### LAGOON

## Leicester Algebra and Geometry Open ONline Seminar

### Thursdays 12pm-1pm (UK time GMT+1/UTC+1 )

Organisers: Frank Neumann and Sibylle Schroll

The LAGOON webinar series has a strong focus on Representation Theory and Algebraic Geometry and their many interactions covering topics such as homological mirror symmetry, stability conditions, derived categories, dg-categories, Hochschild cohomology of algebras, moduli spaces and algebraic stacks, derived algebraic geometry and other topics.

This webinar series is supported as part of the International Centre for Mathematical Sciences (ICMS) and the Isaac Newton Institute for Mathematical Sciences (INI) Online Mathematical Sciences Seminars. Please visit * here* to register and obtain the Zoom link and password for the meetings.

Video recordings of the talks can be viewed **here****.**

The webinar series is also cross-listed on researchseminars.org

**Programme Semester 1 2021**

**Programme Semester 1 2021**

Talks start at 12pm (UK time)

**28 January** Dan Kaplan (Birmingham, UK): Multiplicative preprojective algebras in geometry and topology

**4 February **Merlin Christ (Hamburg, Germany): A gluing construction for Ginzburg algebras of triangulated surfaces

**11 February **Bertrand Toën (Toulouse, France): Foliations on schemes

**18 February **Paolo Stellari (Milano, Italy): Uniqueness of enhancements for derived and geometric categories

**25 February **Claudia Scheimbauer (TU München, Germany): Derived symplectic geometry and AKSZ topological field theories

**4 March ** Sebastian Opper (Prague, Czech Republic): Spherical objects on cycles of projective lines and transitivity

**11 March** Alexandra Zvonareva (Stuttgart, Germany): Derived equivalence classification of Brauer graph algebras

**18 March** Letterio Gatto (Polytechnic University of Turin, Italy): HiDEAs to work with

**25 March** Easter Break - no seminar

**1 April** Easter Break - no seminar

**8 April ** British Mathematical Colloquium - no seminar

**15 April ****3pm** Valery Lunts (Indiana, USA) Subcategories of derived categories on affine schemes and projective curves

**22 April** Severin Barmeier (Freiburg, Germany) Scattering amplitudes from derived categories and cluster categories

**29 April** Okke van Garderen (Glasgow, UK) Stability, duality, and DT invariants for flopping curves

**6 May **Ana Ros Camacho (Cardiff, UK) On the Landau-Ginzburg/conformal field theory correspondence

**13 May** Matt Booth (Antwerp, Belgium) Topological Hochschild cohomology for schemes

**20 May** Sergey Mozgovoy (Trinity College Dublin, Ireland) DT invariants of some 3CY quotients

**27 May **no seminar:** ****VirtARTA2021**** **Online conference in memory of Andrzej Skowronski** **

**3 June **Markus Szymik (NTNU Trondheim, Norway) A homological stroll into the algebraic theories of racks and quandles

**10 June** Fabian Haiden (Oxford, UK) New 3CY categories of topological surfaces

**17 June **David Pauksztello (Lancaster, UK) Functorially finite hearts, simple-minded systems and negative cluster categories

**24 June** Yilin Wu (Université Paris Diderot - Paris 7, France) Derived equivalences from mutations of ice quivers with potential

**1 July** Chelsea Walton (Rice, USA)

**Abstracts: **

**28 January** Dan Kaplan (Birmingham, UK): Multiplicative preprojective algebras in geometry and topology

Abstract: In 2006, Crawley-Boevey and Shaw defined the multiplicative preprojective algebra (MPA) to study certain character varieties. More recently, MPAs appeared in work of Etgü--Lekili in the study of Fukaya categories of 4-manifolds. Nice properties of the (additive) preprojective algebra are expected to hold for MPAs, but most proof techniques are not available. In joint work with Travis Schedler, we define the strong free product property, following older work of Anick. Using this property, we prove MPAs are 2-Calabi--Yau algebras for quivers containing a cycle. Moreover, using a result of Bocklandt--Galluzzi--Vaccarino, we prove the formal local structure of multiplicative quiver varieties is isomorphic to that of a (usual) quiver variety. In this talk, I'll survey these ideas and illustrate them in small examples. **[****Slides****] [****Video****]**

