Matemale Summer School 2023
SEIBERG-WITTEN THEORY
June 19-23
This year, we will study Seiberg-Witten theory and some of its applications to 4-manifolds topology. The school is designed for Master students, PhD students and researchers, younger and older. Participants are encouraged to choose a (not yet attributed) talk in the list. We are grateful to Raphaël Zentner for elaborating the program.
The event is supported by the GDR Tresses, the Mathematics Institute of Marseille, the Mathematics Institute of Toulouse, the ERC project NEDAG.
We will study Seiberg-Witten theory and some of its applications to 4-manifold topology. Some of these results have previously been obtained by instanton gauge theory, for instance Donaldson's diagonalisation theorem, however the proof using Seiberg-Witten is significantly simpler in technical terms. The main simplification arises from the fact in the setup for Seiberg-Witten theory connections with abelian structure group appear, whereas in instanton gauge theory one is dealing with a non-abelian structure group. As a consequence, the moduli space of Seiberg-Witten monopoles is compact, whereas the moduli space of instantons is not, in general.
The configuration space of Seiberg-Witten theory consists of sections in a bundle, called the spinor bundle, defined over an oriented Riemannian four-manifold X, and a connection in that bundle compatible with the Levi-Civita connection. A connection in a bundle is an additional structure that allows to differentiate sections of that bundle (there is no natural identification of different fibres of that bundle).
More details and references here: program.
List of talks:
Connections, characteristic classes, Chern-Weil theory (Anthony Saint-Criq)
Intersection forms (Paolo Farina)
Spin^c-structures, Dirac operators (Julien Dardennes)
Hodge theory, elliptic operators, elliptic regularity (Julien Korinman)
Seiberg-Witten equations, a priori bound, compactness of moduli space (Christopher Johnson)
Linearisation of the equations, local structure of the moduli space, expected dimension (Yohan Mandin-Hublé)
Transversality (Léo Bénard)
Elkies' theorem about short characteristic elements in non-diagonal definite lattices (Florian Deloup)
Local structure around reducibles, proof of Donaldson's diagonalisation theorem (Léo Schelstraete and Benoît Van Vaerenbergh)
Definition of the Seiberg-Witten invariants, vanishing result for 4-manifolds with positive scalar curvature metric (Marco Golla)
Non-vanishing results for Kähler and symplectic manifolds (Jean-François Barraud)
Neck stretching and adjunction inequality (Raphaël Zentner)
Trisections, intersection forms and the Torelli group (Eva Horvat)
Organizers : Thomas Fiedler (Institut de Mathématiques de Toulouse)
Delphine Moussard (Institut de Mathématiques de Marseille)
If you are willing to participate, please contact us by email.
We will be accommodated in the Centre de vacances "La Capcinoise", more precisely in their gites ruraux (https://www.lacapcinoise.fr/). This is just for information, you don't need to make a reservation by yourself. We will cover the stay of the participants in the limit of available funding.
List of participants:
Ismael Ahlouche
Soheil Azarpendar
Rhea Palak Bakshi
Jean-François Barraud
Léo Bénard
Julien Dardennes
Florian Deloup
Paolo Farina
Zsombor Feher
Livio Ferretti
Thomas Fiedler
Vincent Florens
Marco Golla
Eva Horvat
Christopher Johnson
Julien Korinman
Yohan Mandin-Hublé
Gregor Masbaum
Lucas Morel
Delphine Moussard
Stepan Orevkov
Anthony Saint-Criq
Léo Schelstraete
Benoît Van Vaerenbergh
Gabriel Montoya Vega
Vladimir Vershinin
Tony Zaayter
Raphaël Zentner
Butian Zhang