Book

  • Dirichlet branes and mirror symmetry with P.Aspinwall, T.Bridgeland, M.Douglas, M.Gross, A.Kapustin, G.Moore, G.Segal, B.Szendroi, P.M.H.Wilson,
      Clay Mathematics Monographs 4. AMS, Providence, RI, 2009. x+681 pp.

This book introduces the notion of Dirichlet brane in the context of topological quantum field theories. After showing how notions of branes arose in string theory, it turns to an introduction to the algebraic geometry, sheaf theory, and homological algebra needed to define and work with derived categories. The physical existence conditions for branes are discussed and compared in the context of mirror symmetry, culminating in Bridgeland's definition of stability structures, and applications to the McKay correspondence and quantum geometry. The book continues with detailed treatments of the Strominger-Yau-Zaslow conjecture, Calabi-Yau metrics and homological mirror symmetry, and discusses more recent physical developments. This book is suitable for graduate students and researchers with either a physics or mathematics background, who are interested in the interface between string theory and algebraic geometry.

Publications and preprints

  14.   Derived Reid's recipe for abelian subgroups of SL(3,C) with Sabin Cautis and Timothy Logvinenko
          Preprint arXiv:1205.3110.
  13.   Second syzygies of monomial submodules from walks with Alex Quintero Velez
          Preprint arXiv:1111.6018, submitted.
  12.   Mori Dream Spaces as fine moduli of quiver representations with Dorothy Winn
          Preprint arXiv:1104.2490, submitted.
  11.   Cellular resolutions of noncommutative toric algebras from superpotentials with Alex Quintero Velez
          Advances in Mathematics 229 (2012), no. 3., 1516-1554.
  10.   Quiver flag varieties and multigraded linear series
          Duke Mathematical Journal 156 (2011), no. 3., 469-500.
    9.   The Special McKay correspondence as an equivalence of derived categories
          Quarterly Journal of Mathematics 62 (2011), 573-591.
    8.   Projective toric varieties as fine moduli spaces of quiver representations with Greg Smith,
          American Journal of Mathematics 130 (2008), no. 6, 1509-1534.
    7.   Moduli of McKay quiver representations II: Grobner basis techniques with Diane Maclagan and Rekha Thomas,
          Journal of Algebra 316 (2007), no. 2, 514-535.
    6.   Fiber fans and toric quotients with Diane Maclagan,
          Discrete and Computational Geometry 37 (2007), no. 2, 251-266.
    5.   Moduli of McKay quiver representations I: the coherent component with Diane Maclagan and Rekha Thomas,
          Proceedings of the London Mathematical Society 95 (2007), no. 1, 179-198.
    4.   An explicit construction of the McKay correspondence for A-Hilb C^3,
          Journal of Algebra 285 (2005), no. 2, 682-705.
    3.   Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient with Akira Ishii,
          Duke Mathematical Journal 124 (2004), no. 2, 259-307.
    2.   An introduction to motivic integration,
          In Strings and Geometry, 203-225, Clay Mathematical Proceedings 3, AMS, Providence, RI, 2004.
    1.   How to calculate A-Hilb C^3 with Miles Reid,
          In Geometry of toric varieties, 129--154, S'eminaires et Congr`es 6, SMF, Paris, 2002.
    0.   The McKay correspondence and representations of the McKay quiver
           University of Warwick Ph.D thesis, xviii + 134 pp, June 2001.

See Mathscinet for the math reviews.

Published lecture notes and reports

Unpublished lecture notes