**4 February **Merlin Christ (Hamburg, Germany): A gluing construction for Ginzburg algebras of triangulated surfaces

Abstract: Ginzburg algebras associated to triangulated surfaces are a class of 3-Calabi-Yau dg-algebras which categorify the cluster algebras of the underlying marked surfaces. In this talk, we will discuss a description of these Ginzburg algebras in terms of the global sections of a constructible cosheaf of dg-categories (modelling a perverse Schober). This cosheaf description shows that the Ginzburg algebras arise via the gluing of relative versions of Ginzburg algebras associated to the faces of the triangulation along their common edges. The definition of the cosheaf is inspired by a result of Ivan Smith, by which the finite derived category of such a Ginzburg algebra embeds into the Fukaya category of a Calabi-Yau 3-fold equipped with a Lefschetz fibration to the surface. **[****Slides****] [****Video****]**

**11 February **Bertrand Toën (Toulouse, France): Foliations on schemes

Abstract: In this talk I will present a notion of foliations on arbitrary schemes (possibly of positive or mixed characteristics), based on techniques from derived algebraic geometry. As an instance of application I will explain how Baum-Bott's existence of residues for singular holomorphic foliations can be extended to the positive characteristic setting. **[****Slides****] [****Video****]**

**18 February **Paolo Stellari (Milano, Italy): Uniqueness of enhancements for derived and geometric categories

Abstract: In this talk we address several open questions and generalize the existing results about the uniqueness of enhancements for triangulated categories which arise as derived categories of abelian categories or from geometric contexts. If time permits, we will also discuss applications to the description of exact equivalences. This is joint work with A. Canonaco and A. Neeman.** [****Slides****] [****Video****]**

**25 February **Claudia Scheimbauer (TU München, Germany): Derived symplectic geometry and AKSZ topological field theories

Abstract: Derived algebraic geometry and derived symplectic geometry in the sense of Pantev-Toen-Vaquié-Vezzosi allows for a reinterpretation/analog of the classical AKSZ construction for certain $\sigma$-models. After recalling this procedure I will explain how it can be extended to give a fully extended oriented TFT in the sense of Lurie with values in a higher category whose objects are $n$-shifted symplectic derived stacks and (higher) morphisms are (higher) Lagrangian correspondences. It is given by taking mapping stacks with a fixed target building and describes ``semi-classical TFTs". This is joint work in progress with Damien Calaque and Rune Haugseng. **[****Slides****]**

**4 March** Sebastian Opper (Prague, Czech Republic): Spherical objects on cycles of projective lines and transitivity

Abstract: Polishchuk showed that spherical objects in the derived category of any cycle of projective lines yield solutions of the associative Yang-Baxter equation which raises the question whether one can classify spherical objects. He further posed the question whether the group of derived auto-equivalences of a cycle acts transitively on isomorphism classes of spherical objects. Partial solutions to both problems were given in works of Burban-Kreussler and Lekili-Polishchuk. A theorem of Burban-Drozd establishes a connection between the derived category of any cycle of projective lines with the derived category of a certain gentle algebra which can be modeled by a (toplogical) surface and which allows us to translate algebraic information in the derived category such as objects into geometric information on the surface such as curves. I will explain how the result of Burban-Drozd can be used to find a similar model for the derived category of a cycle. Afterwards we discuss how this can be exploited to classify spherical objects and establish transitivity. Further applications include a description of the group of derived auto-equivalences of a cycle and faithfulness of a certain group action as defined by Sibilla. ** [****Slides****] [****Video****]**

**11 March** Alexandra Zvonareva (Stuttgart, Germany): Derived equivalence classification of Brauer graph algebras

Abstract: In this talk, I will explain the classification of Brauer graph algebras up to derived equivalence. These algebras first appeared in representation theory of finite groups and can be defined for any suitably decorated graph on an oriented surface. The classification relies on the connection between Brauer graph algebras and gentle algebras and the classification of the mapping class group orbits of the homotopy classes of line fields on surfaces. We consider A-infinity trivial extensions of partially wrapped Fukaya categories associated to surfaces with boundary, this construction naturally enlarges the class of Brauer graph algebras and provides a way to construct derived equivalences. This is based on joint work with Sebastian Opper. **[****Slides****] [****Video****]**

