https://www.math.snu.ac.kr/board/files/attach/images/701/ff97c54e6e21a4ae39315f9a12b27314.png
Extra Form
Lecturer 김영훈
Dept. 서울대학교
date Apr 10, 2014
In 1980s, Donaldson discovered his famous invariant of 4-manifolds which was subsequently proved to be an integral on the moduli space of semistable sheaves when the 4-manifold is an algebraic surface. In 1994, the Seiberg-Witten invariant was discovered and conjectured to be equivalent to the Donaldson invariant (still open). In late 1990s, Taubes  proved that the Seiberg-Witten invariant also counts pseudo-holomorphic curves.
The Donaldson-Thomas invariant of a Calabi-Yau 3-fold Y (complex projective manifold of dimension 3 with nowhere vanishing holomorphic 3-form) can be thought of as a generalization of the Donaldson invariant. It was defined by a virtual integral on the moduli space of stable sheaves on Y and expected to count algebraic curves in Y. The categorification conjecture due to Kontsevich-Soibelman, Joyce-Song, Behrend-Bryan-Szendroi and others claims that there should be a cohomology theory on the moduli space of stable sheaves whose Euler number coincides with the Donaldson-Thomas invariant.
I will talk about recent progress about the categorification conjecture by using perverse sheaves. Locally the moduli space is the critical locus of a holomorphic function on a complex manifold called a Chern-Simons chart and we have the perverse sheaf of vanishing cycles on the critical locus. By constructing suitable Chern-Simons charts and homotopies using gauge theory, it is possible to glue the perverse sheaves of vanishing cycles to obtain a globally defined perverse sheaf whose hypercohomology is the desired categorified Donaldson-Thomas invariant. As an application, we can provide a mathematical theory of the Gopakumar-Vafa (BPS) invariant. 

Atachment
Attachment '1'
  1. Green’s function for initial-boundary value problem

  2. Mechanization of proof: from 4-Color theorem to compiler verification

  3. On the distributions of partition ranks and cranks

  4. Q-curvature in conformal geometry

  5. Zeros of the derivatives of the Riemann zeta function

  6. Geometry, algebra and computation in moduli theory

  7. Gromov-Witten-Floer theory and Lagrangian intersections in symplectic topology

  8. High dimensional nonlinear dynamics

  9. What is model theory?

  10. Essential dimension of simple algebras

  11. Restriction theorems for real and complex curves

  12. Recommendation system and matrix completion: SVD and its applications (학부생을 위한 강연)

  13. Deformation spaces of Kleinian groups and beyond

  14. Idempotents and topologies

  15. Recent progress on the Brascamp-Lieb inequality and applications

  16. Existence of positive solutions for φ-Laplacian systems

  17. Riemann-Hilbert correspondence for irregular holonomic D-modules

  18. Normal form reduction for unconditional well-posedness of canonical dispersive equations

  19. Random conformal geometry of Coulomb gas formalism

  20. 11Apr
    by 김수현
    in Math Colloquia

    Categorification of Donaldson-Thomas invariants

Board Pagination Prev 1 2 3 4 5 6 7 8 9 10 11 Next
/ 11