Previous Talks

Video recordings

Recordings of the past talks are available here

Titles and abstracts of past talks

2022

15 Dec 2022   Jasper van de Kreeke   (Amsterdam, Netherlands)   Deformation theory of gentle algebras

Abstract. Gentle algebras are discrete models for Fukaya categories of punctured surfaces. How to deform them? We will start with one specific deformation, motivated by Seidel's relative Fukaya categories. We will classify the entire deformation theory of gentle algebras and prove that their Hochschild DGLA is formal. We will comment on deformed mirror symmetry and related work of Barmeier–Schroll–Wang. On a technical level, this talk teaches us A∞-deformations, L∞-algebras, Kadeishvili constructions and Koszul duality. This work is part of my PhD thesis supervised by Raf Bocklandt. [Slides]

08 Dec 2022   Zhengfang Wang   (University of Stuttgart, Germany)   A-infinity deformations of extended Khovanov arc algebras and Stroppel's Conjecture

Abstract.  Extended Khovanov arc algebras Kₘⁿ are introduced by Stroppel when studying parabolic category O. They naturally appear in different subjects (including representation theory and symplectic geometry) and have many nice algebraic properties, for example they are Koszul and quasi-hereditary. In this talk, we will first give the diagrammatic description of Kₘⁿ due to Brundan–Stroppel. Then, by writing Kₘⁿ as the path algebra of a quiver with relations, we show that the Koszul dual of Kₘⁿ admits a natural reduction system satisfying diamond condition, by relating the number of the associated "irreducible paths" to the Kazhdan–Lusztig polynomials. We also explain how to apply this reduction system to study Stroppel's Conjecture. As a result, we show that Kₘⁿ is not intrinsically formal for m, n > 1. This is joint work with S. Barmeier. [Slides] [Video]

01 Dec 2022   Anna Barbieri   (University of Padova, Italy)   Categories associated with weighted marked surfaces and their stability manifold

Abstract. In a paper in 2015, Bridgeland and Smith identified some moduli spaces of meromorphic quadratic differentials with simple zeroes on a Riemann surface with some spaces of stability conditions on certain categories. This identification passes through associating a quiver with potential and a Ginzburg category to a triangulation of a marked bordered surface defined by a quadratic differential. I will review this correspondence and discuss how the picture changes when quadratic differentials with zeroes of arbitrary order are considered. This involves the study of Verdier quotients of Ginzburg categories and their t-structures. The talk is based on a joint work with M. Moeller, Y. Qiu, and J. So. [Slides]

24 Nov 2022   Chris Brav   (Higher School of Economics Moscow, Russia)   Poisson brackets and the string Lie algebra of a Calabi–Yau category

Abstract. Goldman defined a symplectic structure on the moduli space of local systems on a closed oriented surface, constructed a collection of natural Hamiltonians on the moduli space by taking trace of monodromy around loops on the surface, and computed Poisson brackets among these Hamiltonians in terms of what is now called the Goldman bracket on free homotopy classes of loops. Chas and Sullivan generalized the Goldman bracket to a string bracket on the degree-shifted equivariant homology of the free loop space of a closed oriented manifold of any dimension, but the compatibility with the corresponding shifted-symplectic geometry on the moduli space of local systems remained mostly conjectural. We generalize these results of Goldman and of Chas–Sullivan to higher dimensional "non-commutative" closed oriented manifolds in the form of smooth Calabi–Yau categories. Our main results are the description of a chain-level "string Lie bracket" on cyclic chains of a smooth Calabi–Yau category and the intertwining of this string Lie bracket on cyclic chains with the shifted Poisson bracket on functions on the moduli space of objects in the category. [Slides] [Video]

