# Hodge Club

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 meet at **JCMB room 5323**. If you are attending virtually, we will **send an email with the Zoom link every week**. 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 2021/22 academic year is organised by Lucas Buzaglo and Hannah Dell.

### Current Schedule of talks for 2021/22

#### Summer series

13th May | Jeff Hicks | An introduction to A_\infty algebras |

Abstract: An A_\infty algebra is like a differential graded algebra (think: de Rham complex for a manifold) but instead of associativity of the product holding on the nose, we require only that associativity hold up to homotopy; and that the associativity of this homotopy only hold up to a higher homotopy, and so on and so on. In this talk, we'll: (1) introduce the basic definitions for these objects, (2) outline the proof of the "homological perturbation lemma" which shows that we can upgrade quasi-isomorphisms of chain complexes to A_\infty maps --- this can't be done with maps of differential graded algebras! (3) time permitting, look at some examples of where you may encounter these in the wild.
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20th May | Tom Adams (University of Cambridge) | Realising The Smooth Representations of GL(2,Zp) |

Abstract: The character table of GL(2,Fq), for a prime power q, was constructed over a century ago. Many of these characters were determined via the explicit construction of a corresponding representation, but purely character-theoretic techniques were first used to compute the so-called discrete series characters. It was not until the 1970s that Drinfeld was able to explicitly construct the corresponding discrete series representations via the l-adic étale cohomology groups of what is now known as the Drinfeld curve. This work was later generalised by Deligne and Lusztig to all finite groups of Lie type, giving rise to Deligne-Lusztig theory. In a similar vein, we would like to construct the representations affording the (smooth) characters of compact groups like GL(2,Zp), where Zp is the ring of p-adic integers. Deligne-Lusztig theory suggests hunting for these representations inside certain cohomology groups. In this talk, I will consider one such approach using a non-archimedean analogue of de Rham cohomology. I will not assume any background in number theory or non-archimedean geometry.
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27th May | William Bevington | Cohomology theories, Chern classes and elliptic curves |

Abstract: I will introduce the Eilenberg-Steenrod axioms, answering the question "what is a (generalised) cohomology theory?" once and for all. After a few examples of cohomology theories we'll look at how they interact, and how we can define Chern classes in an arbitrary cohomology theory. Finally, time permitting, we'll look at elliptic curves and how they fit into this equation and why they're such an amazing area of interest; if you've heard of the group law then buckle your seatbelt, this'll be fun! There will be few prerequisites for this talk as I will define very little, as is the beauty of being a higher category theorist.
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3rd June | Álvaro Muñiz Brea | Generalized Lagrangian correspondences |

Abstract: Symplectic manifolds are the basic object of study in symplectic topology. The natural morphisms between them are called symplectomorphisms: diffeomorphisms that preserve the symplectic form. One could form a “symplectic category” like this, but this would be too restrictive (for instance, there would be no morphisms between symplectic manifolds of different dimension). In the 80’s, Alan Weinstein introduced Lagrangian correspondences — which are just Lagrangians inside product manifolds — to try to enlarge the morphism spaces between symplectic manifolds. We will introduce this notion and its limitations, and finally arrive to a new notion of the symplectic category using generalised Lagrangian correspondences. Lastly, we will discuss how these (generalised) Lagrangian correspondences induce functors between Fukaya categories, mirroring Fourier-Mukai transforms in algebraic geometry.
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10th June | Arman Sarikyan | On Rationality of Fano-Enriques with terminal cyclic quotient singularities |

Abstract: A three-dimensional non-Gorenstein Fano variety with at most canonical singularities is called a Fano-Enriques threefold if it contains an ample linear system, whose generic element is an Enriques surface with at most canonical singularities. Although there is no complete classification of Fano-Enriques threefolds yet, but there are some partial results. For instance, L. Bayle has classified Fano-Enriques threefolds with terminal cyclic quotient singularities in terms of their canonical covers, which are smooth Fano threefolds in this case. The rationality of Fano-Enriques threefolds is an open classical problem that goes back to the works of G. Fano and F. Enriques. In this talk we will discuss the rationality of Fano-Enriques threefolds with terminal cyclic quotient singularities.
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24th June | Karim Réga | Higgs bundles: fun for the whole family |