**18 March** Letterio Gatto (Polytechnic University of Turin, Italy): HiDEAs to work with

Abstract: HiDEA is the acronym of *Higher Derivations on Exterior Algebra*, a project I am currently working on together with many collaborators, such as O. Behzad & A. Nasrollah Nejad (Iran), L. Rowen & I. Scherbak (Israel), A. Contiero, P. Salehyan & R. Vidal Martins (Brasil), S. Amukugu, M. Mugochi & G. Marelli (Namibia). Originally introduced by Hasse & Schmidt (1937) to extend Taylor expansions of analytic functions and Wronskians in in the realm of positive characteristic commutative algebra, the notion of Higher Order derivations (*Hasse-Schmidt derivation* in the sequel) provides an extremely rich theory when applied to the super--commutative situation supplied by exterior algebras of free modules. The purpose of this talk is to advertise HiDEAs practise, focusing on its main tool, the so-called *integration by parts* formula. The latter shows how the theory is concerned with multilinear algebra (via an extension of the Cayley-Hamilton theorem for possible infinite dimensional vector spaces), with intersection theory of Grassmannians (Schubert Calculus via Pieri's & Giambelli's formula), with Representation Theory and Mathematical Physics, given the spontaneously arising of the vertex operators occurring in the boson-fermion correspondence from the so-called *Schubert Derivations*. The talk aims to be general, non specialistic and self--contained, requiring no more than basics in multilinear algebra (exterior algebras), elementary calculus (Taylor expansions) and a little routine combinatorics (formal power series, partitions, symmetric functions). **[****Slides****]**

**15 April ****3pm** Valery Lunts (Indiana, USA): Subcategories of derived categories on affine schemes and projective curves

Abstract: I will report on my joint recent work with Alexey Elagin (arXiv:2007.02134 , arXiv:2002.06416, arXiv:1711.01492). The famous theorem of Hopkins-Neeman gives a simple geometric classification of thick subcategories of the category Perf(X) for an affine noetherian scheme X. It is natural to ask if there is a similar classification of thick subcategories of D^b(cohX) (for an affine X). I will discuss some positive and some negative results in this direction. In a different situation: surprisingly one is able to classify (up to equivalence) all thick subcategories of D^b(cohC) for a smooth projective curve.

**22 April** Severin Barmeier (Freiburg, Germany): Scattering amplitudes from derived categories and cluster categories

Abstract: Scattering amplitudes are physical observables which play a central role in interpreting scattering experiments at particle colliders. In recent years a new perspective on scattering amplitudes has revealed a fascinating link to various mathematical structures, such as positive Grassmannians and cluster algebras. In this talk I will explain this connection from the point of view of derived and cluster categories of type A quivers, from which the formulae for scattering amplitudes can be obtained from projectives of hearts of intermediate t-structures. This talk is based on arXiv:2101.02884 joint with Koushik Ray. **[****Slides****] [****Video****]**

**29 April** Okke van Garderen (Glasgow, UK) : Stability, duality, and DT invariants for flopping curves

Abstract: Threefold flops are birational surgeries on a contractible curve that connect minimal models of threefolds, and are therefore crucial to the minimal model program. To examine these flops one would like to compute their Donaldson-Thomas invariants, which are virtual counts of semistable objects in the derived category. In this talk I will explain how to determine the semistable objects supported on a flopping curve by showing that their K-theory classes are dual to a hyperplane arrangement induced by tilting complexes. I will also show how this duality can be categorified to give a full description of the (3-Calabi-Yau) deformation theory of these objects, which has various implications for the DT theory. **[****Slides****] [****Video****]**

**6 May** Ana Ros Camacho (Cardiff, UK): On the Landau-Ginzburg/conformal field theory correspondence

Abstract: The Landau-Ginzburg/conformal field theory (LG/CFT) correspondence is a result from the theoretical physics literature dating back to the late 80s-early 90s, which in particular predicts a certain relation between categories of matrix factorizations and categories of representations of vertex operator algebras. Currently we lack a precise mathematical statement for this physics result, but fortunately we have some examples available that we will review during this talk, as well as some current work in progress towards more. This is joint work with I. Runkel, A. Davydov et al. **[****Slides****] [****Video****]**

**13 May** Matt Booth (Antwerp, Belgium): Topological Hochschild cohomology for schemes

Abstract: Topological Hochschild cohomology is a sort of refinement of usual Hochschild cohomology that incorporates data from stable homotopy theory. Instead of working over a base ring, one works over the sphere spectrum, which is a commutative ring in an appropriate sense. I'll give a quick introduction to spectral algebra and THH^*. Then I'll define the THH^* of a scheme in a `derived noncommutative' way - i.e. using appropriate dg categories of sheaves - and explain some invariance results, which in the non-topological setting are due to Lowen and Van den Bergh via Keller. I'll discuss some toy non-affine computations, and time permitting I'll talk about the relationship to deformation theory, especially in positive characteristic. This is joint work with Dmitry Kaledin and Wendy Lowen.** [****Slides****] [****Video****]**