17 Nov 2022   Anya Nordskova   (Hasselt, Belgium)   Derived Picard groups of representation-finite symmetric algebras

Abstract. I will talk about my work on computing the derived Picard groups of a particular class of symmetric representation-finite algebras of type D. We obtain an explicit description of these groups via generators and relations. In particular, we prove that these groups are generated by spherical twists along a collection of 0-spherical objects, the shift and, roughly speaking, outer automorphisms of the algebras. One of the key ingredients in the proof is the faithfulness of braid group actions by spherical twists along ADE configurations of 0-spherical objects. Another part of the strategy is based on the fact that symmetric representation-finite algebras are tilting-connected. To apply this result we in particular develop a combinatorial-geometric model for silting mutations in type D, generalising the classical concepts of Brauer trees and Kauer moves. I will also discuss possible directions in which the result might be extended to a more general categorical context. [Slides] [Video]

10 Nov 2022   Victoria Hoskins   (Radboud University Nijmegen, Netherlands)   Motivic mirror symmetry for Higgs bundles

Abstract. Moduli spaces of Higgs bundles for Langlands dual groups are conjecturally related by a form of mirror symmetry. For SLₙ and PGLₙ, Hausel and Thaddeus conjectured a topological mirror symmetry given by an equality of (twisted orbifold) Hodge numbers, which was proven by Groechenig–Wyss–Ziegler and also Maulik–Shen. We lift this to an isomorphism of Voevodsky motives, and thus in particular an equality of (twisted orbifold) rational Chow groups. Our method is based on Maulik and Shen's approach to the Hausel–Thaddeus conjecture, as well as showing certain motives are abelian, in order to use conservativity of the Betti realisation on abelian motives. This is joint work with Simon Pepin Lehalleur. [Slides] [Video]

03 Nov 2022   Hanwool Bae   (Seoul, Korea)   Cluster categories from Fukaya categories

Abstract. The wrapped Fukaya category of the plumbing X of the cotangent bundles of spheres along a tree T is shown to be quasi-equivalent to the dg category of dg modules over the Ginzburg dg algebra associated to a quiver Q whose underlying graph is T.  In this talk, I will first discuss that the quotient of the wrapped Fukaya category W of X by its compact Fukaya category F is equivalent to the Amiot–Guo–Keller cluster category associated to Q, and a certain generator L of W becomes a cluster-tilting object of W/F. Then using the minimal model of the Ginzburg dg algebra computed by Hermes, in the case the tree T is given by a Dynkin diagram of type A, D or E, I will explain how to show that the endomorphism algebra of L in the quotient category W/F is isomorphic to the path algebra of a certain quiver with relations. This talk is based on a joint work with Wonbo Jeong and Jongmyeong Kim. [Slides] [Video]

27 Oct 2022   Xiaofa Chen   (Paris, France)   On exact dg categories

Abstract. In this talk, we propose a framework of dg enhancements, which we call exact dg categories, for a certain class of extriangulated categories, which we call algebraic. The motivation comes from typical examples like Yilin Wu’s Higgs categories and Haibo Jin’s categories of dg Cohen–Macaulay modules. We will present several results concerning the dg nerve, the dg derived category, tensor products and functor dg categories with exact target. We will conclude by computing the lattice of all exact structures on dg categories satisfying certain strong finiteness conditions. [Slides]

20 Oct 2022   Ilaria Di Dedda   (King's College, London)   A symplectic interpretation of Auslander correspondence

Abstract. Auslander correspondence establishes a bijection between the class of algebras of finite representation type and the class of Auslander algebras, with both families considered up to Morita equivalence. This allows one to study the representation theory of the former via the homological properties of the latter. The aim of this talk is to give a symplectic interpretation to this correspondence when the algebra of finite representation type is the path algebra of the quiver of Dynkin type Aₙ. This result relies on a realisation of Auslander algebras of type A as Fukaya–Seidel categories of a family of Lefschetz fibrations. [Slides] [Video]