Abstract: Higgs Bundles were introduced by Hitchin through studying dimensional reductions of the 4d self dual Yang Mills equation. They gave rise to some beautiful connections between the fields of algebraic geometry, differential geometry and representation theory through the Non-abelian Hodge theory, and also to natural examples of algebraically completely integrable systems. In this talk, we will study the definition and properties of Higgs bundles, their moduli spaces, the Hitchin system and non-abelian Hodge theory, and if time permits, discuss some natural generalizations.
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1st July | Hannah Dell | Bridgeland stability manifolds via group actions |

Abstract: One of the things algebraic geometers most like to do is classify algebraic varieties: for example curves (by genus), surfaces (by Kodaira dimension), Fano varieties (by deformation type) etc. Going one step further we also like to classify objects on algebraic varieties, such as linear subspaces, curves on surfaces, vector bundles, and more generally coherent sheaves. An algebraic geometer’s dreams will come true if you study the spaces of all objects of one of these “types” and it turns out to be an algebraic variety itself – which we call a moduli space. Unfortunately these spaces are often too wild to have any chance of being an algebraic variety. But there’s hope, and that hope is called stability. There are many different notions of stable objects in algebraic geometry (slope, GIT, King/quiver stability, K-Stability …) the slogan being that spaces of stable objects are much nicer (and often in fact algebraic varieties). In this talk I will discuss Bridgeland stability which generalises many existing notions of stability for vector bundles and moreover extends the definition to any triangulated category. I will be particularly interested in the space of Bridgeland stability conditions on a given triangulated category which (you guessed it) turns out to also be a complex manifold.
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8th July | Isambard Goodbody (University of Glasgow) | Stable homotopy theory and spectra |

Abstract: Freudenthal's Suspension Theorem tells us there is a pattern in the homotopy groups of spheres if one looks at a sufficiently high dimension. This led to the definition of the stable homotopy category of spectra where spaces are studied "up to suspension". Another reason to define this category is that cohomology theories (singular, K-theory, Bordism) and some algebraic invariants (algebraic K-theory, the representation ring of a group) are represented by objects in this category. Unlike for spaces, the homotopy category of spectra is tensor-triangulated and provides a motivation for many problems in tensor-triangulated geometry. I'm going to give the classical definition of this category not assuming any knowledge of homotopy theory and if there's time mention some things about homotopical algebra.
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15th July | Lucas Buzaglo | Universal enveloping algebras of infinite-dimensional Lie algebras |

Abstract: Universal enveloping algebras of finite-dimensional Lie algebras are fundamental examples of well-behaved noncommutative rings. On the other hand, enveloping algebras of infinite-dimensional Lie algebras remain mysterious. For example, it is widely believed that they are never noetherian, but there are very few examples whose noetherianity is known. In this talk, I will outline my recent work on this problem, where I considered Lie algebras of vector fields on affine varieties. This generalises work of Sierra and Walton on the Witt algebra. |

#### Semester 2

28th January | Lucas Buzaglo | Deformations of Lie algebras |

Abstract: We can use deformation theory to study a specific object by deforming it into a family of "similar" objects, giving us a richer picture about the original object itself. In many different contexts, there is a close relationship between deformation theory and cohomology. This is the case, for example, in the context of compact complex manifolds. One might expect this relationship to also apply to deformations of Lie algebras. Indeed, this is the case, but we have to be a bit more careful. In this talk, I will introduce deformations of Lie algebras and explain their intricate relationship to Lie algebra cohomology. I will also give some examples of deformations of a subalgebra of the Witt algebra. To end the talk, I will introduce some non-trivial deformations of the Witt algebra, despite the well-known fact that the Witt algebra is rigid (cannot be deformed) and explain how we can understand this paradox.
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4th February | Šarūnas Kaubrys | Towards loop spaces in algebraic geometry |

