# Seminars this semester

Oct 17 | Tue | Alexander Vishik (Nottingham) | |

14:00 | J11 | Subtle Stiefel-Whitney classes | |

Oct 18 | Wed |
Thomas Prince (Imperial) | |

15:00 |
J11 | From period integrals to toric degenerations of Fano manifolds | |

Abstract:Given a Fano manifold we will consider two ways of attaching a (usually infinite) collection of polytopes, and a certain combinatorial transformation relating them, to it. The first is via Mirror Symmetry, following a proposal of Coates-Corti-Kasprzyk-Galkin-Golyshev. The second is via symplectic topology, and comes from considering degenerating Lagrangian torus fibrations. We then relate these two collections using the Gross--Siebert program. I will also comment on the situation in higher dimensions, noting particularly that by 'inverting' the second method (degenerating Lagrangian fibrations) we can produce topological constructions of Fano threefolds. |
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Oct 24 | Tue | Pierrick Bousseau (Imperial) | |

14:00 | J11 | Quantum mirrors of log Calabi-Yau surfaces | |

Abstract:I will start describing the Gross-Hacking-Keel realization of mirror symmetry for log Calabi-Yau surfaces: the mirror variety is constructed by gluing elementary pieces together according to some gluing functions determined by counting rational curves in the original variety. I will then explain how to construct non-commutative deformations of these mirrors by including contributions of counts of higher genus curves in the original variety. |
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Nov 7 | Tue | Soheyla Feyzbakhsh (Edinburgh) | |

14:00 | J11 | Reconstructing a K3 surface from a curve via wall-crossing | |

Abstract:In 1997, Mukai introduced a geometric program to reconstruct a K3 surface from a curve on that surface. The idea is to first consider a Brill-Noether locus of vector bundles on the curve. Then the K3 surface containing the curve can be obtained uniquely as a Fourier-Mukai partner of the Brill-Noether locus. Mukai carried out this program for curves of genus 11. I will explain how wall-crossing with respect to Bridgeland stability conditions implies that the Mukai's strategy works for curves of higher genera. |
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Nov 21 | Tue | Davide Masoero (Lisbon) | |

14:00 | J11 | The isomonodromic deformation method for Painleve I and meromorphic functions with 5 transcendental singularities | |

Abstract:I will introduce the isomonodromic deformation method for Painleve I and the corresponding Riemann-Hilbert problem in term of Stokes multipliers. I will then use a theory due to R. Nevanlinna, [Ueber Riemannsche Flaechen mit endlich vielen Windungspunkten, Acta Math 1932] to give an alternative construction of the monodromy manifold, and a proof of the surjectivity of the monodromy map. Finally, I will comment on some applications of the same method to other Painleve equations: in particular, I will show how to compute the numer of real roots of the rational solutions of the fourth Painleve equations. |
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Nov 28 | Tue | Emilie Dufresne (Nottingham) | |

14:00 | J11 | Separating invariants and local cohomology | |

Abstract:The study of separating invariants is a new trend in Invariant Theory and a return to its roots: invariants as a classification tool. For a finite group acting linearly on a vector space, a separating set is simply a set of invariants whose elements separate the orbits o the action. Such a set need not generate the ring of invariants. In this talk, we give lower bounds on the size of separating sets based on the geometry of the action. These results are obtained via the study of the local cohomology with support at an arrangement of linear subspaces naturally arising from the action. (Joint with Jack Jeffries) |
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Dec 5 | Tue | Gregory Stevenson (Glasgow) | |

14:00 | J11 | A^1-homotopy invariants of singularity categories | |

Dec 12 | Tue | Marta Mazzocco (Loughborough) | |

14:00 | J11 | Colliding holes in Riemann surfaces | |

Abstract:In 1997 Hitchin proved that the Riemann Hilbert correspondence between Fuchsian systems and conjugacy classes of representations of the fundamental group of the punctured sphere is a Poisson map. Since then, some generalisations of this result to the case of irregular singularities have been proposed by Boalch and by Gualtieri, Li and Pym. In this talk we interpret irregular singularities as the result of collisions of boundaries in a Riemann surface and show that the Stokes phenomenon corresponds to the presence of "bordered cusps". We introduce the concept of decorated character variety of a Riemann surface with bordered cusps and construct a generalised cluster algebra structure and cluster Poisson structure on it. We define the quantum cluster algebras of geometric type and show that they provide an explicit canonical quantisation of this Poisson structure. |
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Jan 30 | Tue | Yu Qiu (Chinese University of Hong Kong) | |

14:00 | J11 | Q-stability conditions on Calabi-Yau-X categories of quivers with superpotential | |

Abstract:We introduce X-stability conditions on Calabi-Yau-X categories and spaces of their specializations, the q-stability conditions. The motivating example comes from the Calabi-Yau-X category D(S) associated to a graded marked surface S, constructed from quivers with superpotential. We show that the cluster category of D(S) is Haiden-Katzarkov-Kontsevich's topological Fukaya category C(S) and Bridgeland-Smith type Calabi-Yau-N categories are the orbit quotients of D(S). Moreover, we show that stability conditions on C(S) induce q-stability conditions on D(S). Finally, we are constructing moduli space to realize the fiber of the spaces of q-stabilty conditions for given complex number s. |
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