13 Oct 2022   Nicolò Sibilla   (SISSA, Italy)   Fukaya category of surfaces and pants decomposition

Abstract. In this talk I will explain some results joint with James Pascaleff on the Fukaya category of Riemann surfaces. I will explain a local-to-global principle which allows us to reduce the calculation of the Fukaya category of surfaces of genus g > 1 to the case of the pair-of-pants, and which holds both in the punctured and in the compact case. The starting point are the sheaf-theoretic methods which are available in the exact setting, and which I will review at the beginning of the talk. This result has several interesting consequences for HMS and geometrization of objects in the Fukaya category. I will conclude the talk hinting at more recent developments, related to work of Max Jeffs on the Fukaya category of singular surfaces, and conjectures of Lekili–Ueda. The talk is based on arXiv 1604.06448, 2103.03366 and 2208.03896. [Slides] [Video]

Summer break

07 Jul 2022   Mikhail Gorsky   (Versailles)   Cluster structures on braid varieties

Abstract. A braid variety is a certain affine algebraic variety associated with a simply-connected simple algebraic group G and a positive braid of the corresponding type. These varieties generalise open Richardson varieties and appear in the context of symplectic topology and in the study of  link invariants such as HOMFLY-PT polynomials and Khovanov–Rozansky homology. In this talk, I will give a proof of the existence of cluster A-structures and cluster Poisson structures on any braid variety. I will sketch an explicit construction of cluster seeds involving the  diagrammatic calculus of weaves and a tropicalization of Lustig's coordinates. These cluster algebras are local acyclic and equal their upper cluster algebras. The main result also proves the conjecture of B. Leclerc on the existence of cluster algebra structures on the coordinate rings of open Richardson varieties in simply laced types. The talk is based on joint work with Roger Casals, Eugene Gorsky, Ian Le, Linhui Shen, and José Simental. [Slides] [Video]

30 Jun 2022   Andy Tonks   (Malaga)   On some generalisations of Baues–Wirsching cohomology 

Abstract. I will discuss the (co)homology of categories of Baues–Wirsching, and how it is related to several other (co)homologies, new and old: of decomposition spaces, of 2-categories (and higher) and of simplicial sets. Joint work and work in progress with I. Gálvez-Carrillo, F. Neumann and S. Paoli. [Slides] [Video]

23 Jun 2022   Jongmyeong Kim   (IBS Center for Geometry and Physics, South Korea)   Categorical entropy, (co-)t-structures and ST-triples

Abstract. In this talk, we study a dynamical property of an exact endofunctor F of a triangulated category D using the notion of categorical entropy introduced by Dimitrov–Haiden–Katzarkov–Kontsevich. In particular, we consider the following question: Given full triangulated subcategories A and B of D such that F restricts to A and B, how are the categorical entropies of the restricted functors related? To answer this question, we will introduce new entropy-type invariants using bounded (co-)t-structures and see their basic properties. We then apply these results to answer our question for the case where our A and B form an ST-triple in which case A has a bounded t-structure and B has a bounded co-t-structure which are, in some sense, dual to each other. [Slides] [Video]

26 May 2022   Shengyuan Huang   (Birmingham, UK)   The HKR isomorphism and Hochschild cohomology 

Abstract. For a smooth scheme X, the HKR isomorphism identifies the Hochschild cohomology of X with the cohomology of polyvector fields as vector spaces. This isomorphism is known as the classical HKR isomorphism for the diagonal embedding: X ⊂ X x X. We discuss the generalization of the HKR isomorphism to arbitrary closed embeddings and the corresponding functoriality property. Then we recall the generalization of the HKR isomorphism to orbifolds. We apply all the results above to study the product structure of orbifold Hochschild cohomology. [Slides] [Video]

19 May 2022   Imma Gálvez Carrillo   (Barcelona, Spain)   Combinatorial bialgebras and decomposition spaces

Abstract. Decomposition spaces (introduced independently by Dyckerhoff and Kapranov under the name 2-Segal spaces) are simplicial spaces with a certain exactness property that models coassociativity in the same way that usual Segal spaces capture associative composition. I will talk about joint work with Joachim Kock and Andrew Tonks on the application of decomposition spaces in "categorified" or "objective" algebra.  As motivating examples, I will discuss decomposition spaces for many classical combinatorial Hopf algebras, such as those of symmetric functions.