Abstract: The free loop space of a topological space X is the space of maps from the circle to X equipped with an appropriate topology. The loop space carries a natural action of the circle by rotating the domain. One can then construct isomorphisms between the singular and equivariant singular cohomology of the loop space and the Hochschild and cyclic homology of the original space. After briefly mentioning this topological story, I will try to explain how one can try to construct a loop space of schemes or stacks. In the process we will naturally see that we need derived algebraic geometry to make sense of such a space. If time permits, I may say something about how one can view differential forms in derived geometry as certain functions on this algebraic geometric loop space.
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11th February | Ben Brown | Index Theory for Orbifolds |

Abstract: The nlab entry on Generalized Smooth Spaces says, "Manifolds are fantastic spaces. It’s a pity that there aren’t more of them." Fortunately, we may pity no more, for orbifolds are generalisations of manifolds in that they locally look like the quotient of Euclidean space by a finite group. They were first introduced in 1956 by Kawasaki, and have since made appearances in many areas of geometry, mathematical physics, as well as in music theory. Mutatus mutandi, we can apply our favourite adjectives to orbifolds; smooth, complex, symplectic, etc., as we may do with manifolds. In this talk, I would like to introduce orbifolds along with some examples, before discussing how one may extend equivariant localisation formulae to symplectic toric orbifolds, and how one may combinatorially extract the data required for the formulae from convex rational simple polytopes.
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18th February | Karim Réga | Introduction to Urban Architecture and Parahoric Groups |

Abstract: Semisimple Lie groups over the complex numbers are classified by the possible Dynkin diagrams. Similarly, there is a classification of reductive groups by root data. In this talk, I will explain the basic ingredients for a similar classification over local fields, focusing on the case of Laurent fields over the complex numbers. The explicit example of GL_3 will be discussed in detail. We will see how this leads to interesting subgroups not present in the classical case, the parahoric subgroups. If time permits, we will see some applications of parahoric subgroups and parahoric group schemes in algebraic geometry.
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4th March | Theodoros Lagiotis | TQFTs and their classification |

Abstract: Topological quantum field theories (TQFTs for short) were introduced by Atiyah and Segal as an attempt to formalize a specific class of quantum field theories (QFTs). One can think of QFT in a very non-rigorous way as a map from ‘geometry’ to ‘algebra’. I will briefly talk about the physical motivation for the definition of a TQFT and then go on to describe explicitly what a TQFT looks like in low dimensions. We will actually see that low dimensional TQFTs are completely classified. We will then check whether such a classification is possible for all dimensions and how to address the potential problems that arise.
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11th March | Benjamin Haïoun | Fully Extended skein TQFT |

Abstract: In this talk I will introduce Walker's skein categories and how (according to the folklore) they form a fully extended TQFT. I will focus on the example of the Kauffman bracket skein relations and the Temperley-Lieb category. I will briefly discuss monoidal, braided and ribbon categories and how they come into play in the target this TQFT, which is a Morita category of E_2-algebras in Cat. We will try not to get lost in the details and stay informal, with as many drawings as one can make.
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18th March | Dora Puljic | Braces and Classification Struggles |

Abstract: A brace consists of a set with two operations, one forming an abelian group and the other a group, along with a certain distribution law. Braces were introduced by Wolfgang Rump in 2007 to help the study of non-degenerate, involutive, set theoretic solutions to the Yang-Baxter equation. Connections to other objects have been found since - braid groups with an involutive braiding operator, bijective 1-cocycles, quantum groups, trusses, etc. In this talk I will touch on the origins of braces and give an exposition of past and current classification efforts and methods.
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25th March | Álvaro Muñiz Brea | Lagrangian cobordisms in mirror symmetry |

