a conjecture due to Maxim Kontsevich, attempting to explain the previously studied versions of mirror symmetry as consequences of an equivalence between certain triangulated categories: the Fukaya category of a symplectic manifold X and the bounded derived category of coherent sheaves on the mirror complex variety Y. Unfortunately, coming up with a precise mathematical formulation of the conjecture is a very laborious task, not to mention proving it. Despite this, it presents us with an interesting philosophy, which has motivated several key developments in symplectic and algebraic geometry in the last couple of decades.

In this talk, I will give you a gist of HMS by working through a relatively simple example of the mirror pair (T*S^1,C*). Time permitting, I might also briefly discuss compactifications on the B-side and mirror symmetry for P^1.

**19th April** | Duncan Laurie | Young wall realizations for representations of (affine) quantum groups

Abstract: Kashiwara’s theory of crystal bases provides a powerful tool for studying representations of quantum groups. Crystal bases retain much of the structural information of their corresponding representations, whilst being far more straightforward and ‘stripped-back’ objects (coloured digraphs). Their combinatorial description often enables us to obtain concrete realizations which shed light on the representations, and moreover turn challenging questions in representation theory into far more tractable problems.

After reviewing the construction and basic theory regarding quantum groups, I will introduce and motivate crystal bases as ‘nice q=0 bases’ for their representations. I shall then present (in both finite and affine types) the construction of Young wall models in the important case of highest weight representations. Time permitting, I will finish by discussing some applications across algebra and geometry.

This talk aims to be a gentle and accessible introduction to the area, with few prerequisites!

**12th April** | Barthelemy Neyra | Interpolation of open-closed TQFTs

Abstract: TQFTs are known to produce invariant of manifolds. In particular, an ordinary (oriented) closed 2d-TQFT with target the category of complex vector spaces will associate a complex number to each closed surface of genus g, giving thus a complex-valued sequence indexed by the natural numbers N. The sequences one could obtain that way are constrained by the nature of the target category. A work by Khovanov, Ostrik and Kononov explains how to interpolate between target categories in order to build TQFTs affording arbitrary sequences. I will show, using tools developed by my supervisor E. Meir, how to extend their work to the case of (oriented) open-closed 2d-TQFTs, and talk about the properties of these interpolating categories.

**5th April** | Julia Bierent | Dualisation and topological quantum fields theories

Abstract: Why do we care about dualisation in the context of topological quantum field theories? We will introduce extended quantum field theories and the cobordism hypothesis in order to justify dualisation. We will finally go through the notions of duality and rigid categories.

**29th March** | Tudur Lewis | A Dehn surgery approach to the Birman—Craggs and Sato maps.

Abstract: We provide a unified framework for studying two families of maps: the Birman—Craggs maps of the Torelli subgroup, and Sato’s maps of the level 2 congruence subgroup of the mapping class group. These maps completely determine torsion in the first homology of these subgroups. We discuss a relation between an extension of the Birman—Craggs maps to the level 2 congruence subgroup, and Meyer’s signature cocycle.

**22th March** | Joseph Malbon | Quotients in Algebraic Geometry

Abstract: In an undergraduate course in algebraic geometry, one normally doesn’t encounter group quotients of algebraic varieties. This is a shame, because they appear everywhere across the subject, and are integral in the construction of moduli spaces. In this talk, starting with finite groups, I will introduce quotient varieties, give many examples, and then after progressing to general reductive algebraic groups I will give a crash-course in geometric invariant theory, illustrating its utility in constructing the moduli spaces of smooth and stable elliptic curves.

**15th March** | Subrabalan Murugesan | Screened Vertex Operators and representation theory of quantum groups

Abstract: A certain class of 2d quantum field theories known as conformal field theories are interesting to physicists because they are more often than not physical theories that are exactly solvable. If you know any physics, then you would know that this is very very rare. At the same time, it has also captured the imagination of mathematicians thanks to its relation to Teichmuller theory, isomonodromic deformations, quantum groups, vertex algebras, so on and so forth.

In this talk, I will address the connection between 2d CFTs and quantum groups. In particular, I will restrict myself to showing that there is a one-one correspondence between certain CFT operators, known as screened vertex operators, and highest weight representations of the quantum group Uq(sl2). If time permits, I will say why physicists care about this.

**8th March** | Gianni Gagliardo | Topological T-duality

Abstract: In this talk we shall explore topological aspects of T-duality. In string theory, T-duality relates two a priori unrelated string backgrounds, which nevertheless behave identically from a physical point of view. We shall show how to formulate this duality using the language of principal $U(1)$-bundles with non-trivial $H$-flux and how this induces isomorphisms on their twisted cohomology groups. Then we shall study some simple examples of T-duality such as the Hopf fibration and nilmanifolds and if time permits discuss connections to generalised geometry and mirror symmetry.