**20 May** Sergey Mozgovoy (Trinity College Dublin, Ireland): DT invariants of some 3CY quotients

Abstract: Given a finite subgroup of SL3, the corresponding quotient singularity has a natural non-commutative crepant resolution, the skew group algebra. By the result of Ginzburg, this crepant resolution is Morita equivalent to the Jacobian algebra of the McKay quiver equipped with a canonical potential. We will discuss refined DT invariants of such Jacobian algebras for the cases of finite subgroups of SL2 and SO3, where the quotient singularity admits a small crepant resolution and the McKay quiver is symmetric. **[****Slides****] **

**3 June ** Markus Szymik (NTNU Trondheim, Norway) A homological stroll into the algebraic theories of racks and quandles

Abstract: Racks and quandles are rudimentary algebraic structures akin to groups and tied to symmetry. I will begin my presentation with an introduction to these concepts, focussing on their ubiquity in geometry and topology. Current developments illustrate how an interplay between conceptual curiosity and computational aspiration can substantially progress our understanding of such structures. I will take a homological vantage point and weave a narrative around some recent joint work with Tyler Lawson and Victoria Lebed.** [****Slides****]**

**10 June** Fabian Haiden (Oxford, UK) New 3CY categories of topological surfaces

Abstract: To a topological surface, perhaps with certain markings, one can attach several different triangulated categories whose objects are, roughly speaking, curves on the surface. One such example is the Fukaya category of the surface, another is the 3-d Calabi-Yau (3CY) category of an ideal triangulation. These have proven useful, among other things, in the study of Bridgeland stability conditions and the representation theory of finite-dimensional algebras. In the recent preprint arXiv:2104.06018 I introduce yet another class of triangulated A-infinity categories of surfaces. The motivation for constructing them was to extend the work of Bridgeland-Smith on stability conditions and quadratic differentials to the finite area case (e.g. holomorphic differentials). They are closely related to the existing triangulated categories of surfaces and clarify the relation between them. Their construction involves some algebraic tricks, such as twisted complexes and modules over curved A-infinity categories, which will be discussed in detail. **[****Slides****]**

**17 June **David Pauksztello (Lancaster, UK) Functorially finite hearts, simple-minded systems and negative cluster categories

Abstract: Simple-minded systems (SMSs) were introduced by Koenig-Liu as an abstraction of nonprojective simple modules in stable module categories: the idea was to use SMSs as a way to get around the lack of projective generators to help develop a Morita theory for stable module categories. Recent developments have shown that SMSs in negative Calabi-Yau categories admit mutation theories and combinatorics that are highly suggestive of cluster-tilting theory. In this talk, we explain one such development: that negative Calabi-Yau orbit categories of bounded derived categories of acyclic quivers serve as categorical models of positive Fuss-Catalan combinatorics and one can think of SMSs as negative cluster-tilting objects. Along the way, we will make use of the rather surprising observation that in a triangulated category of finite homological dimension, functorial finiteness of the heart of a t-structure is related to the property of the heart having enough injectives and enough projectives. This is surprising because it says that some feature of how a heart behaves within an ambient triangulated category can be detected intrinsically in the heart. This talk is based on joint work with Raquel Coelho Simoes and David Ploog.

**24 June** Yilin Wu (Université Paris Diderot - Paris 7, France) Derived equivalences from mutations of ice quivers with potential

Abstract: In 2009, Keller and Yang categoriﬁed quiver mutation by interpreting it in terms of equivalences between derived categories. Their approach was based on Ginzburg’s Calabi--Yau algebras and on Derksen-Weyman-Zelevinsky’s mutation of quivers with potential. Recently, Matthew Pressland has generalized mutation of quivers with potential to that of ice quivers with potential. We will explain how his rule yields derived equivalences between the associated relative Ginzburg algebras, which are special cases of Yeung’s deformed relative Calabi–Yau completions arising in the theory of relative Calabi--Yau structures due to Toën and Brav--Dyckerhoff. We will illustrate our results on examples arising in the work of Baur—King--Marsh on dimer models and cluster categories of Grassmannians.

