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# 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 William Bevington, Álvaro Muñiz Brea and Karim Rega.

### Current Schedule of talks for 2021/22

#### Semester 1

##### Upcoming talks
 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. 9th December Isambard Goodbody (University of Glasgow) Abstract:

##### Past talks
 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). 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. 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. 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. 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. . 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. 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. 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.