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 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

Upcoming talks

29 May 2024   David Favero   (University of Minnesota)

26 Jun 2024   tba

summer break

28 Aug 2024   tba

Recent talks

24 Apr 2024   Pieter Belmans   (University of 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]

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]

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]

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   (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]