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.
Organized in partnership with the Clay Mathematics Institute
8 January 2024 to 12 January 2024
22 January 2024 to 26 January 2024
18 March 2024 to 22 March 2024
13 May 2024 to 17 May 2024
17 June 2024 to 21 June 2024
|Tuesday 16th January 2024|
|10:00 to 16:00||Room 2|
|Wednesday 17th January 2024|
|10:00 to 16:00||Room 2|
|Thursday 18th January 2024|
|10:00 to 16:00||Room 2|
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The Isaac Newton Institute is a national and international visitor research institute. It runs research programmes on selected themes in mathematics and the mathematical sciences with applications over a wide range of science and technology. It attracts leading mathematical scientists from the UK and overseas to interact in research over an extended period.
INI has a vital national role, building on many strengths that already exist in UK universities, aiming to generate a new vitality through stimulating and nurturing research throughout the country.During each scientific programme new collaborations are made and ideas and expertise are exchanged and catalysed through lectures, seminars and informal interaction, which the INI building has been designed specifically to encourage.
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