12 May 2022   Severin Barmeier   (Cologne, Germany)   Deformations of categories of coherent sheaves via quivers with relations

Abstract. In this talk I will explain how to use the deformation theory of path algebras of quivers with relations to give a concrete and complete description of the deformation theory of the Abelian category coh(X) of coherent sheaves on any separated Noetherian scheme X. Deformations of coh(X) are a generous source of interesting examples of noncommutative schemes arising as an amalgamation of classical deformations of X, quantizations of Poisson structures on X and twists of the structure sheaf. Lowen and Van den Bergh showed that the deformation theory of coh(X) is equivalent to the deformation theory of a certain associative algebra, namely the "diagram algebra" associated to the restriction of the structure sheaf to any affine open cover. When X is Noetherian, this associative algebra and its deformations can be described concretely via a finite quiver with relations. This talk is based on arXiv:2107.07490 and arXiv:2002.10001 both joint with Zhengfang Wang. [Slides] [Video]

5 May 2022   Chrysostomos Psaroudakis   (Thessaloniki, Greece)   Homological invariants of the arrow removal operation

Abstract. Several homological conjectures in representation theory are related with specific problems concerning the homological behaviour and the structure theory of finite dimensional algebras. In joint work with Edward Green and Øyvind Solberg we developed new reduction techniques (vertex removal operation and arrow removal operation) on quotients of path algebras for testing the finiteness of the finitistic dimension. In this talk, we will review these operations and we will focus on the arrow removal operation. In particular, we will show that Gorensteinness, singularity categories and the finite generation condition Fg for the Hochschild cohomology are invariants under the arrow removal operation for a finite dimensional algebra. This is joint work with Karin Erdmann and Øyvind Solberg. [Slides] [Video]

07 Apr 2022   Pavel Safronov   (Edinburgh, UK)   Euler structures and noncommutative volume forms

Abstract. Calabi–Yau structures on dg categories provide a noncommutative analog of symplectic structures. In this talk I will introduce a noncommutative analog of volume forms called noncommutative Euler structures. I will give some examples of these and relate noncommutative Euler structures to string topology-type operations. An application of these ideas is the proof that the Goresky–Hingston string coproduct on the homology of free loop space is not homotopy invariant. If I have time, I will also discuss how Euler structures give rise to volume forms on derived mapping stacks. This is a report on work in progress joint with Florian Naef. [Slides] [Video]

31 Mar 2022   Asilata Bapat   (Canberra, Australia)   The sphere of spherical objects

Abstract. Consider the 2-Calabi–Yau triangulated category arising from the zigzag algebra of the Aₙ quiver. The braid group acts on this category by twists in spherical objects. Given a Bridgeland stability condition, we describe how to realise the spherical objects as a dense subset of a piecewise-linear manifold. This manifold is canonically associated to the category, and the braid group acts on it piecewise-linearly. We also describe how the manifold transforms under wall-crossings of the stability condition. The talk is based on joint work with Anand Deopurkar and Anthony M. Licata. [Slides] [Video]

24 Mar 2022   Fernando Muro   (Sevilla, Spain)   Enhanced n-angulated categories

Abstract. These categories were introduced by Geiss, Keller, and Oppermann in order to encode the behaviour of n-cluster tilting subcategories of triangulated categories. In this talk I will define differential graded enhancements for these categories, analogous to those of Bondal and Kapranov for triangulated categories. I will present an existence and uniqueness theorem for n-angulated enhancements which holds, for instance, for the additivization of an n-cluster tilting object. As a corollary, I will deduce that a triangulated category with a (higher) cluster tilting object has a unique triangulated enhancement. This is joint work with Gustavo Jasso (Lund). [Slides] [Video]