**Programme Semester 2 2020**

**Programme Semester 2 2020**

**All talks from 1 October on will start at 12pm (UK time). **

**3 September** Andrea Solotar (Buenos Aires, Argentina): A cup-cap duality in Koszul calculus

Abstract: In this talk I will introduce a cup-cap duality in the Koszul calculus of N-homogeneous algebras following https://arxiv.org/abs/2007.00627. As an application of this duality, it follows that the graded symmetry of the Koszul cap product is a consequence of the graded commutativity of the Koszul cup product. I will also comment on a conceptual approach to this problem that may lead to a proof of the graded commutativity, based on derived categories in the framework of DG algebras and DG bimodules. This is joint work with Roland Berger. **[****Slides****] [****Vid****eo****]**

**10 September** Zhengfang Wang (Stuttgart, Germany): Deformations of path algebras of quivers with relations

Abstract: In this talk, we provide a very explicit method to describe deformations of path algebras of quivers with relations. This method is based on a combinatorial description of an L-infinity algebra constructed from Chouhy-Solotar’s projective resolution. As an application, we show that the variety associated to monomial algebras constructed by Green-Hille-Schroll is actually given by the Maurer--Cartan equation of the L-infinity algebra. This is joint work with Severin Barmeier. **[****Slid****es****]**** ****[****Video****]**

**17 September** No Seminar: Categorifications in Representation Theory Conference at Leicester

**24 September **Ailsa Keating (Cambridge, UK): Homological mirror symmetry for log Calabi-Yau surfaces

Abstract: Given a log Calabi-Yau surface Y with maximal boundary D, I'll explain how to construct a mirror Landau-Ginzburg model, and sketch a proof of homological mirror symmetry for these pairs when (Y,D) is distinguished within its deformation class (this is mirror to an exact manifold). I'll explain how to relate this to the total space of the SYZ fibration predicted by Gross--Hacking--Keel, and, time permitting, explain ties with earlier work of Auroux--Katzarkov--Orlov and Abouzaid. Joint work with Paul Hacking. [**Vid*** eo*]

**1 October **Karin Baur (Leeds, UK and Graz, Austria): Structure of Grassmannian cluster categories

Abstract: The category of Cohen Macaulay modules over a quotient of a preprojective algebra provides an additive categorification of Scott’s cluster algebra structure of the coordinate ring of the Grassmannian of k-subspaces in n-space, by work of Jensen, King and Su. Under this correspondence, rigid indecomposable objects map to cluster variables. A special role is played by rank 1 indecomposables which correspond bijectively to Plücker coordinates. These are in fact all indecomposables in case k=2. In the other finite types (i.e. $(k,n)\in \{(3,6),(3,7),(3,8)\}$), there are also rank 2 and rank 3 rigid indecomposables. In general, the Grassmannian categories are not well understood. We provide characterisations for these low rank modules in infinite types. This is joint work with Dusko Bogdanic and Ana Garcia Elsener and with Bogdanic, Garcia Elsener and Jianrong Li. [* Slides*] [

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**Video****8 October **Andrey Lazarev (Lancaster, UK): Koszul duality for dg-categories and infinity-categories

Abstract: Differential graded (dg) Koszul duality is a certain adjunction between the category of dg algebras and conilpotent dg coalgebras that becomes an equivalence on the levels of homotopy categories. More precisely, this adjunction is a Quillen equivalence of the corresponding closed model categories. Various versions of this result exist and play important roles in rational homotopy theory, deformation theory, representation theory and other related fields. We extend it to a Quillen equivalence between dg categories (generalizing dg algebras) and a class of dg coalgebras, more general than conilpotent ones. As applications we describe explicitly and conceptually Lurie’s dg nerve functor as well as its adjoint and characterize derived categories of (\infty,1)-categories as derived categories of comodules over simplicial chain coalgebras. (joint work with J. Holstein) [* Slides*] [

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**Video****15 October **Fabrizio Catanese (Bayreuth, Germany): Topologically trivial automorphisms of compact Kähler surfaces and manifolds

Abstract: The abstract can be downloaded * here*. [

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**Video****22 October **Lang Mou (HMI Bonn, Germany): Caldero–Chapoton formulas for generalized cluster algebras from orbifolds

