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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 곡선의 정의란 무엇인가? file 서울대학교 김영훈
Math Colloquia Fano manifolds of Calabi-Yau Type file 서울대학교 Atanas Iliev
Math Colloquia 젊은과학자상 수상기념강연: From particle to kinetic and hydrodynamic descriptions to flocking and synchronization file 서울대학교 하승열
Math Colloquia Noncommutative Geometry. Quantum Space-Time and Diffeomorphism Invariant Geometry file 서울대학교 Raphael Ponge
Math Colloquia 정년퇴임 기념강연회: 숙제 file 서울대학교 지동표
Math Colloquia Randomness of prime numbers file 서울대학교 임선희
Math Colloquia Non-commutative Lp-spaces and analysis on quantum spaces file 서울대학교 이훈희
Math Colloquia 학부생을 위한 ε 강연회: Sir Isaac Newton and scientific computing file 서울대학교 신동우
Math Colloquia 학부생을 위한 ε 강연회: Mathematics from the theory of entanglement file 서울대학교 계승혁
Math Colloquia Combinatorial Laplacians on Acyclic Complexes file 서울대학교 국웅
Math Colloquia 정년퇴임 기념강연: Volume Conjecture file 서울대학교 김혁
Math Colloquia Fefferman's program and Green functions in conformal geometry file 서울대학교 Raphaël Ponge
Math Colloquia Subgroups of Mapping Class Groups file 서울대학교 김상현
Math Colloquia Categorification of Donaldson-Thomas invariants file 서울대학교 김영훈
Math Colloquia Random conformal geometry of Coulomb gas formalism file 서울대학교 강남규
Math Colloquia Brownian motion and energy minimizing measure in negative curvature file 서울대학교 임선희
Math Colloquia Seeded Ising Model for Human Iris Templates and Secure Distributed Iris Recognition file 서울대학교 최형인
Math Colloquia <학부생을 위한 ɛ 강연> 기하와 대수의 거울대칭 file 서울대학교 조철현
Math Colloquia W-algebras and related topics file 서울대학교 서의린
Math Colloquia <학부생을 위한 ɛ 강연> Geometry and algebra of computational complexity file 서울대학교 현동훈
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