17 Mar 2022   Daniel Labardini-Fragoso   (UNAM, Mexico City / Cologne, Germany)   Gentle algebras arising from surfaces with orbifold points

Abstract. Some years ago, Diego Velasco and I associated a gentle algebra to each triangulation of a polygon with one orbifold point of order three, and showed that the τ-tilting combinatorics of this gentle algebra coincides with the combinatorics of flips of triangulations. Moreover, we showed that whenever one mutates support τ-tilting pairs, the corresponding Caldero–Chapoton functions obey a generalized cluster exchange formula, which means that the Caldero–Chapoton algebra is isomorphic to a generalized cluster algebra of Chekhov–Shapiro. Generalizing the aforementioned work, in an ongoing collaboration Lang Mou and I associate a gentle algebra to each triangulation of any unpunctured surface with orbifold points of order three. We are able to define generalized reflection functors and DWZ-like mutations of representations. This is somewhat surprising, since the quivers we consider are allowed to have loops, and the matrix-mutation classes of their skew-symmetrizable matrices may fail to have acyclic representatives. [Video]

10 Mar 2022   Nick Sheridan   (Edinburgh, UK)   The Gamma and SYZ conjectures

Abstract. I will give some background on the Gamma Conjecture, which says that mirror symmetry does not respect integral cycles: rather, the integral cycles on a complex manifold correspond to integral cycles on the symplectic mirror, multiplied by a certain transcendental characteristic class called the Gamma class. In the second part of the talk I will explain a new geometric approach to the Gamma Conjecture, which is based on the SYZ viewpoint on mirror symmetry. We find that the appearance of ζ(k) in the asymptotics of period integrals arises from the codimension-k singular locus of the SYZ fibration. This is based on joint work with Abouzaid, Ganatra, and Iritani. [Slides] [Video]

3 Mar 2022   Eleonore Faber   (Leeds, UK)   Matrix factorizations of some discriminants

Abstract. In this talk, we consider discriminants of complex reflection groups. We identify certain matrix factorizations, whose corresponding Cohen-Macaulay modules give a noncommutative resolution of the discriminant. We will in particular consider the family of pseudo-reflection groups G(r,p,n), for which one can explicitly determine these matrix factorizations that are indexed by partitions, using higher Specht polynomials (work in progress with Colin Ingalls, Simon May, and Marco Talarico). [Slides] [Video]

24 Feb 2022   Sira Gratz   (Glasgow, UK)   When aisles meet

Abstract. Discrete cluster categories of type A are an exciting playing field on which to learn about infinite rank cluster combinatorics: On the one hand, they combinatorially behave, in many ways, in a familiar finite type A way. On the other hand, they exhibit new phenomena for which finite type A is "too small". One such phenomenon is the existence of t-structures. In this talk, we describe the classification of t-structures in discrete cluster categories of type A via decorated non-crossing partitions and explain how they form a lattice under inclusion of aisles – an unusual occurrence for t-structures in a triangulated category. This classification of t-structures can inform our understanding of posets of t-structures more generally and help to tackle completions of triangulated categories from a combinatorial perspective. This talk is based on joint work with Alexandra Zvonareva, and with Thorsten Holm and Peter Jørgensen. [Slides] [Video]

17 Feb 2022   Sergey Mozgovoy   (Dublin, Ireland)   Wall-crossing structures arising from surfaces

Abstract: Families of Bridgeland stability conditions induce families of stability data (DT invariants), wall-crossing structures and scattering diagrams on the motivic Hall algebra. These structures can be transferred to the quantum torus if the stability conditions of the family have global dimension at most 2. I will discuss geometric stability conditions on a surface with nef anticanonical bundle. These stability conditions have global dimension 2, hence induce a family of stability data. I will also discuss the relationship of this family to the family of stability data associated to a quiver with potential, with an emphasis on the projective plane. [Slides] [Video]

