# 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 normally meet at **Bayes 5.46** (exceptions announced by e-mail). 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 2022/23 academic year is organised by Willow Bevington, Álvaro Muñiz Brea and Karim Rega.

### Current Schedule of talks for 2021/22

#### Semester 2

##### Upcoming talks

7th April | Joe Malbon | Rational Algebraic Surfaces |

Abstract: The theory of complex algebraic surfaces is a beautiful and well-developed area of mathematics. In this talk we will give a brief introduction to the subject, set in the modern language of sheaves and topology, and touching on intersection theory, cohomology and minimal models. We will focus our efforts towards rational surfaces, whose theory is particularly simple. Time permitting, we will prove the rationality of smooth cubic surfaces and quartic complete intersections. |

##### Past talks

31st March | Marina Purri Brant Godinho (University of Glasgow) | A infinity algebras arising in geometry |

Abstract: Stashef introduced A infinity algebras to study "group-like" topological spaces in the beginning of the 1960s. In more recent years, these algebras have become relevant for algebra, geometry and mathematical physics. In this talk, I will introduce A infinity algebras from an algebraic point of view, draw examples from geometry, and discuss their applications in algebraic geometry.
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24th March | Luke Naylor | Bridgeland Stabilities and Finding Walls |

Abstract: As many are aware, quite a few of us in the school are concerned with the business of stability conditions on sheaves. If you’ve talked to these people before, you very quickly realise that this is actually short for “stability conditions on the derived category of coherent sheaves”. In this talk I introduce this topic, particularly, the explicit stability conditions constructed by Bridgeland for K3 surfaces. But also how they relate to the more classical Mumford and Gieseker stabilities (which do not involve the derived). Later on, some stabilities on threefolds too. Along the way we will have a look at the notion of walls and why we want to find them. This will also expose the benefit of a couple of different viewpoints to find restrictions on the set of possible walls, some numeric, but also the geometry of some characteristic curves on the space of stability conditions.
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10th March | Charlotte Llewellyn (University of Glasgow) | NCCRs and Tilting Modules |

Abstract: The notion of an NCCR was introduced in 2002 by Van den Bergh as a noncommutative analogue of a crepant resolution. This concept provided a link between birational geometry and categorical representation theory, which was later exploited by Wemyss to obtain a recursive method for producing the minimal models of a complete local Gorenstein 3-fold at the level of derived categories. This talk will be an introduction to this theory. I’ll begin with some background on NCCRs. Then I’ll explain how the Bridgeland-Chen flop functor can be described as the derived equivalence induced by a tilting module.
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3rd March | Yan Yau Cheng | What prime is your shoelace? The answer may surprise you! (Linking, Knots and Primes) |

Abstract: In the 1960s Barry Mazur pointed out an analogy between the behaviour of prime ideals of a number field and knots in a 3-manifold. This observation birthed the field of Arithmetic Topology, which is the study of Number Theory through this perspective. The goal of this talk is to show the audience snippets of this analogy, in particular the analogy between linking numbers of knots and the power residue symbol of primes.
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24th February | Malthe Sporring | Topological K-Theory |

Abstract: Topological K-Theory is a generalized cohomology theory that captures information about vector bundles over a space. To build it, we first consider the Groethendieck group of C(X)-algebras, then extend this to a cohomology theory using suspensions. Alternatively, like any (co)homology theory, we can build a spectrum that represents it. The latter view illuminates the following important property of K-Theory: it controls the second cohomology theory in an infinite approximation of the stable homotopy groups of spheres.
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17th March | Julie Rasmusen (University of Warwick) | THR of Poincaré infinity-categories |

Abstract: In recent years work by Calmés-Dotto-Harpaz-Hebestreit-Land-Moi-Nardin-Nikolaus-Steimle have moved the theory of Hermitian K-theory into the framework of stable infinity-categories. I will introduce the basic ideas and notions of this new theory and introduce a tool which can help us understand this better: Real Topological Hochschild Homology. I will then explain the ingredients that goes into constructing the geometric fixed points of this THR as a functor, generalising the formula for ring spectra with anti-involution of Dotto-Moi-Patchkoria-Reeh.
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3rd February | Danil Kozevnikov | Tropical pants and very affine hypersurfaces |