Abstract: Homological mirror symmetry conjectures a deep relation between symplectic and algebraic geometry. Roughly speaking, for a pair of “mirror” spaces — one symplectic and the other complex — one expects to obtain an equivalence between the (symplectic) Fukaya category and the (complex) category of coherent sheaves. An example of such a mirror pair is the 2-torus together with the elliptic curve, and homological mirror symmetry has been proven in this case. In this talk I will try to motivate the conjecture for this pair and arrive to a particular consequence, namely an isomorphism between the Lagrangian cobordism group and the Grothendieck group of the category of coherent sheaves. By the end of the talk, we will have related the geometry of Lagrangian surgery to the algebra of short exact sequences of sheaves.
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1st April | Andrew Beckett | Symplectic folklore: Homogeneous spaces and central extensions of Lie groups |

Abstract: A homogeneous symplectic space is a symplectic manifold with a transitive action by a Lie group which preserves the symplectic structure. In physics, such spaces are sometimes known as "elementary systems" because they are a classical analogue of elementary particles. In this talk, I will demonstrate a new proof of a "folkloric theorem" about simply-connected homogeneous symplectic space — namely, that they are universal covers of coadjoint orbits of one-dimensional central extensions of the group (I promise to explain all of these words!) — and hopefully show that this proof unveils some unexpected links between the existence of Lie group central extensions and symplectic geometry via symplectic cohomology. This talk is based on our recent preprint.
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8th April | Shivang Jindal | Mixed Hodge Theory for Budding Mathematicians |

Abstract: It's a theorem by Hodge that any smooth complex projective variety has a certain canonical direct sum decomposition, called the Hodge Decomposition. This is too good to be true for an arbitrary variety, However in 1970's, P. Deligne generalized the Hodge decomposition to any arbitrary variety X over the complex numbers. Deligne's main theorem asserts that cohomology of X carries a certain structure, called the Mixed Hodge Structure which recovers the Hodge decomposition when X is smooth and projective. The slogan of my talk is that these structure are so nice that you don't really need to know the construction to work with them. In this talk we will see their definition, some examples and properties and finally use a sandwiching argument to show that the cohomology of the Hilbert Scheme of Points on a plane is pure.
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15th April | Girish Vishwa | Semi-Infinite Cohomology of Graded Lie Algebras |

Abstract: On an infinite dimensional Lie algebra, one can construct the space of semi-infinite forms, which are wedge products of forms that are unbounded in one direction. By constructing an appropriate differential on this space, one can construct what is known as the semi-infinite cohomology of the Lie algebra, which defers greatly from the regular Lie algebra cohomology. In this talk, I will discuss the basics of this construction for graded Lie algebras and present one useful result/feature that one can obtain from semi-infinite cohomology. I will also explain qualitatively present the link that semi-infinite cohomology has to string theory, through the application of these ideas to the Virasoro algebra.
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22nd April | Luke Naylor | Conway’s approach to symmetries |

Abstract: When I first saw the classification of wallpaper groups, as I'm sure it was the same with many others, it was a very algebraic approach. For example considering the translation subgroup, the point group, and ways they can fit together. This past semester I had the opportunity to tutor for Toby Bailey's course "Symmetry and Geometry", which is based on material from a book co-authored by one of his past lecturers John Conway titled "The Symmetries of things". The approach in this text 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 summary and advert for the topics in this area. |

#### Semester 1

1st October (Bayes) | Sebastian Schlegel Mejia | An interactive stroll towards the E-polynomial of the moduli stack of rank two degree zero Higgs bundles |

Abstract: The subject of my talk is the calculation of the E-polynomial of the moduli stack of rank two degree zero Higgs bundles. However, the aim of the talk is not to reach the final calculation nor to give super fancy reasons why you should care about the calculation. Instead, we focus on giving a feeling for the concepts and techniques involved in the calculation. These include (moduli of) Higgs bundles, stacks, motivic measures, and lambda-rings. All kinds of questions and interruptions are strongly encouraged and will be seen as pleasant diversions on our field trip through the land of Higgs bundles and its surroundings.
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8th October (Bayes) | Augustinas Jacovskis | Geometry from derived categories |