10 Feb 2022   Matt Pressland   (Glasgow, UK)   Grassmannian twists categorified

Abstract. The Grassmannian of k-dimensional subspaces of an n-dimensional space carries a birational automorphism called the twist (or sometimes the Donaldson–Thomas transformation), defined by Berenstein–Fomin–Zelevinsky and Marsh–Scott. This automorphism respects the cluster algebra structure on the coordinate ring, being a quasi-cluster automorphism in the sense of Fraser. By work of Muller–Speyer, similar results hold for positroid strata in the Grassmannian. The cluster algebras in this picture have been categorified, by Jensen–King–Su in the case of the full Grassmannian, and by myself for more general (connected) positroid varieties. In this talk I will report on joint work with İlke Çanakçı and Alastair King, in which we describe the twist in terms of these categorifications. The key ingredient is provided by perfect matching modules, certain combinatorially defined representations for a quiver "with faces", and I will also explain this construction. [Video]

3 Feb 2022   Alberto Canonaco   (Pavia, Italy)   Dg enhancements of triangulated categories and their uniqueness

Abstract. It is well known that often some intrinsic defects of triangulated categories can be avoided by enhancing them to suitable higher categorical structures, like (pretriangulated) dg categories. However, despite significant progress made in recent years, the relation between the triangulated and the dg level is not yet completely clear, and some foundational questions have received only partial answers so far. After presenting the general picture, I will report on a joint work (partly in progress) with Neeman and Stellari about uniqueness of dg enhancements. [Slides] [Video]

2021

09 Dec 2021   Jeremy Brightbill   (Santa Barbara, USA)   Higher simple-minded systems in negative Calabi–Yau categories.

Abstract. Higher simple-minded systems are collections of objects in a negative Calabi–Yau category whose behavior mimics that of simple modules.  Under certain hypotheses the collection of all simple-minded systems admits a theory of mutations.  In this talk, we shall discuss how to construct many examples of negative Calabi–Yau categories using the so-called "dg-stable category".  For a concrete example, we consider the dg-stable category of a negatively-graded Brauer tree algebra.  Using a combinatorial model, we classify the simple-minded systems of this category and describe its mutation theory. [Slides] [Video]

25 Nov 2021   Marcy Robertson   (Melbourne, Australia)   A topological characterization of the Kashiwara–Vergne groups

Abstract. Solutions to the Kashiwara–Vergne equations in noncommutative geometry are a "higher dimensional" version of Drinfeld associators. In this talk we build on work of Bar-Natan and Dancso and identify solutions of the Kashiwara–Vergne equations with isomorphisms of (completed) wheeled props of "welded tangled foams" — a class of knotted surfaces in R⁴. As a consequence, we identify the symmetry groups of the Kashiwara–Vergne equations with automorphisms of our (completed) wheeled props.  This talk is aimed at a general audience and I will not assume familiarity with the Kashiwara–Vergne equations, Drinfeld associators or wheeled props. Includes joint work with Z. Dancso and I. Halacheva. [Slides] [Video]

18 Nov 2021   Tobias Dyckerhoff   (Hamburg, Germany)   Perverse sheaves and schobers on Riemann surfaces

Abstract. Reporting on joint work with M. Kapranov, V. Schechtman, and Y. Soibelman, I will explain how to describe the derived constructible category of a stratified Riemann surface as representations of the so-called paracyclic category of the surface. This allows for geometric depictions of the various t-structures of interest (including the perverse one) and their interplay with Verdier duality. We will then discuss how this leads to an approach to categorified perverse sheaves (perverse schobers) and provide some examples. [Slides] [Video]

11 Nov 2021   Bruno Vallette   (Université Sorbonne Paris Nord, France)   Deformation theory of Cohomological Field Theories

