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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. 

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List of Articles
Category Subject Dept. Lecturer
Math Colloquia Random conformal geometry of Coulomb gas formalism file 서울대학교 강남규
Math Colloquia Categorification of Donaldson-Thomas invariants file 서울대학교 김영훈
Math Colloquia Noncommutative Surfaces file 서강대학교 Jens Hoppe
Math Colloquia The Shape of Data file Stanford University Gunnar E. Carlsson
Math Colloquia Topological aspects in the theory of aperiodic solids and tiling spaces file Georgia Institute of Technology, School of Mathematics and School of Physics Jean V. Bellissard
Math Colloquia Subgroups of Mapping Class Groups file 서울대학교 김상현
Math Colloquia Analytic torsion and mirror symmetry file Kyoto University Ken-ichi Yoshikawa
Math Colloquia Fefferman's program and Green functions in conformal geometry file 서울대학교 Raphaël Ponge
Math Colloquia 정년퇴임 기념강연: Volume Conjecture file 서울대학교 김혁
Math Colloquia Connes's Embedding Conjecture and its equivalent file RIMS Narutaka Ozawa
Math Colloquia Connectedness of a zero-level set as a geometric estimate for parabolic PDEs file KAIST 김용정
Math Colloquia Combinatorial Laplacians on Acyclic Complexes file 서울대학교 국웅
Math Colloquia 학부생을 위한 ε 강연회: Mathematics from the theory of entanglement file 서울대학교 계승혁
Math Colloquia L-function: complex vs. p-adic file 충북대학교 선해상
Math Colloquia 학부생을 위한 ε 강연회: Sir Isaac Newton and scientific computing file 서울대학교 신동우
Math Colloquia A brief introduction to stochastic models, stochastic integrals and stochastic PDEs file 고려대학교 김경훈
Math Colloquia Mixed type PDEs and compressible flow file POSTECH 배명진
Math Colloquia Freudenthal medal, Klein medal 수상자의 수학교육이론 file 서울대 수학교육과 권오남
Math Colloquia Compressible viscous Navier-Stokes flows: Corner singularity, regularity file POSTECH 권재룡
Math Colloquia 학부생을 위한 ε 강연회: Constructions by ruler and compass together with a conic file 건국대/서울대 최인송
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