Abstract: To a marked bordered surface with orbifold points of order 3, we associated a quiver (with loops) with potential. We then connect the cluster structure of the corresponding skew-symmetrizable matrix with the stability conditions and the \tau-tiliting theory of the Jacobian algebra. Finally we provide Caldero–Chapoton type formulas for cluster monomials of the generalized cluster algebra of Chekhov and Shapiro associated to the surface. This is joint work with Labardini-Fragoso. [* Slides*] [

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**Video****29 October **Michael Wemyss (Glasgow, UK): Contraction algebras, plumbings and flops

Abstract: I will explain how certain symmetric Nakayama algebras (under the disguise of "contraction algebras") control and prove theorems about geometric objects on both sides of mirror symmetry. As part of this, I will explain our symplectic geometry model, our algebraic geometry model, and then how the contraction algebra relates them. The cohomology of objects in the underlying categories are naturally modules for the associated contraction algebra, and I will explain how to use this information to obtain otherwise tricky results, such as a classification of spherical (and more generally, fat-spherical) objects. This has purely topological corollaries. One feature, which I will probably gloss over but is actually fundamental, is that our categories have a dependence on the characteristic of the ground field. This is joint work with Ivan Smith (arXiv:2010.10114). [* Slides*] [

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**Video****5 November **Yukinobu Toda (Kavli IPMU, Tokyo, Japan): On d-critical birational geometry and categorical DT theories

Abstract: In this talk, I will explain an idea of analogue of birational geometry for Joyce's d-critical loci, and categorical Donaldson-Thomas theories on Calabi-Yau 3-folds. The motivations of this framework include categorifications of wall-crossing formulas of DT invariants and also a d-critical analogue of D/K conjecture in birational geometry. The main result is to realize the above story for local surfaces. I will show the window theorem for categorical DT theories on local surfaces and apply it to categorify wall-crossing invariance of genus zero GV invariants, MNOP/PT correspondence, etc. [* Slides*] [

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**Video****12 November **Amihay Hanany (Imperial, UK): Coulomb branch

Abstract: The Coulomb branch is a symplectic singularity that appears in the physics study of gauge theories (more precisely in 3d N=4 supersymmetric gauge theories). A recent (2013) progress in understanding the Coulomb branch was when a combinatorial formula for this singularity was introduced, named the monopole formula. This raised excitement both in physics and in mathematics. It plays an important role in a collection of physical phenomena which were hard to solve previously, and it gives a new construction of geometric singularities that opens new directions of study in representation theory. This talk will focus on the monopole formula for a quiver and will discuss the different objects and features which arise from the quiver. [* Slides*][

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**Video****19 November **John Greenlees (Warwick, UK): The singularity category of C^*(BG)

Abstract: [joint work with G.Stevenson and D.Benson] For an ordinary commutative Noetherian ring R we would define the singularity category to be the quotient of the (derived category of) finitely generated modules modulo the (derived category of) fg projective modules [``the bounded derived category modulo compact objects’’]. For a ring spectrum like C^*(BG) (coefficients in a field of characteristic p) it is easy to define the module category and the compact objects, but finitely generated objects need a new definition. The talk will describe the definition and show that the singularity category is trivial exactly when G is p-nilpotent. We will go on to describe the singularity category for groups with cyclic Sylow p-subgroup. [* Slides*] [

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**Video****26 November **Evgeny Shinder (Sheffield, UK): Birationality centers, rationality problems and Cremona groups

Abstract: I will introduce a framework to account for the ambiguity of stable birational types of a sequence of centers for birational transformations. I will explain in which settings the introduced invariants are nonvanishing, and give applications to the structure of Cremona groups over various fields. This is joint work in progress with Hsueh-Yung Lin and Susanna Zimmermann. [* Slides*] [

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**Video****3 December !!5-6pm!! **Giulia Saccà (Collège de France and Columbia University, USA): Hodge numbers of OG10 via Ngô strings

Abstract: I will talk on joint work with M. de Cataldo and A. Rapagnetta, in which we compute the Hodge numbers of the 10-dimensional hyperkähler manifold known as OG10. The main technique is the use of Ngô's support theorem, applied to a natural Lagrangian fibration on a certain projective model of OG10, together with the study of the geometry of the fibration itself. [* Slides*] [

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**Video****10 December **Ralph Kaufmann (Purdue, USA): Categorical Interactions in Algebra, Geometry and Representation Theory

