## Grant Objectives

*The description below is taken directly from our grant application; some other details are here.*

We aim to make major progress on fundamental problems in representation theory, algebraic geometry and noncommutative algebra, and to apply the new methods we develop to answer important questions at their interfaces. Our objectives combine the world-leading algebra expertise we have in the team through Gordon, Smoktunowicz and Wemyss with the geometric know-how of Bayer and Bridgeland. Never before for a programme grant has a team with this range of skills come together. While our skills have proven again and again to be powerful on their own in addressing some of the most exciting problems in algebra and geometry, it is through their combination that we will create common powerful methodology across a swathe of algebra and geometry. We will apply our common methodology to solve major problems in each of our areas, and bring it to a form that will be accessible to many researchers.

Our specific objectives are as follows:

**Strand 1. Geometric Representation Theory. **

Establish a theory of microlocal mixed Hodge modules on conic symplectic resolutions, developing the underlying geometry sufficiently to allow the gluing of simple categories of DQ-modules. Use these modules and their construction to study gradings on category O's. Use large volume limit of symplectic resolutions to study combinatorics of representation categories. Study the representation theory attached to the quantization of shifted symplectic structures, establishing an algebro-geometric theory of D_q-modules. Apply this theory to higher rank Hilbert schemes and their - to be found - representation theory.

**Strand 2. Geometry. **

Construct stability conditions on smooth projective threefolds. Determine wall-crossing behaviour in stability for higher dimensions, using in particular constraints arising from shifted symplectic structures. Prove the simple-connectedness of the space of stability conditions on CY threefolds, deducing a description of the group of derived autoequivalences. Through mirror symmetry, apply these results to symplectic topology. Make significant progress on the K-equivalence/D-equivalence conjecture, particularly in the hyperkaehler and wall-crossing cases, and develop the use of Hodge modules in this context.

**Strand 3. Homological and Noncommutative Algebra. **

Control the structure and deformations of contraction algebras in low GK dimension, and relate these to curve invariants and properties of the geometric contraction. Prove the noncommutative Bondal-Orlov conjecture for symplectic singularities, using developments from categorical and diagrammatic representation theory. Develop the theory of braces, motivated by connections with nil rings, and pursue the connections to R-matrices and the QYBE as well as other parts of mathematics.

**Strand 4. Applications. **

Develop the theory of noncommutative enhancements of positive dimensional moduli spaces to produce new autoequivalences of derived categories. Create a more geometric approach to the Bezrukavnikov-Okounkov programme, giving faithfulness of fundamental group actions as derived autoequivalences. Use the theory of microlocal mixed Hodge modules to connect DQ-derived equivalences with stability conditions for symplectic resolutions, and extend beyond the symplectic case.

Crucially, to deliver these objectives we will significantly develop the theory and common methodology in key areas that cut across all the above strands. These areas are organised by our three themes of moduli, deformations, and stability.