supported by Universität zu Köln (Germany) & Università di Pavia (Italy)
Organisers: Severin Barmeier, Frank Neumann and Sibylle Schroll
YouTube Channel: Max Kaipel
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, moduli spaces and algebraic stacks, derived algebraic geometry and other topics.
This global webinar series was supported as part of the International Centre for Mathematical Sciences (ICMS) and the Isaac Newton Institute for Mathematical Sciences (INI) Online Mathematical Sciences Seminars. We thank the ICMS for their continued support over two years from May 2020 to April 2022.
Register here to obtain the Zoom link and password for the upcoming meetings.
Video recordings as well as a list of titles and abstracts of past talks is available here.
The webinar series is also cross-listed on researchseminars.org
05 Nov 2025 Sira Gratz (Aarhus University, Denmark)
26 Nov 2025 Timothy Logvinenko (Cardiff University, UK)
24 Sep 2025 Jon Pridham (Hodge Institute, University of Edinburgh, UK) Derived deformation functors, Koszul duality, and Maurer–Cartan spaces
Abstract. For Koszul dual pairs (L, C) of differential graded operads (the motivating case being Lie and commutative), I will give an overview of the equivalence between dg L-algebras up to quasi-isomorphism and functors on Artinian dg C-algebras satisfying some exactness conditions. The latter tend to arise as derived versions of natural deformation functors. [Slides]
25 Jun 2025 Travis Schedler (Imperial College London, UK) Creating quantum projective spaces by deforming q-symmetric algebras
Abstract. I will explain how to construct new "quantum projective spaces", in the form of Koszul, Calabi–Yau algebras with the Hilbert series of a polynomial ring. To do so we deform the relations of toric ones — q-symmetric algebras — using a diagrammatic calculus. Such deformations are unobstructed under suitable nondegeneracy conditions, which also guarantee that the algebras are Kontsevich's canonical quantizations of corresponding quadratic Poisson structures. This produces the first broad class of quadratic Poisson structures for which his quantization can be computed and shown to converge, as he conjectured in 2001. On the other hand, we also give examples of purely noncommutative deformations, which cannot be obtained by quantizing Poisson structures. This is joint work with Mykola Matviichuk and Brent Pym. [Slides 1, 2]
28 May 2025 Nick Williams (University of Cambridge, UK) Steenrod operations via higher Bruhat orders
Abstract. The cohomology of a topological space has a ring structure via the cup product. The cup product is defined at the level of cochains, where it is not commutative, but it becomes commutative at the cohomology level. At the cochain level, the lack of commutativity is resolved homotopically by an infinite tower of higher products, known as the Steenrod cup-i products. This additional structure provides more refined information which can be used to tell apart non-homotopy-equivalent spaces. In this talk, I will explain recent work with Guillaume Laplante-Anfossi, where we show how conceptual proofs of the key properties of Steenrod's cup-i products can be given using the higher Bruhat orders of Manin and Schechtman. [Slides]
30 Apr 2025 Michael Wemyss (University of Glasgow, UK) The classification of 3-fold flops via Jacobi algebras
Abstract. The talk will give an overview of the analytic classification of smooth, simple, 3-fold flops. There are three main aspects: (1) reducing the problem to the classification of certain noncommutative finite dimensional algebras, (2) a full understanding of those algebras, then lastly (3) building the associated geometry for each algebra in that class. There are various bonus corollaries. The talk will be algebraic, and so will focus mostly on (1) and (2), where new techniques in A∞ algebras and in noncommutative standard bases will be explained. Perhaps the main point is that a new invariant of Jacobi algebras on the two-loop quiver called the "algebraic length" will be introduced, and I will speculate on how general this construction really is. Part (1) is joint with Joe Karmazyn and Emma Lepri, the rest is joint with Gavin Brown. [Slides] [Video]
26 Feb 2025 Julian Holstein (University of Hamburg, Germany) Curvature, Koszul duality and Calabi–Yau structures
Abstract. I will talk about two aspects of Koszul duality. Firstly, Koszul duality for dg categories provides a way of modelling dg categories as certain curved coalgebras. This is a linearization of the correspondence of simplicial categories as simplicial sets (quasi-categories) and curved coalgebras have better formal properties than dg categories. Secondly, Koszul duality exchanges two self-dualities: smooth and proper Calabi–Yau structures. This is a generalization and conceptual explanation of the following phenomenon: For a topological space X with the homotopy type of a finite complex having an oriented Poincaré duality structure (with local coefficients) is equivalent to having a smooth Calabi–Yau structure on the dg algebra of chains on the based loop space of X. A similar phenomenon occurs for Lie algebras. This is joint work with Andrey Lazarev and with Manuel Rivera, respectively. [Slides] [Video]
