# Seminars this semester

Oct 16 | Tue | Kohei Kikuta (Osaka) | |

16:00 |
J11 | On categorical entropy | |

Abstract:The categorical entropy of triangulated endofunctors was defined by Dimitrov-Haiden-Katzarkov-Kontsevich motivated by classical dynamical theory. In this talk, I'll explain relations to classical entropy theory, basics on categorical entropy, many examples and future directions. |
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Oct 23 | Tue | Nebojsa Pavic (Sheffield) | |

16:00 |
J11 | Grothendieck groups and singularity categories of quotient singularities | |

Abstract:We study the K-theory of the Buchweitz-Orlov singularity category for quasi-projective algebraic schemes. Particularly, we show for isolated quotient singularities with abelian isotropy groups that the Grothendieck group of the singularity category is finite torsion and that rational Poincare duality is satisfied on the level of Grothendieck groups. We consider also consequences for the resolution of singularities of such quotient singularities and study dual properties in this setting, more concretely we prove a conjecture of Bondal and Orlov in the case of quotient singularities. |
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Nov 20 | Tue | Eoin Murphy (Sheffield) | |

16:00 |
J11 | Simultaneous deformations of Hall algebras | |

Abstract:In this talk, we discuss how Ringel-Hall algebras, an algebra associated to suitably finite Abelian categories, can be viewed in certain cases as simultaneously deforming two simpler algebras. One of these algebras is the universal enveloping algebra of a Lie algebra, while the other is a Poisson algebra. Time permitting we also discuss an analogous deformation picture for a generalization of Ringel-Hall algebras due to Bridgeland. |
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Nov 27 | Tue | Caitlin McAuley (Sheffield) | |

16:00 |
J11 | Stability conditions of the Kronecker quiver | |

Abstract:To a quiver $Q$, we can associate a sequence of Calabi--Yau-$n$ triangulated categories. The spaces of stability conditions of these categories can then be computed. I will give a description of these stability manifolds, and discuss the relationship between them and the Frobenius structure of the quantum cohomology of the projective line. |
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Dec 4 | Tue | Cristina Manolache (Imperial) | |

15:00 |
J11 | The enumerative content of Gromov-Witten theory | |

Abstract:I will discuss old and new ways of answering questions in enumerative geometry. New methods have many advantages and one major drawback. In this talk I will discuss this drawback. I will introduce Gromov--Witten invariants and I will give evidence that they do not give correct curve counts. I will introduce new enumerative invariants from curves with cusps and I will argue that cuspidal invariants have a better enumerative meaning. In the end, I will highlight one application. This is based on work in collaboration with L Battistella, F Carocci and T Coates. |
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Dec 4 | Tue | Andrea Brini (Birmingham) | |

16:00 |
J11 | Structures in Gromov-Witten theory | |

Abstract:I will survey the existence of hidden recursive structures in the Gromov--Witten (GW) theory of a complex projective variety. I will discuss a characterisation of recursions for genus zero invariants in terms of associative deformations of the cup product in cohomology, and some alternative presentations of the deformed cup product motivated by singularity theory (the celebrated "mirror symmetry"). In some happy (and central) instances mirror symmetry is often a tool powerful enough to determine recursively all GW invariants starting from minimal input data. I will consider one application of these ideas to low-dimensional topology, which is partly joint work with Gaetan Borot, relating a class of smooth invariants of 3-manifolds to recursions for GW invariants. |
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Dec 11 | Tue | Dominic Joyce (Oxford) | |

16:00 |
J11 | A Ringel-Hall type construction of vertex algebras | |

Abstract:Vertex algebras are complicated algebraic structures coming from Physics, which also play an important role in Mathematics in areas such as monstrous moonshine and geometric Langlands. I will explain a new geometric construction of vertex algebras, which seems to be unknown. The construction applies in many situations in algebraic geometry, differential geometry, and representation theory, and produces vast numbers of new examples. It is also easy to generalize the construction in several ways to produce different types of vertex algebra, quantum vertex algebras, representations of vertex algebras, … It seems to be related to work by Grojnowski, Nakajima and others, which produces representations of interesting infinite-dimensional Lie algebras on the homology of moduli schemes such as Hilbert schemes. Suppose A is a nice abelian category (such as coherent sheaves coh(X) on a smooth complex projective variety X, or representations mod-CQ of a quiver Q) or T is a nice triangulated category (such as D^bcoh(X) or D^bmod-CQ) over C. Let M be the moduli stack of objects in A or T, as an Artin stack or higher stack. Consider the homology H_*(M) over some ring R. Given a little extra data on M, for which there are natural choices in our examples, I will explain how to define the structure of a graded vertex algebra on H_*(M). By a standard construction, one can then define a graded Lie algebra from the vertex algebra; roughly speaking, this is a Lie algebra structure on the homology H_*(M^{pl}) of a “projective linear” version M^{pl} of the moduli stack M. For example, if we take T = D^bmod-CQ, the vertex algebra H_*(M) is the lattice vertex algebra attached to the dimension vector lattice Z^{Q_0} of Q with the symmetrized intersection form. The degree zero part of the graded Lie algebra contains the associated Kac-Moody algebra. There is also a differential-geometric version, involving putting a vertex algebra structure on homology of moduli stacks of connections on a compact manifold X equipped with an elliptic complex E. |
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Jan 8 | Tue | Josep Alvarez-Montaner (Universitat Politecnica de Catalunya) | |

14:00 | J11 | Local cohomology of binomial edge ideals and their generic initial ideals | |

Abstract:The aim of this talk is to give a detailed study of local cohomology modules of binomial edge ideals. Our main result is a Hochster type decomposition formula for these modules. As a consequence, we obtain a simple criterion for the Cohen-Macaulayness of these ideals and we describe their Castelnuovo-Mumford regularity and their Hilbert series. We also prove a conjecture of Conca, De Negri and Gorla relating the graded components of the local cohomology modules of binomial edge ideals and their generic initial ideals. |
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Jan 16 | Wed |
Atsushi Takahashi (Osaka) | |

16:00 |
J11 | On a full exceptional collection in the category of maximally graded matrix factorizations of an invertible polynomial of chain type | |

Abstract:In ’77 Orlik-Randell asked about the existence of a certain distinguished basis of vanishing cycles in the Milnor fiber associated to an invertible polynomial of chain type. With my student, Daisuke Aramaki we transport their conjecture to the category of matrix factorizations by the (conjectural) homological mirror symmetry equivalence and then prove the resulting statement. |
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