Projects per year

## Personal profile

### Research interests

My research interests lie in pure mathematics, and specifically in algebraic geometry and representation theory. Much of my research applies explicit methods to improve our understanding of moduli spaces of quiver representations and their derived categories.

One motivating question is to ask for nice moduli descriptions of well known varieties. This is now understood for projective toric varieties and, more generally, for Mori Dream Spaces. This leads naturally to the study of multigraded linear series (also known as framed quiver varieties or quiver flag varieties) as ambient spaces in noncommutative algebraic geometry.Another theme of my research is to understand which algebraic varieties carry tilting bundles, thereby providing an explicit understanding of their derived category of coherent sheaves. The first known examples were provided by Beilinson and Kapranov, while many of the more recent examples have been influenced by the McKay correspondence in one form or another. Examples include framed quiver varieties, G-Hilbert schemes, moduli spaces arising in the study of consistent dimer model algebras, and smooth toric Fano varieties in low dimensions. These constructions lead to interesting questions about mutation, wall crossing phenomena and Bridgeland stability manifolds.

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## Projects 2013 2016

- 1 Finished

### Alastair Craw - Bridgeland Stability and the Moveable Cone

3/04/13 → 2/10/16

Project: Research council

## Research Output 2002 2018

### Birational geometry of symplectic quotient singularities

Bellamy, G. & Craw, A., 2018, (Submitted) In : Unspecified. 49 p.Research output: Contribution to journal › Article

### Multigraded linear series and recollement

Craw, A., Ito, Y. & Karmazyn, J., 1 Jun 2018, In : Mathematische Zeitschrift. 289, 1-2, p. 535-565Research output: Contribution to journal › Article

### Reconstructing toric quiver flag varieties from a tilting bundle

Craw, A. & Green, J., 1 Mar 2018, In : European Journal of Mathematics. 4, 1, p. 185–199 12 p.Research output: Contribution to journal › Article

### Derived Reid's recipe for abelian subgroups of SL_{3}(ℂ)

Cautis, S., Craw, A. & Logvinenko, T., 1 Jun 2017, In : Journal Fur Die Reine Und Angewandte Mathematik. 727, p. 1-48 48 p.Research output: Contribution to journal › Article

### Nef divisors for moduli spaces of complexes with compact support

Bayer, A., Craw, A. & Zhang, Z., Apr 2017, In : Selecta Mathematica. 23, 2, p. 1507-1561 55 p.Research output: Contribution to journal › Article