29 Jan 2025 Anna Barbieri (University of Verona, Italy) A compactification of the stability space for the Aₙ-quiver
Abstract. The space of Bridgeland stability conditions is a non-compact complex manifold attached to a triangulated category D, parametrizing some t-structures of the category. In this talk I will propose a notion of multi-scale stability conditions that gives a smooth compactification of an appropriate quotient of the stability manifold of the Ginzburg category of type Aₙ and a partial compactification for other Ginzburg categories attached to quivers with potential from triangulated marked Riemann surfaces. Based on a joint work with M. Möller and J. So. [Slides]
18 Dec 2024 Igor Burban (Paderborn University, Germany) Exceptional hereditary non-commutative curves and real curve orbifolds
Abstract. An exceptional hereditary non-commutative curve over an algebraically closed field is a weighted projective line of Geigle and Lenzing. However, over arbitrary fields, the theory of exceptional curves is significantly richer. In my talk I am going to explain the definition, examples and key properties of these classes of non-commutative curves, including their invariants and relation to squid algebras and canonical algebras. [Slides] [Video]
27 Nov 2024 Jonas Schnitzer (University of Pavia, Italy) Global homotopies for differential Hochschild cohomologies
Abstract. It is well known that the Hochschild–Kostant–Rosenberg map from multivector fields to differential Hochschild cochains is a quasi-isomorphism. Nevertheless, it is sometimes desirable to not only know the cohomology itself, but also find reasonable formulas for primitives of exact cochains. In my talk I will explain how to construct a quasi-inverse of the Hochschild–Kostant–Rosenberg map and a homotopy to turn all the maps into a deformation retract. The difference to already existing attempts is that our construction is performed globally and enjoys nice properties regarding symmetries. Moreover, the idea of this construction allows for generalizations in various directions. This is a joint work with Marvin Dippell, Chiara Esposito and Stefan Waldmann. [Video]
30 Oct 2024 Alekos Robotis (Cornell University, USA) Partial compactification of the stability manifold by categorical decompositions
Abstract. I will discuss new geometric-categorical structures on triangulated categories that give a partial compactification of spaces of stability conditions. This is based on joint work with Daniel Halpern-Leistner. [Slides]
26 Jun 2024 Merlin Christ (Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG), France) Relative graded Brauer graph algebras and stability conditions
Abstract. We consider a class of dg-algebras which generalize Brauer graph algebras, called relative graded Brauer graph algebras. Their Koszul dual dg-algebras are given by finite group quotients of relative Ginzburg algebras associated with n-angulated surfaces. By constructing a partial geometric model of the derived category, we fully describe the finite heart tilting theory. This leads to a description of the space of Bridgeland stability conditions in terms of quadratic differentials. Based on joint work with Y. Qiu and F. Haiden. [Video]
29 May 2024 David Favero (University of Minnesota, USA) Homotopy path algebras and resolutions
Abstract. A homotopy path algebra is like a directed version of the group ring on a fundamental group. One can imagine a directed graph (quiver) embedded in a topological space and consider the path algebra up to homotopy. Alternatively, one can think of homotopy classes of directed paths in a stratified topological space. I will introduce homotopy path algebras and describe their connections to mirror symmetry and resolutions of coherent sheaves on toric varieties. [Video]
24 Apr 2024 Pieter Belmans (University of Luxembourg, Luxembourg) Hochschild cohomology of Hilbert schemes of points
Abstract. I will present a formula describing the Hochschild cohomology of symmetric quotient stacks, computing the Hochschild–Kostant–Rosenberg decomposition of this orbifold. Through the Bridgeland–King–Reid–Haiman equivalence this allows the computation of Hochschild cohomology of Hilbert schemes of points on surfaces. These computations explain how this invariant behaves differently from say Betti or Hodge numbers, which have been studied intensively in the past 30 years, and it allows for new deformation-theoretic results. This is joint work with Lie Fu and Andreas Krug. [Slides] [Video]
27 Mar 2024 Emanuele Pavia (SISSA, Trieste, Italy) Sheaves of categories over topological and coaffine stacks
Abstract. Starting from any sufficiently nice topological space X, one can produce two different (derived) stacks defined over a field k of characteristic 0: the Betti stack X_B, which bears information on the underlying homotopy type of X, and the coaffine stack cSpec(C*(X; k)) on the commutative algebra C*(X; k) of k-valued singular cochains on X, which behaves as the affinization of the Betti stack X_B.