**1st March** | Samuel Lewis | An introduction to real variations of stability

Abstract: In the study of Bridgeland stability conditions on a triangulated category D, the contractibility of the stability manifold StabD is still an open problem. The usual approach for this involves expressing (a connected component of) StabD as a covering space of something, often a hyperplane arrangement H controlled by combinatorial data. In this talk I discuss an alternative perspective using real variations of stability conditions, a notion introduced by Anno, Bezrukavnikov, and Mirković which explicitly packages H into the definition. I will give an outline of my work to construct these stability conditions on 2-Calabi–Yau triangulated categories associated with graphs of affine and hyperbolic type. This has applications to noncommutative K3 surfaces.

**23rd February** | Tuan Pham | Local representations of Witt algebra

Abstract: The orbit method is a fundamental tool to study a finite dimensional solvable Lie algebra g. It relates the annihilators of simple U(g)-module to the coadjoint orbits of the adjoint group on g^*

*.*In my talk, I will extend this story to the Witt algebra – a simple (non-solvable) infinite dimensional Lie algebra which is important in physics and representation theory. I will construct an induced module from an element of W^*, show that it is simple, and establish some nice properties of the category of these induced modules under tensor product. I will also construct an algebra homomorphism that allows one to relate the orbit method for W to that of a finite dimensional solvable algebra.

**16th February** | Karim Rega | The moduli space of Higgs bundles

Abstract: Higgs bundles are a fascinating area of study related to many different fields of mathematics, i.e. representation theory, algebraic geometry, symplectic geometry, mathematical physics,… A proof of the existence of a moduli space for semistable Higgs bundles was already formulated by Nitsure using Geometric Invariant Theory. However, recently Alpe, Halpern-Leistner and Heinloth introduced criteria for when an algebraic stack admits a good moduli space. After showing how to interpret stability stack-theoretically and defining the stack of semistable Higgs bundles, we will show that it satisfies these criteria and thus prove the existence of a moduli space via these new methods.

**9th February** | Alvaro Gonzalez Hernandez | Explicit theory of Kummer surfaces in characteristic two

Abstract: One of the landmark theorems of number theory, the Mordell–Weil theorem, states that in an elliptic curve the set of rational points is a finitely generated group. If we look at the next easiest type of curve, genus 2 curves, then, it turns out that the set of rational points is finite (and no longer a group), but we can still study this set by analysing the group of rational points of a surface related to the curve known as the Jacobian. One of the issues of this approach is that computing the equations describing the Jacobian of a curve is quite a difficult matter, so number theorists have resorted instead to study a related kind of surfaces known as Kummer surfaces. In this talk, I will explain the geometric theory that allow us to compute equations for Kummer and Jacobian surfaces, explain why we would want to work with these surfaces over finite fields and, finally, motivate why working in characteristic two, specifically, is such a challenge.

**2nd February** | Willow Bevington | Galois theory applied to… Everything?Abstract: Galois theory has been an essential tool in the algebraist’ toolkit for decades by allowing an analogy between normal subgroups and field extensions to be made explicit. It’s a beautiful example of why category theory is so important, by making analogies rigorous and extending the domain of use of ideas.

We’ll have a look at this story to refresh our memories of what a Galois extension is, then generalise the notion to various other constructs we know and love in maths; commutative rings, topological spaces, and schemes! The main goal of this talk is actually to emphasise how easy higher algebra can be, and we’ll discuss the Galois theory of so-called “E_∞-rings”. But don’t worry, you don’t need any knowledge of higher category theory, this is a purely expository talk!

**26th January** | Theodoros Lagiotis | A Hopf coend

Abstract: Given a category $\mathcal{C}$ and a functor F: \mathcal{C}^{op}/times\mathcal{C} -> \mathcal{C}, one can define the so called coend of F. Hopf algebras on the other hand, are of interest to representation theorists, due to their nicely behaving categories of modules.

In this talk, I want to review how these two notions meet when restricting the coend construction to a certain class of categories, and a certain functor. In this case, such a coend exists, and carries the structure of a Hopf algebra (internal to the original category we started with).

Time permitting, we will discuss why this is a “natural” thing to expect, due to TQFT considerations.

The choice of topic was made in an attempt to appeal to Hodgies with different interests, so the aim is for everything to be as accessible as possible.

###### Semester 1

**8th December** | Simone Castellan | Automorphisms of deformations and quantizations of Kleinian singularities

Abstract: Given a non-commutative algebra Q and its semiclassical limit A, an intriguing question has always been “Do the properties of Q always reflect the (Poisson) properties of A?”. Of particular interest is the behaviour of automorphisms. The most famous example is the Belov-Kanel-Kontsevich Conjecture, which predicts that the group of automorphisms of the nth-Weyl algebra A_n is isomorphic to the group of Poisson automorphisms of the polynomial algebra C[x_1,…,x_2n]. In this talk, I will explain the theory of filtered deformations and quantizations for symplectic quotient singularities, and present my work on a problem similar to the BKK Conjecture, in the case of Kleinian singularities.