Abstract. In this talk, I will develop the deformation theory of Cohomological Field Theories (CohFTs), that is algebras over the moduli spaces of stable curves with marked points. This will lead to two new natural extensions of the notion of a CohFT: homotopical (necessary to structure chain-level Gromov–Witten invariants) and quantum (with examples found in the works of Buryak–Rossi on integrable systems). I will introduce a new version of Kontsevich's graph complex, enriched with tautological classes, and I will use it to study a new universal deformation group which acts naturally on the moduli spaces of quantum homotopy CohFTs.  This group is shown to contain both the prounipotent Grothendieck–Teichmüller group and the Givental group. (Joint work with Vladimir Dotsenko, Sergey Shadrin, Arkady Vaintrob arXiv:2006.01649.) [Slides] [Video]

04 Nov 2021  Véronique Bazier-Matte   (UConn, USA)   Triangulations of the Möbius strip and its connections with quasi-cluster algebras

Abstract.  In 2015, Dupont and Palesi defined quasi-cluster algebras, which are cluster algebras arising from surfaces, orientable or not. They proved that the only quasi-cluster algebras with a finite number of clusters are the ones arising from the Möbius strip. In this talk, we will define quasi-cluster algebras, list some of their properties and count the number of clusters in a quasi-cluster algebra arising from a Möbius strip, i.e. the number of triangulations of the Möbius strip. [Slides] [Video]

28 Oct 2021   Amnon Neeman   (ANU, Australia)   Finite approximations as a tool for studying triangulated categories

Abstract. A metric on a category assigns lengths to morphisms, with the triangle inequality holding. This notion goes back to a 1974 article by Lawvere. We'll begin with a quick review of some basic constructions, like forming the Cauchy completion of a category with respect to a metric. And then will begin a string of surprising new results. It turns out that, in a triangulated category with a metric, there is a reasonable notion of Fourier series, and an approximable triangulated category can be thought of as a category where many objects are the limits of their Fourier expansions. And then come two types of theorems: (1) theorems providing examples, meaning showing that some category you might naturally want to look at is approximable, and (2) general structure theorems about approximable triangulated categories. And what makes it all interesting is (3) applications. These turn out to include the proof of a conjecture by Bondal and Van den Bergh, a major generalization of a theorem of Rouquier's, and a short, sweet proof of Serre's GAGA theorem. [Slides]

21 Oct 2021   Xiao-Wu Chen   (Hefei, China)   The dg Leavitt path algebra, singular Yoneda category and singularity category

Abstract. We prove that, for any finite dimensional algebra given by a quiver with relations, its dg singularity category is quasi-equivalent to the perfect dg derived category of the dg Leavitt path algebra of its radical quiver. This result might be viewed as a deformation of the known description of the dg singularity category of a radical-square-zero algebra in terms of a Leavitt path algebra. The main ingredient is  a new dg enhancement of the singularity category, namely the singular Yoneda dg category, which is obtained by a new strict dg localization inverting a natural transformation. This is joint with Zhengfang Wang. 

14 Oct 2021   Shunsuke Kano   (Tohoku, Japan)   Categorical dynamical systems arising from sign-stable mutation loops

Abstract. A pair formed by a triangulated category and an autoequivalence is called a categorical dynamical system. Its complexity is measured by the so-called categorical entropy. In this talk, I will present a computation of the categorical entropies of categorical dynamical systems obtained by lifting a sign-stable mutation loop of a quiver to an autoequivalence of the derived category of the corresponding Ginzburg dg algebra. The notion of sign-stability is introduced as ananalogy of the pseudo-Anosov property of mapping classes of surfaces. If time permits, we will discuss the pseudo-Anosovness of the autoequivalences constructed. [Slides]

