Tom Bridgeland

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Tom Bridgeland

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Thomas Andrew Bridgeland [1]

1973 (age 4950)
Education Shelley High School [1]
Alma mater
Scientific career
Thesis Fourier-Mukai transforms for surfaces and moduli spaces of stable sheaves  (2002)
Doctoral advisor Antony Maciocia [2]

Thomas Andrew Bridgeland FRS [3] (born 1973) is a Professor of Mathematics at the University of Sheffield. [2] [4] [5] [6] [7] [1] He was a senior research fellow in 2011–2013 at All Souls College, Oxford and, since 2013, remains as a Quondam Fellow. He is most well-known for defining Bridgeland stability conditions on triangulated categories.



Bridgeland was educated at Shelley High School [7] in Huddersfield and Christ's College, Cambridge, where he studied the Mathematical Tripos in the University of Cambridge, graduating with a first class degree in mathematics in 1994 and a distinction in Part III the following year. He completed his PhD [8] at the University of Edinburgh, where he also stayed for a postdoctoral research position.[ citation needed ]

Research and career

Bridgeland's research interest is in algebraic geometry, focusing on properties of derived categories of coherent sheaves on algebraic varieties. [9] [10] His most-cited papers are on stability conditions, on triangulated categories [11] and K3 surfaces; [12] in the first he defines the idea of a stability condition on a triangulated category, and demonstrates that the set of all stability conditions on a fixed category form a manifold, whilst in the second he describes one connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic K3 surface.

Bridgeland's work helped to establish the coherent derived category as a key invariant of algebraic varieties and stimulated world-wide enthusiasm for what had previously been a technical backwater. [3] His results on Fourier–Mukai transforms solve many problems within algebraic geometry, and have been influential in homological and commutative algebra, the theory of moduli spaces, representation theory and combinatorics. [3] Bridgeland's 2002 Annals paper introduced spaces of stability conditions on triangulated categories, replacing the traditional rational slope of moduli problems by a complex phase. This far-reaching innovation gives a rigorous mathematical language for describing D-branes and creates a new area of deep interaction between theoretical physics and algebraic geometry. It has been a central component of subsequent work on homological mirror symmetry. [3]

Bridgeland's research has been funded by the Engineering and Physical Sciences Research Council (EPSRC). [13]

Awards and honours

Bridgeland won the Berwick Prize in 2003, the Adams Prize in 2007 and was elected a Fellow of the Royal Society (FRS) in 2014. [3] He was an invited speaker at the International Congress of Mathematicians, Madrid in 2006.

Related Research Articles

In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.

In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology.

The mathematical term perverse sheaves refers to a certain abelian category associated to a topological space X, which may be a real or complex manifold, or a more general topologically stratified space, usually singular. This concept was introduced in the thesis of Zoghman Mebkhout, gaining more popularity after the (independent) work of Joseph Bernstein, Alexander Beilinson, and Pierre Deligne (1982) as a formalisation of the Riemann-Hilbert correspondence, which related the topology of singular spaces and the algebraic theory of differential equations. It was clear from the outset that perverse sheaves are fundamental mathematical objects at the crossroads of algebraic geometry, topology, analysis and differential equations. They also play an important role in number theory, algebra, and representation theory. The properties characterizing perverse sheaves already appeared in the 75's paper of Kashiwara on the constructibility of solutions of holonomic D-modules.

Homological mirror symmetry is a mathematical conjecture made by Maxim Kontsevich. It seeks a systematic mathematical explanation for a phenomenon called mirror symmetry first observed by physicists studying string theory.

In algebraic geometry, a Fourier–Mukai transformΦK is a functor between derived categories of coherent sheaves D(X) → D(Y) for schemes X and Y, which is, in a sense, an integral transform along a kernel object K ∈ D(X×Y). Most natural functors, including basic ones like pushforwards and pullbacks, are of this type.

In mathematics, especially homological algebra, a differential graded category, often shortened to dg-category or DG category, is a category whose morphism sets are endowed with the additional structure of a differential graded -module.

The SYZ conjecture is an attempt to understand the mirror symmetry conjecture, an issue in theoretical physics and mathematics. The original conjecture was proposed in a paper by Strominger, Yau, and Zaslow, entitled "Mirror Symmetry is T-duality".

In mathematics, a stable vector bundle is a vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable ones using Harder–Narasimhan filtration. Stable bundles were defined by David Mumford in Mumford (1963) and later built upon by David Gieseker, Fedor Bogomolov, Thomas Bridgeland and many others.

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In mathematics, specifically algebraic geometry, Donaldson–Thomas theory is the theory of Donaldson–Thomas invariants. Given a compact moduli space of sheaves on a Calabi–Yau threefold, its Donaldson–Thomas invariant is the virtual number of its points, i.e., the integral of the cohomology class 1 against the virtual fundamental class. The Donaldson–Thomas invariant is a holomorphic analogue of the Casson invariant. The invariants were introduced by Simon Donaldson and Richard Thomas (1998). Donaldson–Thomas invariants have close connections to Gromov–Witten invariants of algebraic three-folds and the theory of stable pairs due to Rahul Pandharipande and Thomas.