Abstract: Tropical geometry provides a tool that allows us to understand a great deal about the geometry of complex subvarieties of (C*)n through studying combinatorics of piecewise linear objects in Rn. In this talk, I will introduce some basic concepts from tropical geometry and discuss tropical localization, a technique that is used to describe how complex manifolds can degenerate to a tropical limit. The main goal will be showing how this framework naturally leads to a generalization of pants decomposition of surfaces to hypersurfaces in (C*)n. If time permits, I will also discuss some applications of these ideas to problems in symplectic topology and mirror symmetry.
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27th January | Nikola Tomic | Homotopy Type Theory |

Abstract: Homotopy Type Theory (HoTT) is a new foundation for modern mathematics that was introduced by Vladimir Voedvodsky. He figured out, while working on some stuff about homotopy theory, that the object he was studying had unexpected type-theoretic interpretations. In this talk, I will introduce basic knowledge about Classical Logic, then upgrade it to Type Theory and finally introduce the Homotopy Type interpretation of Type Theory. Why should you care about HoTT? Here are some answers that are related to modern maths: (1) HoTT is totally suited to encode homotopy types on a computer and makes proofs of homotopy theory theorems (e.g. computation of homotopy groups of spheres) relatively easy to write down on a proof assistant (like Lean/Coq/Agda). (2) If you are interested in Higher Category Theory, one should know that ∞-topoi are models of HoTT: HoTT is the syntax of ∞-topoi and that gives objects of those topoi a fresher and (I believe) easier interpretation. (3) HoTT can give you nice intuitions of what homotopy (co)limits you are computing should look like. (4) If you are interested in (Constructive) Logic of Functional Programming you may be interested as well because it gives new interpretations of Types.
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20th January | Karim Rega | Filtrations Steeped in Lemon Juice |

Abstract: We will begin by reviewing the notion of the filtration of a vector space, module over a ring, and a quasicoherent sheaf on a scheme. Using the Rees contruction, these can be thought of as coming from equivariant vector bundles over the product of a scheme with an affine line. This viewpoint allows us to generalise the notion to other situations, and we will see how this relates to the construction of various stratifications of interest in moduli problems, using Theta-stratifications as introduced by Halpern-Leistner. |

#### Semester 1

##### Past talks

9th December | Isambard Goodbody (University of Glasgow) | Finite Dimensional Differential Graded Algebras |

Abstract: A DGA is a chain complex with a compatible multiplication. They appear in representation theory and algebraic geometry. We say a DGA is finite dimensional if its underlying algebra is finite dimensional. This property is not homotopy invariant and in fact Greg Stevenson and Theo Raedschelders have shown that every proper connective DGA is quasi-isomorphic to a finite dimensional DGA. A result of Orlov says that if the underlying algebra of fd DGA is semisimple then it is also semisimple as a DGA. We hope that this will provide a way of answering some open questions on fd DGAs (eg what is K0?).
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2nd December | Arman Sarikyan | Topic G-Fano Threefolds |

Abstract: Let G be a finite group acting biregularly on a threefold X. if X is Fano with terminal Q-factorial singularities and the G-invariant part of the Picard group is isomorphic to Z, then we say that X is G-Fano. Such varieties appear naturally as an end product of the G-invariant Minimal Model Program. During this talk, we will briefly discuss the G-equivariant Minimal Model Program, toric geometry and classify toric G-Fano threefolds. The classification will not require any knowledge of birational geometry, as we will do it purely by playing with polytopes.
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18th November | Sebastian Schlegel-Mejía | On BPS cohomology |

Abstract: Since its inception 20 years ago as an enumerative theory of sheaves on Calabi-Yau 3-folds, Donaldson-Thomas theory has evolved and reached far beyond its original domain with applications in subjects such as quantum groups, cluster algebras, and nonabelian Hodge theory. A key role in many of these applications is played by BPS invariants and BPS cohomology which intuitively are a count of simple objects in certain abelian categories of homological dimension no more than 3. I intend my talk to be an introduction to BPS cohomology. There will be examples and computations of BPS cohomology. Depending on what I actually prepare I might explain applications of BPS cohomology. Most likely these will have something to do with the $\chi$-independence phenomenon in enumerative geometry (I think this is super cool).
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11th November | Sarunas Kaubrys | Moduli of local systems |