**1st December** | Misha Schmalian | Surgery and its Effect on Hyperbolic 3-Manifolds

Abstract: Surgery is a method to modify a 3-manifold by removing a solid torus and gluing it back in a different way. This innocuous-looking procedure is shockingly powerful, namely any two 3-manifolds are connected by a sequence of surgeries. Hence, studying the behaviour of 3-manifolds under surgery is a key theme in low-dimensional topology with many classical results and open conjectures. This talk will give an introduction to hyperbolic 3-manifolds, surgery, and the relation between these. The focal point of the talk will be the classical result that gluing a solid torus to the boundary of a hyperbolic 3-manifold almost always gives a manifold that is still hyperbolic. No prerequisite knowledge of 3-manifolds will be needed. We will also mention recent results relating surgery to the contact topology of a hyperbolic 3-manifold.

**24th November** | Shivang Jindal | Mckay Correspondence and Algebraic Structures

Abstract: In 1982, given any finite subgroup G of SL_2 or equivalently a platonic solid, Mckay constructed a graph associated to the representations of G and observed a strange relation with the ADE classification of semisimple lie algebras. In this talk, after reviewing this correspondence to different levels of generality, we will explain how given G, in fact one can geometrically recover the Lie algebra and so much more!

**17th November** | Hannah Dell | An etymological tour through mathematics.

Abstract: Have you ever wondered what a derived category has in common with a river? Or how corollaries and flower crowns are related? In this talk we’ll explore the meaning and origin behind some of the mathematical terms that Hodge club members use day to day.

**10th November** | Luke Naylor | Conway’s approach to symmetry

Abstract: When I first saw the classification of wallpaper groups, it was a very algebraic approach. It involved considering the translation subgroup, the point group, and ways they can fit together. However the approach introduced by Conway in “the symmetry of things” is, instead of considering the groups themselves, to consider the space (which has the symmetries) quotiented by the group of symmetries giving so called orbifolds. This method generalises well to spherical and hyperbolic symmetry and gives clean picturesque ways of classifying certain types of tilings. This talk is a 50 minute picturesque exposition of the topics in this area.

**3rd November** | David Cueto Noval | A Cluster-Theoretic Approach to Quantum Multiplicative Quiver Varieties

Abstract: Quiver varieties have been studied for decades and have found numerous applications including to instanton moduli spaces. After reviewing some basic concepts in the classical setting, we will discuss the notion of quantum multiplicative quiver varieties introduced by D. Jordan. We believe these algebras can be studied via quantum cluster algebras and explain a possible approach.

**27th October** | Patrick Kinnear | An invertible sheaf on the character stack!

Abstract: In this talk I will describe the construction of an invertible sheaf of vector spaces (i.e. a line bundle) on the character stack of a 3-manifold, which is the moduli stack of G-local systems. In fact, this is just one part of a larger functorial structure called a TQFT which has a nice invertibility property. At the level of surfaces we obtain an invertible sheaf of categories over the character stack, which is expected to relate to a sheaf of algebras whose global sections are the skein algebra of the surface. Invertibility of the sheaf of categories should relate to an invertibility property of the skein algebra called being Azumaya. I will explain how these constructions follow from algebraic statements about braided tensor categories which we regard as local statements, and can be “integrated” to statements for general manifolds via technologies such as the cobordism hypothesis and factorization homology.

**20th October** | Harvey Yau | When every place lets you down

Abstract: Given a variety over Q, oftentimes looking modulo various primes can give valuable information about the distribution of rational points. But sometimes, no prime is sufficient to explain why there are no points on the variety. This talk will introduce the Brauer-Manin obstruction, a method that can be applied to prove the non-existence of rational points, and give examples of its applications to curves and surfaces.

**13th October** | Lucas Buzaglo | A classification of subalgebras of the one-sided Witt algebra

Abstract: We give a classification of Lie subalgebras of the one-sided Witt algebra. As applications of the classification, we prove that universal enveloping algebras of these infinite-dimensional subalgebras are not noetherian, and one case of a conjecture of Kaiming Zhao with links to the Jacobian conjecture. This is joint work with Jason Bell.

**6th October** | Parth Shimpi | What does the A2 singularity have in common with an octopus?

Abstract: Bridgeland (2007) showed that the stability manifold of a Kleinian surface singularity is a Galois cover of the hyperplane complement of the same ADE type, thus adding another link in the network of ideas connecting algebraic geometry, homological algebra, representation theory, and geometric group theory. In this talk I will sketch some of these ideas, and exhibit a computation which adds marine biology to this network as promised in the title.

**29th September** | Adrián Doña Mateo | Magnitude, magnitude homology, and Euclidean sets

Abstract: Magnitude is an invariant of enriched categories that generalises some well-known concepts (such as Euler characteristic), but also specialises to previously unknown ones (such as the magnitude of a metric space). Magnitude homology is a homology theory that, under favourable conditions, categorifies magnitude, and hence provides an algebraic perspective of this invariant. I will begin by introducing both of these notions, with some examples. In the second half, I will talk about joint work with Tom Leinster about the magnitude homology of subspaces of Euclidean space. In particular, I will give a geometric characterisation of when two closed subsets of R^n are ‘magnitude homology equivalent’.

**22th September** | Social Meeting