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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.
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INI is a creative collaborative space which is occupied by up to fifty-five mathematical scientists at any one time (and many more when there is a workshop). Some of them may not have met before and others may not realise the relevance of other research to their own work.
INI is especially important as a forum where early-career researchers meet senior colleagues and form networks that last a lifetime.
Here you can learn about all activities past, present and future, watch live seminars and submit your own proposals for research programmes.
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INI and its programme participants produce a range of publications to communicate information about activities and events, publish research outcomes, and document case studies which are written for a non-technical audience. You will find access to them all in this section.
The Isaac Newton Institute aims to maximise the benefit of its scientific programmes to the UK mathematical science community in a variety of ways.
Whether spreading research opportunities through its network of correspondents, offering summer schools to early career researchers, or hosting public-facing lectures through events such as the Cambridge Festival, there is always a great deal of activity to catch up on.
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“A world famous place for research in the mathematical sciences with a reputation for efficient management and a warm welcome for visitors”
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.
For INI’s knowledge exchange arm, please see the Newton Gateway to Mathematics.
The Institute depends upon donations, as well as research grants, to support the world class research undertaken by participants in its programmes.
Fundraising activities are supported by a Development Board comprising leading figures in academia, industry and commerce.
Visit this section to learn more about how you could play a part in supporting INI’s groundbreaking research.
In this section you can find contact information, staff lists, maps and details of how to find INI’s main building in Cambridge.
Our administrative staff can help you with any queries regarding a prospective or planned visit. If you would like to discuss a proposed a research programme or other event, our senior management team will be happy to help.