New equivariant methods in algebraic and differential geometry
EMG
3 January 2024 to 28 June 2024
Programme theme
This programme is concerned with symmetries of spaces which arise naturally in various algebraic
and geometric contexts. It aims to explore applications of the new machinery of non-reductive
geometric invariant theory (NRGIT) to different areas of geometry and beyond.
Actions of linear algebraic groups are of fundamental importance across algebraic and differential
geometry and also in the applications of geometry to the rest of mathematics. Classical
geometric invariant theory (GIT) was developed by David Mumford in the 1960s to construct
and study quotients of algebraic varieties by linear actions of reductive groups, with the main application
to the construction of moduli spaces in algebraic geometry. Since then, GIT has been
linked to symplectic geometry via the theory of Hamiltonian group actions and moment maps, and
infinite-dimensional analogues of GIT have provided motivation for crucial advances in differential
geometry, in particular in gauge theory and other areas motivated by mathematical physics,
and more recently K-stability. Important links also arise via toric and tropical geometry to convex
geometry and combinatorics; here the flow of ideas has gone in both directions over the decades.
Recently, GIT has been extended to cover a large class of linear actions by algebraic groups
which are not necessarily reductive. This new ‘non-reductive’ version of GIT is providing applications
in different areas of geometry (such as singularity theory) and potential applications in others
(such as K-stability), as well as new connections with symplectic and hyperkähler geometry.
Many other potential applications exist and remain to be more thoroughly explored. The aim
of this programme is to bring together a broad spectrum of researchers in diverse areas of algebraic
and differential geometry to collaborate on possible new applications mainly focused around the
following themes, which will be the topics of the workshops: applied algebraic geometry, global
singularity theory, complex geometry and hyperbolicity, symplectic and hyperkähler implosion,
K-stability, moduli stacks and enumerative geometry.
In particular, the programme will focus on:
(i) the interactions of GIT with applied and computational algebraic geometry (including toric
geometry, algebraic statistics and geometric group theory);
(ii) the existence of holomorphic curves contained in projective algebraic varieties (‘hyperbolicity’),
which is a fundamental problem in complex geometry;
(iii) the study of singularities of maps between complex manifolds, which is central to global
singularity theory and enumerative geometry;
(iv) the geometry and topology of Hilbert schemes of points on surfaces and higher dimensional
manifolds, which play a crucial role in enumerative geometry;
(v) the construction of moduli spaces of polarised projective schemes and related constructions
in K-stability, and the description of their geometry and topology;
(vi) the relationship between concepts of stability and wall-crossing in different settings such as
geometric invariant theory, Bridgeland stability, K-stability and moduli stacks.
Workshops
Equivariant methods in geometry
8 January 2024 to 12 January 2024
Applied and computational algebraic geometry
22 January 2024 to 26 January 2024
Singularity theory and hyperbolicity
18 March 2024 to 22 March 2024
K-stability and moment maps
13 May 2024 to 17 May 2024
Moduli stacks and enumerative geometry
17 June 2024 to 21 June 2024
Organisers
- Gergely Berczi Aarhus Universitet
- Damian Brotbek Université de Lorraine
- Ruadhaí Dervan University of Glasgow
- Victoria Hoskins Radboud Universiteit Nijmegen
- Frances Kirwan University of Oxford
- Vidit Nanda University of Oxford
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