Abstract: There are several fundamental interactions between combinatorics, algebra and geometry, where the combinatorial structures give representations and suitably interpreted encode cells for a geometric realization. A prime example of this is Deligne's conjecture, where the representation of certain graphs yields actions on the Hochschild complex and geometrically these graphs can be considered as graphs dual to a system of arcs on a surface.There is a way to encode the combinatorial structures into categorical ones, the so-called Feynman categories. The representations in this setting functors out of them. More generally they yield the representations can also be algebras of certain types. In the functorial formalism one has restriction, reduction and Frobenius reciprocity. To make these geometric, one can use a so-called W-construction. For trees and graphs, this program leads to the construction of moduli spaces of graphs and Riemann surfaces. These are versions of the commutative and associative geometries studied by Kontsevich. Staying inside the algebraic world, one can use functors to enrich Feynman categories. The enriched categories play the role of algebras and the representations are modules - all with possible higher operations. The enrichment is made by using a plus construction, which has a connection to bi-algebras and Hopf algebras based on the morphisms of a Feynman category. [* Slides*] [

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**Video****Programme Semester 1 2020**

**Programme Semester 1 2020**

**Talks start at 1pm (UK time) .**

**14 May ** Hipolito Treffinger (University of Leicester, UK): Representation theoretic aspects of scattering diagrams

Abstract: The notion of algebraic scattering diagram associated to any finite dimensional algebra was recently introduced by Bridgeland as an algebraic construction of the celebrated cluster scattering diagrams of Gross, Hacking, Keel and Kontsevich. In this talk, after briefly recalling the construction of scattering diagrams given by Bridgeland, we will show how the homological aspects of the module category determine several properties of the support of the scattering diagrams. In particular, we will show that chambers in the scattering diagram of an algebra are in one-to-one correspondence with certain torsion pairs in its module category. This is joint work with Thomas Brustle and David Smith. Based on this characterisation, we will discuss how the study of torsion pairs in the module category of algebras can play a key role in the calculation of Donaldson-Thomas invariants for certain Calabi-Yau threefolds. [* Slides*]

**21 May** Alex Takeda (IHES, France): Gluing relative stability conditions along pushouts

Abstract: In this talk I will discuss the results of arXiv:1811.10592 and some later developments, concerning how to produce Bridgeland stability conditions on certain categories from using a local-to-global principle. The example of particular interest will be the topological Fukaya category of a marked surface, and the description of the local data is inspired by the construction of stability conditions on such categories using quadratic differentials by Haiden, Katzarkov and Kontsevich. As an application of this method, we show that one can understand all the components of the stability space of such categories, and that in suitable cases the whole space is described by these HKK stability conditions. **[Slides****]**** [****Video****]**

**28 May** Bernhard Keller (Université Paris Diderot - Paris 7, France): Grassmannian braiding categorified

Abstract: Chris Fraser has discovered an action of the extended affine braid group on d strands on the Grassmannian cluster algebra of k-subspaces in n-space, where d is the least common divisor of k and n. We lift this action to the corresponding cluster category first constructed by Geiss-Leclerc-Schröer in 2008. For this, we use Jensen-King-Su's description of this category as a singularity category in the sense of Buchweitz/Orlov. We conjecture an action of the same braid group on the cluster algebra associated with an arbitrary pair of Dynkin diagrams whose Coxeter numbers are k and n. This is a report on ongoing joint work with Chris Fraser. **[****Slides****] [****Video****]**

**4 June** Balazs Szendroi (University of Oxford, UK): Hilbert schemes of points on singular surfaces: combinatorics, geometry, and representation theory

Abstract: Given a smooth algebraic surface S over the complex numbers, the Hilbert scheme of points of S is the starting point for many investigations, leading in particular to generating functions with modular behaviour and Heisenberg algebra representations. I will explain aspects of a similar story for surfaces with rational double points, with links to algebraic combinatorics and the representation theory of affine Lie algebras. I will in particular recall our 2015 conjecture concerning the generating function of the Euler characteristics of the Hilbert scheme for this singular case, and aspects of more recent work that lead to a very recent proof of the conjecture by Nakajima. Joint work with Gyenge and Nemethi, respectively Craw, Gammelgaard and Gyenge. [* Slides*][

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**Video****11 June** Qiu Yu (Tsinghua University Beijing, China): Graded decorated marked surfaces: Calabi-Yau-X categories of gentle algebras