In this talk, I shall apply the technical machinery of sheaves of (∞-)categories on derived stacks as developed in Gaitsgory’s work to the case of Betti stacks and their associated coaffine stacks. We shall see how sheaves of categories on X_B are intimately related to categorified local systems over the original space X and to homotopy-coherent actions of topological groups on k-linear ∞-categories. At the very end, I shall briefly describe how sheaves of categories on X_B and on cSpec(C*(X; k)) interact, and how such interaction can be interpreted as an instance of higher Koszul duality.
This talk is based on upcoming joint work with J. Pascaleff and N. Sibilla. [Slides] [Video]
28 Feb 2024 Nadia Romero (Universidad de Guanajuato, Mexico) Hochschild cohomology for functors on linear symmetric monoidal categories
Abstract. Let X be an essentially small symmetric monoidal category enriched in R-Mod, with R a commutative ring with identity. Under these conditions, the category F, of R-linear functors from X to R-Mod, becomes an abelian symmetric monoidal category, also enriched in R-Mod. The fact that F is monoidal and abelian at the same time allows for a nice theory of modules over the monoids in F, in particular it allows for a nice and easy definition of an internal hom functor. In this talk, we will see how this internal hom is the key to define a Hochschild cohomology theory in F. [Slides] [Video]
31 Jan 2024 Lucy Yang (Columbia University, New York, USA) Categorical dynamics on stable module categories
Abstract. Given a mathematical object X and an endomorphism f of X, entropy assigns to this pair a number h(f) measuring the dynamical complexity of f. Initially defined for measure spaces and topological spaces, it has also been generalized by Dimitrov–Haiden–Katzarkov–Kontsevich to measure the complexity of endomorphisms of stable ∞-categories. I will discuss a result showing that the categorical polynomial entropy of a twist functor on stable module categories of certain algebras A of cohomology operations over a field k reflects the complexity of A: it is at least one less than the Krull dimension of H*(A; k), generalizing results of Fan–Fu–Ouchi. We will then discuss work in progress to bring this dynamical perspective to homotopy theory.
20 Dec 2023 Jaiung Jun (State University of New York, USA) Quiver representations over F₁
Abstract. A quiver is a directed graph, and a representation of a quiver assigns a vector space to each vertex and a linear map to each arrow. Quiver representations over F₁, "the field with one element", can be considered as a combinatorial model of quiver representations over a field, where vector spaces and linear maps are replaced by F₁-vector spaces and F₁-linear maps. I will introduce several aspects of quiver representations over F₁, and its potential applications. This is joint work with Jaehoon Kim and Alex Sistko. [Slides] [Video]
29 Nov 2023 Laura Pertusi (University of Milano, Italy) Non-commutative abelian surfaces and generalized Kummer varieties
Abstract. A hyperkähler manifold is a compact complex simply connected Kähler manifold whose space of holomorphic two-forms is generated by a symplectic form, unique up to scalar multiplication. Together with complex tori and irreducible Calabi–Yau manifolds, they are building blocks for compact Kähler manifolds with trivial first Chern class. In dimension two hyperkähler manifolds are K3 surfaces, while finding examples in higher dimensions is a challenging problem. In this talk we will construct new families of hyperkähler manifolds of generalized Kummer type via moduli spaces of stable objects in a non-commutative deformation of the bounded derived category of an abelian surface. This work in progress is joint with Arend Bayer, Alex Perry and Xiaolei Zhao. [Slides]
25 Oct 2023 Yu Qiu (Tsinghua University, China) On cluster braid groups
Abstract. We introduce cluster braid groups, with motivations coming from the study of stability conditions and quadratic differentials. In the Coxeter–Dynkin case, they are naturally isomorphic to the corresponding braid groups (1407.5986 and 2310.02871). In the surface case, they are naturally isomorphic to braid twist groups (1407.0806, 1703.10053 and 1805.00030). [Slides] [Video]
27 Sep 2023 Naoki Koseki (University of Liverpool, UK) Symmetric products of dg categories and semi-orthogonal decompositions
Abstract. The notion of symmetric products of a dg category was introduced by Ganter and Kapranov. I will explain how a semi-orthogonal decomposition (SOD) of an original dg category induces an SOD on the symmetric products. This is a generalization of the direct sum decomposition of the symmetric product of a direct sum of two vector spaces. The main application is the construction of various interesting SODs on the derived categories of the Hilbert schemes of points on surfaces. [Slides] [Video]