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In differential geometry, algebraic geometry, and gauge theory, the Kobayashi–Hitchin correspondence relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. The correspondence is named after Shoshichi Kobayashi and Nigel Hitchin, who independently conjectured in the 1980s that the moduli spaces of stable vector bundles and Einstein–Hermitian vector bundles over a complex manifold were essentially the same.

Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras, simplicial commutative rings or -ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness of the structure sheaf. Grothendieck's scheme theory allows the structure sheaf to carry nilpotent elements. Derived algebraic geometry can be thought of as an extension of this idea, and provides natural settings for intersection theory of singular algebraic varieties and cotangent complexes in deformation theory, among the other applications.

The article "Sur quelques points d'algèbre homologique" by Alexander Grothendieck, now often referred to as the Tôhoku paper, was published in 1957 in the Tôhoku Mathematical Journal. It revolutionized the subject of homological algebra, a purely algebraic aspect of algebraic topology. It removed the need to distinguish the cases of modules over a ring and sheaves of abelian groups over a topological space.

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Mark William Gross is an American mathematician, specializing in differential geometry, algebraic geometry, and mirror symmetry.

In mathematics, derived noncommutative algebraic geometry, the derived version of noncommutative algebraic geometry, is the geometric study of derived categories and related constructions of triangulated categories using categorical tools. Some basic examples include the bounded derived category of coherent sheaves on a smooth variety, , called its derived category, or the derived category of perfect complexes on an algebraic variety, denoted . For instance, the derived category of coherent sheaves on a smooth projective variety can be used as an invariant of the underlying variety for many cases. Unfortunately, studying derived categories as geometric objects of themselves does not have a standardized name.

In mathematics, and especially algebraic geometry, a Bridgeland stability condition, defined by Tom Bridgeland, is an algebro-geometric stability condition defined on elements of a triangulated category. The case of original interest and particular importance is when this derived category is the derived category of coherent sheaves on a Calabi–Yau manifold, and this situation has fundamental links to string theory and the study of D-branes.

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  1. 1 2 3 Anon (2017). "Bridgeland, Prof. Tom Andrew" . Who's Who (online Oxford University Press  ed.). Oxford: A & C Black. doi:10.1093/ww/9780199540884.013.U281971.(Subscription or UK public library membership required.)
  2. 1 2 Tom Bridgeland at the Mathematics Genealogy Project
  3. 1 2 3 4 5 Anon (2014). "Professor Tom Bridgeland FRS". Royal Society . Retrieved 2 May 2014. One or more of the preceding sentences incorporates text from the website where:
    "All text published under the heading 'Biography' on Fellow profile pages is available under Creative Commons Attribution 4.0 International License." --Royal Society Terms, conditions and policies at the Wayback Machine (archived 2016-11-11)
  4. Tom Bridgeland publications indexed by Google Scholar OOjs UI icon edit-ltr-progressive.svg
  5. Tom Bridgeland publications indexed by the Scopus bibliographic database. (subscription required)
  6. Bridgeland, T. (2002). "Flops and derived categories". Inventiones Mathematicae . 147 (3): 613–632. arXiv: math/0009053 . Bibcode:2002InMat.147..613B. doi:10.1007/s002220100185. S2CID   53059980.
  7. 1 2 Bridgeland, Tom (2017). "Tom Bridgeland CV" (PDF). Archived from the original (PDF) on 4 March 2016.
  8. Bridgeland, Thomas Andrew (1998). Fourier-Mukai Transforms for Surfaces and Moduli Spaces of Stable Sheaves (PhD thesis). University of Edinburgh. hdl:1842/12070. OCLC   606214894. EThOS
  9. Bridgeland, T.; King, A.; Reid, M. (2001). "The McKay correspondence as an equivalence of derived categories" (PDF). Journal of the American Mathematical Society . 14 (3): 535. doi: 10.1090/S0894-0347-01-00368-X . S2CID   15808151.
  10. Bridgeland, T. (2005). "T-structures on some local Calabi–Yau varieties". Journal of Algebra . 289 (2): 453–483. arXiv: math/0502050 . Bibcode:2005math......2050B. doi:10.1016/j.jalgebra.2005.03.016. S2CID   14101159.
  11. Bridgeland, Tom (2002). "Stability conditions on triangulated categories". arXiv: math/0212237v3 .
  12. Bridgeland, T. (2008). "Stability conditions on K3 surfaces". Duke Mathematical Journal. 141 (2): 241–291. arXiv: math/0212237 . doi:10.1215/S0012-7094-08-14122-5. S2CID   16083703.
  13. "UK Government Grants awarded to Tom Bridgeland". Swindon: Research Councils UK.

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