Abstract: I will start by introducing local systems, which are representations of the fundamental group of some topological space. After giving some motivation to study these, I will try to explain some examples and properties of the moduli of local systems (or character variety). For us this is some algebraic geometric space such that the points of the space correspond to (iso classes of ) local systems. In particular, the character variety of a surface is well known to have a symplectic structure. Finally, we will consider two enhancements of the character variety: one given by the theory of stacks and one by derived geometry. Among other things, this technology allows us to make sense of a version of symplectic structures for any closed oriented manifold.
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4th November | Patrick Kinnear | Skein module dimensions of mapping tori of T2 |

Abstract: The skein module of a 3-manifold is a C(q)-vector space. It is a 3-manifold invariant which generalises the Jones polynomial of a knot, and has its roots in topological quantum field theory. The most intensively studied version is the Kauffman bracket skein module, which has local relations given by the representation theory of quantum SL_2, however skein modules can be defined for any reductive group G. It was recently shown that skein modules of closed, compact, oriented 3-manifolds are generically finite dimensional, however the proof is not constructive and one key research goal of quantum topologists is to describe these vector spaces explicitly. To date, the skein module dimension has been computed for the 3-torus, the product of a surface with a circle, and a few other 3-manifolds, usually just for SL_2. In this talk we will introduce skein modules and explain our recent computations of the SL_N skein modules of mapping tori of T^2. This gives the skein dimension for a new and large family of 3-manifolds, in the more general SL_N setting. Our work also suggests how techniques from higher algebra and geometric representation theory can be brought to bear on skein-theoretic problems.
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28th October | Lucas Buzaglo | A survey on 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 not known whether it is possible for such an enveloping algebra to be (left or right) noetherian, a question that has been open since 1974. In this talk, I will give an overview of the recent progress in the study of these enveloping algebras, with a focus on questions of noetherianity.
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21st October | Linden Disney-Hogg | Theta characteristics, Spin Structures, and their Orbits |

Abstract: Theta characteristics are a classical object on Riemann surfaces arising from considerations of theta functions, permuted by the action of the curves automorphism group, but despite having first being studied in 1851 there is still very little known. I will talk about their modern study, both via spin structures (thanks to Atiyah) and via Scorza theory (thanks to Dolgachev and Kanev), involving some lovely geometry, and some highly-symmetrical curves. Time permitting, I'll mention the current computational frontiers. .
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14th October | Ruben Van Belle | Radon-Nikodym derivates and martingales using category theory |

Abstract: We give a categorical proof of the Radon-Nikodym theorem in measure theory. We start from the easier Radon-Nikodym theorem on finite probability spaces and then Kan extend the result to general probability spaces. From this the concept of conditional expectation arises naturally. We then repeat this with everything enriched over complete metric spaces. Using this we (categorically) prove a version of the martingale convergence theorem.
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7th October | Lisa Marquand (Stony Brook University) | Symplectic Birational Involutions of manifolds of OG10 Type |

Abstract: Compact Hyperkähler manifolds are one of the building blocks of kähler manifolds with trivial first chern class, but very few examples are known. One strategy for potentially finding new examples is to look at finite groups of symplectic automorphisms of the known examples, and study the fixed loci or quotient. In this talk, we will obtain a partial classification of birational symplectic involutions of manifolds of OG10 type. We do this from two vantage points: firstly following classical techniques relating birational transformations to automorphisms of the Leech lattice. Secondly, we look at involutions that are obtained from cubic fourfolds via the compactified intermediate Jacobian construction. In this way, we obtain new involutions that could potentially give rise to new holomorphic symplectic varieties. If time permits, we will mention ongoing work to identify the fixed loci in one of these examples.
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30th September | Theodoros Lagiotis | Handling Cobordisms |

Abstract: Handle attachments are a standard construction in topology. In this talk we will discuss how and why one would want to view certain 3d cobordisms as handle attachments. In doing so, we will of course introduce all the relevant definitions, so if these terms mean nothing to you, fear not! The goal is to do all of the above with as many pictures and examples as possible. Time permitting, we will see how this story relates to semisimplicity in 3d TQFTs. |

### Historical schedules

Hodge Club 2021/22

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.