Abstract: A lot of geometric information about a variety X can be recovered from its derived category D(X). If the variety is Fano, then X can in fact be reconstructed up to isomorphism from D(X). This begs the question of whether less information than D(X) can determine X up to isomorphism. In this talk I’ll discuss some known cases when “less information” means a certain subcategory of D(X) called the Kuznetsov component. Time permitting, I’ll discuss joint work with Zhiyu Liu and Shizhuo Zhang which describes the situation for index 1 Fano threefolds.
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15th October (Online) | Guy Boyde (University of Southampton) | Homotopy groups, and how they grow |

Abstract: Homotopy groups are an important invariant of topological spaces – loosely, the n-th homotopy group is a picture of the space taken from the point of view of the n-dimensional sphere. Unfortunately, they are incredibly hard to compute – even for a friendly example like the two-dimensional sphere, we do not know all of them. We might therefore like to ask a coarser question. The one I am going to ask in this talk is “what can be said about the behaviour of the homotopy groups as n goes to infinity?” Asking this question reveals lots of interesting structure, and that’s what I’ll aim to tell you about, starting with the basics and including lots of examples.
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22nd October (Bayes) | Patrick Kinnear | An Invertible Sheaf on the Character Stack? |

Abstract: It is known that the skein algebra of a surface defines a sheaf of algebras on the coordinate ring of the character variety of the surface, which is in a sense the moduli space of local systems on the surface. It is also known that this sheaf satisfies an invertibility property over an open and dense subset of the character variety. Another way to describe local systems on a surface is via the character stack, and we can ask if there is a stacky version of the above invertibility statement. In this talk we will outline how such a statement can (hopefully!) be obtained from the study of invertible morphisms in the Morita category Alg_3, and will mention recent results characterising invertibility in this category. Along the way we will describe the powerful technology of factorization homology, which connects these two viewpoints.
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29th October (JCMB) | Lucien Hennecart | Categorification of Hall algebras |

Abstract: Hall algebras are now ubiquitous objects of geometric representation theory. The rough idea is to build algebra structures on constructible functions, cohomology and K-theory of moduli spaces of objects in certain categories and to study the algebras one obtains. To define them (and to be able to study them), one needs to put strong conditions on the categories under consideration and the examples one should keep in mind are the categories of representations of a quiver or of coherent sheaves on a smooth projective curve (at the 1-dimensional level); modules over the preprojective algebra of a quiver or compactly supported coherent sheaves on a smooth quasiprojective surface (at the 2-dimensional level); modules over the Jacobi algebra of a quiver with potential or coherent sheaves on a Calabi-Yau threefold (at the 3-dimensional level). We will try to clarify these constructions and also show examples and modern questions related to this subject.
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5th November (Online) | Ben Brown | A Survey of Hyperkähler and Hypertoric Geometry |

Abstract: Hyperkähler manifolds are special classes of Kähler manifolds, with three complex structures obeying the algebraic identities of the quaternions, and compatible with a Riemannian metric. These constraints or symmetries give rise to special characteristics of the hyperkähler structure, e.g. the manifolds are 4n-dimensional, its holonomy lies within Sp(k), is Ricci-flat (thus Calabi-Yau), and it is complex-symplectic. It is therefore more difficult to find examples of hyperkähler manifolds than Kähler ones; indeed, there are only two known classes of compact ones, namely the 4-torus and K3 surfaces, yet any complex submanifold of CPⁿ is automatically Kähler. On the other hand, there are many non-compact ones, e.g. the moduli spaces of Higgs bundles, of Nahm equations, and of monopoles, as well as complex cotangent bundles, to name but a few. A construction that yields a vast array of examples of hyperkähler manifolds is that of the hyperkähler quotient, which is the hyperkähler analogue of symplectic reduction. Many of the previous examples can be constructed this way, and the quotient method also establishes various global properties of the quotient. There are several other interesting examples of hyperkähler manifolds that can be constructed just by simply taking the hyperkähler quotient of the flat quaternionic vector space, Hⁿ, with respect to the action of a compact Lie group, G. In this talk, I wish to provide an overview of hyperkähler manifolds and their respective quotient operation, as a prerequisite to introducing toric hyperkähler manifolds, a.k.a. hypertoric manifolds. Hypertoric manifolds are essentially the hyperkähler analogues of toric manifolds and can be studied both geometrically and combinatorially, via hyperplane arrangements in the image of the hyperkähler momentum mapping. This combinatorial description also provides a simple recipe for constructing hyperkähler manifolds (and orbifolds) and as well as identifying their Kähler submanifolds.
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12th November (Bayes) | Alyosha Latyntsev (University of Oxford) | BPS states, vertex algebras and torus localisation |