07 Oct 2021   Pedro Tamaroff   (Leipzig, Germany)   Minimal models for monomial algebras

Abstract. We will explain how to obtain the minimal model of a monomial associative algebra A, as in [1]. This multiplicative resolution has for generators the Anick chains of A, and as a differential a combinatorial "cutting" operation that splits such chains into "smaller" ones. Along with the formalism of Anick chains, our results make use of the algebraic discrete Morse theory of Jöllenbeck–Welker and Sköldberg, and the general theory of A∞-(co)algebras. We aim to also mention certain open questions and conjectures that emerged from [1] and related work [2] with Dotsenko and Gélinas, and how one could begin elucidating similar results for other algebraic structures, where it is known the behaviour of the minimal model is already pathological.
[1] Minimal models for monomial algebras, Homology, Homotopy and Applications Volume 23 (2021) no. 1, pp. 341–366. (arXiv:1804.01435)
[2] Finite generation for Hochschild cohomology of Gorenstein monomial algebras, preprint arXiv:1909.00487

Summer break

01 Jul 2021   Chelsea Walton   (Rice, USA)   Frobenius algebras galore

Abstract. In this talk, I’ll chat about wonderful algebraic structures that were discovered in the early 1900’s: Frobenius algebras. I will survey the 100+ year history of the development and uses of these structures, ending with very recent research results from joint work with Harshit Yadav. [Slides]

24 Jun 2021   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. If time permits, we will also sketch a categorification of mutation at frozen vertices as it appears in recent work of Fraser–Sherman-Bennett on positroid cluster structures.

17 Jun 2021   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 Simões and David Ploog.

10 Jun 2021   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] [Video]

03 Jun 2021   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]

20 May 2021   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] [Video]

13 May 2021   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]

06 May 2021   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]

29 Apr 2021   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]

22 Apr 2021   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]

15 Apr 2021   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 Dᵇ(coh X) (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 Dᵇ(coh C) for a smooth projective curve C.

18 Mar 2021   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 practice, 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-specialist 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] [Video]

11 Mar 2021   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]

04 Mar 2021   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]

25 Feb 2021   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–Toën–Vaquié–Vezzosi allows for a reinterpretation/analog of the classical AKSZ construction for certain σ-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]

18 Feb 2021   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]

11 Feb 2021   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]

04 Feb 2021   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]

28 Jan 2021   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]

2020

10 Dec 2020   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] [Video]

03 Dec 2020   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] [Video]

26 Nov 2020   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] [Video]

19 Nov 2020   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] [Video]

12 Nov 2020   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][Video]

05 Nov 2020   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] [Video]

29 Oct 2020   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] [Video]

22 Oct 2020   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 τ-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] [Video]

15 Oct 2020   Fabrizio Catanese   (Bayreuth, Germany)   Topologically trivial automorphisms of compact Kähler surfaces and manifolds

Abstract. The abstract can be downloaded here. [Slides] [Video]

08 Oct 2020   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 (∞,1)-categories as derived categories of comodules over simplicial chain coalgebras. (joint work with J. Holstein) [Slides] [Video]

01 Oct 2020   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] [Video]

24 Sep 2020   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. [Video]

17 Sep 2020   No Seminar   Categorifications in Representation Theory Conference at Leicester

10 Sep 2020   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. [Slides] [Video]

03 Sep 2020   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 arXiv: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] [Video]

Summer break

23 Jul 2020   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] [Video]

16 Jul 2020   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] [Video]

09 Jul 2020   Alexander Polishchuk   (University of Oregon, USA)   Geometry of the Associative Yang–Baxter equation

Abstract. I will describe the connection, discovered jointly with Yankı 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] [Video]

02 Jul 2020   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. Balázs 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 E78 in general. In this talk, I will emphasize on representation theoretic aspects to minimize overlaps with Balázs's talk. [Slides] [Video]

25 Jun 2020   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 respect 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] [Video]

18 Jun 2020   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] [Video]

11 Jun 2020   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] [Video]

04 Jun 2020   Balázs Szendrői   (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] [Video]

28 May 2020   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]

21 May 2020   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]

14 May 2020   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 Brüstle 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]