Abstract: Motivated by q-deforming of stability conditions and categories, we study the Calabi-Yau-X categories of gentle algebras from graded decorated marked surfaces. The string model in this case unifies the Calabi-Yau-3 case in the prequels and the usual/Calabi-Yau-infinity case (via Lagrangian immersion). This is a joint work with Akishi Ikeda and Yu Zhou. [* Slides*][

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**Video****18 June !!4-5pm!!** Lara Bossinger (UNAM Oaxaca, Mexico): Families of Gröbner degenerations

Abstract: In this talk I will present a construction of one flat family that combines many Gröbner degenerations. More precisely, for a (weighted) homogeneous ideal we consider a maximal cone in its Gröbner fan. Associated to that cone we define a flat family that contains various special fibers associated to the initial degenerations of the cone and all its faces. This construction has several interesting applications. Most surprisingly, it recovers the recursive construction of universal coefficients for cluster algebras in a non-recursive way for the Grassmannians Gr(2,n) and Gr(3,6). If time permits I will present another application explaining how to recover Kaveh-Manon's toric equivariant families arising from a collection of nice cones in the tropicalization of an ideal.This talk is based on joint work in progress with F. Mohammadi and A. Nájera Chávez. [* Slides*][

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**Video****25 June** Wendy Lowen (University of Antwerp, Belgium): Linear quasi-categories as templicial modules (joint work with Arne Mertens)

Abstract: We introduce a notion of enriched infinity categories over a given monoidal category, in analogy with quasi-categories over the category of sets. We make use of certain colax monoidal functors, which we call*templicial objects*, as a replacement of simplicial objects that respects the monoidal structure. We relate the resulting *enriched quasi-categories* to nonassociative Frobenius monoidal functors, allowing us to prove that the free templicial module over an ordinary quasi-category is a linear quasi-category. To any dg category we associate a linear quasi-category, the *linear dg nerve*, which enhances the classical dg nerve, and we argue that linear quasi-categories can be seen as relaxations of dg-categories. [* Slides*][

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**Video****2 July **Hiraku Nakajima (Kavli IPMU, Tokyo, Japan): Euler numbers of Hilbert schemes of points on simple surface singularities and quantum dimensions of standard modules of quantum affine algebras

Abstract: Balazs explained his conjecture with Gyenge and Nemethi on Euler numbers of Hilbert schemes on June 4. I proved it by showing that quantum dimensions of standard modules of quantum affine algebras are always 1. This remarkable property is the simplest case of a conjecture on quantum dimensions of Kirillov-Reshetikhin modules proposed by Kuniba in 93, which is still open for E7,8 in general. In this talk, I will emphasize on representation theoretic aspects to minimize overlaps with Balazs' talk. [* Slides*][

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**Video****9 July !!4-5pm!! ** Alexander Polishchuk (University of Oregon, USA): Geometry of the Associative Yang-Baxter equation

Abstract: I will describe the connection, discovered jointly with Yanki Lekili, between Associative Yang-Baxter equation (AYBE) and pairs of 1-spherical objects in A-infinity categories. I will then explain how such pairs arise from noncommutative orders over singular curves, in particular, how to get all nondegenerate trigonometric solutions of the AYBE in this way. If time allows, I will talk about the Lie analog of this story for the classical Yang-Baxter equation. [* Slides*][

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**Video****16 July** Markus Reineke (Universität Bochum, Germany): Fano quiver moduli

Abstract: We construct a large class of quiver moduli spaces which are Fano varieties, by studying global sections of line bundles on quiver moduli and identifying a special class of stabilities. We discuss several classes of examples (e.g. toric varieties, point configuration spaces, Kronecker moduli). [* Slides*][

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**Video****23 July** Laura Schaposnik (University of Illinois at Chicago, USA): On Generalized Hyperpolygons

Abstract: In this talk we will introduce generalized hyperpolygons, which arise as Nakajima-type representations of a comet-shaped quiver, following a recent work joint with Steven Rayan. After showing how to identify these representations with pairs of polygons, we shall associate to the data an explicit meromorphic Higgs bundle on a genus g Riemann surface, where g is the number of loops in the comet. We shall see that, under certain assumptions on flag types, the moduli space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system. Time permitting, we shall conclude the talk by mentioning some partial results on current work on the construction of triple branes (in the sense of Kapustin-Witten mirror symmetry), and dualities between tame and wild Hitchin systems (in the sense of Painlevé transcendents). [* Slides*]

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