Abstract: To understand the ideas coming from string theory, mathematicians have noticed the importance of moduli spaces (of objects in a category). By this method the physics notions of 1. BPS states/D-branes, 2. a conformal field theory, 3. \Pi stability, ... have been turned into 1. cohomological Hall algebras, 2. vertex/chiral algebras, 3. Bridgeland stability conditions, ... , all of which are extremely rich mathematical objects. In this talk, I will a. explain what a cohomological Hall algebra and vertex algebra is (and why you should care), b. sketch their physics analogues (no physics knowledge required!), c. show how they are connected. The main tool used to prove c. is a new version of the torus localisation which works for singular/derived spaces. Loosely, torus localisation "turns geometry into combinatorics" and is an important tool in Gromov Witten theory, toric geometry, and Donaldson Thomas theory. If time allows I will explain how this works. This is a talk based on my recent paper.
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19th November (JCMB) | Andrew Beckett | Infinitesimal automorphisms of Cartan geometries and filtered deformations |

Abstract: A Cartan geometry is manifold which looks locally like a Klein geometry, ie. like a (connected) space of the form G/H, where G is a Lie group and H is a closed subgroup of G. Cartan geometries generalise a number of different types of geometric structure, including Riemannian and symplectic geometry. The automorphisms of a Cartan geometry can be described infinitesimally by a finite-dimensional Lie algebra, and in the case of a "regular" Cartan geometry, this algebra has a particular structure: it is filtered, and its associated graded algebra is a graded subalgebra of Lie(G) - we say that it is a "filtered subdeformation" of Lie(G). In this talk, I will give an overview of these structures and discuss how a particular cohomology of graded Lie (super)algebras, Spencer cohomology, can help us to understand filtered deformations. If there is enough time, I will say a little about my own work applying this to the infinitesimal automorphisms of some solutions of supergravity theories in physics.
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26th November (Bayes) | Vivek Mistry | Cohomological Hall algebras of character varieties |

Abstract: Cohomological Hall algebras (CoHAs) provide an interesting refinement of Donaldson-Thomas invariants for objects in a 3-CY category. In this talk I will give a brief introduction to CoHAs, focussing mainly on the case of the cohomology of the stack of representations of quivers and Jacobi algebras. I'll then explain a 2d version of this story for character varieties, with the end-goal to compare two CoHA multiplications and explain why they are in fact equal.
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10th December (JCMB) | Jeff Hicks | An introduction to tropical geometry |

Abstract: One of the main ideas of tropical geometry is that much of geometry of complex sub-varieties of the algebraic torus, (C*)ⁿ, can be reduced to understanding combinatorics of piecewise linear objects in Rⁿ. These are the central object of study in tropical geometry. In this talk, we'll see why we might expect such a degeneration from complex geometry to tropical geometry, and look at how (assuming that this degeneration holds) we can obtain the degree-genus formula for curves in P². Then we'll see in what ways tropical geometry fails to capture aspects of complex geometry. |

### Historical schedules

Hodge Club 2020/21

Hodge Club 2019/20

Hodge Club 2018/19

Hodge Club 2017/18

Hodge Club 2016/17

Geometry club 2015/16

Geometry club 2014/15

Geometry club 2013/14

Geometry club 2012/13

You can also visit the old Geometry club website for more historical schedules.