African Institute for Mathematical Sciences (AIMS)

Muizenberg

Cape Town, South Africa

We acknowledge the generous financial support of the Imperial College-AIMS connect partnership seed fund and Elsevier.

David Holgate (AIMS and University of Western Cape)

Karin-Therese Howell (Academic Director, AIMS)

Frank Neumann (University of Pavia and LMS)

Loyiso Nongxa (AIMS and University of the Witwatersrand)

Ambrus Pál (Imperial College London)

Raina Ralaivaosaona (AIMS and Stellenbosch)

Talks and coffee breaks will be in the library at AIMS, Muinzenberg. We leave ample time for discussions in the afternoons.

Mon10 | 09:00-10:00 | Paolo Cascini | Positivity in algebraic geometry: morphisms, divisors, and endomorphisms, I |

10:00-10:30 | Coffee | ||

10:30-11:30 | Charlotte Kestner | An introduction to model theory, I | |

11:30-12:00 | Coffee | ||

12:00-13:00 | Ambrus Pál | Galois invariants, I | |

13:00-13:30 | Lunch |

Tue11 | 10:30-11:30 | Frank Neumann | Arithmetic and geometry of moduli stacks of principal bundles, I |

11:30-12:00 | Coffee | ||

12:00-13:00 | Paolo Cascini | Positivity in algebraic geometry: morphisms, divisors, and endomorphisms, II | |

13:00-13:30 | Lunch |

Wed12 | 10:30-11:30 | Charlotte Kestner | An introduction to model theory, II |

11:30-12:00 | Coffee | ||

12:00-13:00 | Ambrus Pál | Galois invariants, II | |

13:00-13:30 | Lunch |

Thu13 | 09:00-10:00 | Paolo Cascini | Positivity in algebraic geometry: morphisms, divisors, and endomorphisms, III |

10:00-10:30 | Coffee | ||

10:30-11:30 | Sophie Marques | A characterisation of ramification groups via Taylor morphism | |

11:30-12:00 | Coffee | ||

12:00-13:00 | Frank Neumann | Arithmetic and geometry of moduli stacks of principal bundles, II | |

13:00-13:30 | Lunch |

Fri14 | 09:00-10:00 | Charlotte Kestner | An introduction to model theory, III |

10:00-10:30 | Coffee | ||

10:30-11:30 | Ambrus Pál | Galois invariants, III | |

11:30-12:00 | Coffee | ||

12:00-13:00 | Karin-Therese Howell | The weird and wonderful world of near-vector spaces | |

13:00-13:30 | Lunch |

Speaker: Paolo Cascini (Imperial College London)

Title: Positivity in algebraic geometry: morphisms, divisors, and endomorphisms

Abstract: The lectures explore the concept of positivity in algebraic geometry, focusing on the question of when a projective variety admits a morphism. We will introduce divisors and line bundles, and discuss the notions of ampleness and semi-ampleness, which are crucial in characterising projective varieties that admit morphisms. Through examples and some theory, we will study how to determine the positivity of line bundles and their relation to the existence of morphisms. We will also investigate varieties that admit endomorphisms and their connection to positivity.

Speaker: Karin-Therese Howell (AIMS)

Title: The weird and wonderful world of near-vector spaces

Abstract: In this talk I will introduce the concept of a near-vector space, with a focus on those studied by Johannes Andre. I will give an overview of the recent developments in the area and highlight some future ideas for exploration.

Speaker: Charlotte Kestner (Imperial College London)

Title: An introduction to model theory

Abstract: I will give an introduction to model theory starting from types and the compactness theorem, covering quantifier elimination and finishing with an introduction to stability theory.

Speaker: Sophie Marques (Stellenbosch University)

Title: A characterisation of ramification groups via Taylor morphism

Abstract: This talk introduces an analogue of Taylor maps in the category of commutative unitary rings at any prime ideal of an R-algebra. We explore the kernel of these Taylor morphisms for algebras of finite type over a field, particularly when the residue field at the prime ideal considered is separably generated. This computation allows us to characterise the ramification groups of a given action as inertia groups of an induced action on a certain composite of jet algebras. Notably, this approach is functorial, enabling extension of the same construction to algebraic geometry, thereby defining the ramification groups associated with certain group schemes. (This is joint work with Luigi Pagano (Rome).)

Speaker: Frank Neumann (University of Pavia)

Title: Arithmetic and geometry of moduli stacks of principal bundles

Abstract: We will study the moduli stack of principal bundles over a smooth projective algebraic variety and its cohomology algebra. In the first part we will concentrate on the study of the various arithmetic and geometric Frobenius morphisms on the moduli stack of principal bundles over a smooth projective algebraic curve and determine explicitly their actions on the l-adic cohomology in terms of Chern classes of the universal principal bundle. In the second part we will work over the field of complex numbers and determine the rational cohomology and the Hodge-Tate structure of the moduli stack of principal bundles over some class of connected smooth complex projective varieties using the homotopy theory of the underlying topological stack.

Speaker: Ambrus Pál (Imperial College London)

Title: Galois invariants

Abstract: I will start with a gentle introduction to Galois invariants in the sense of Serre, Rost-Merkurjev, Garibaldi, etc. We will concentrate on the case of finite étale algebras and quadratic forms. I will give a recent application to the existence of power structures (joint with J. Pajwani). Then we turn our attention to Galois invariants of higher dimensional varieties which is currently the focus of much research.