Donaldson-Thomas and Gromov-Witten theories
The study of the moduli spaces of algebraic curves and coherent sheaves, and their induced invariants over ambient complex kahler varieties of complex dimension 2, 3 and higher, has been a central source of focus for mathematicians in the past 50 years, due to their profound connections to geometry, topology, number theory as well as fruitful contributions to superstring theory. The course aims at introducing these topics and provides discussion of computations of Gromov-Witten and Donaldson-Thomas invariants of complex algebraic varieties.
Lecturer
Date
2nd March ~ 13th July, 2023
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Thursday | 09:00 - 10:30 | A3-1-101 | ZOOM 08 | 787 662 9899 | BIMSA |
Syllabus
Part 1: GW invariants
1.1 Stable curves
1.2 Families and moduli spaces of smooth and stale pointed curves
1.3 Gromov-Witten classes
1.4 Gromov-Witten invariant
1.5 Virtualfundamentalclasses
1.6 Formal construction of virtual fundamental class
1.7 Behrend-Pantechi Construction of virtual fundamental class
1.8 Computation of virtual fundamental class in some examples
1.9 Propertiesof GW classes
1.10 Tree-level classes and invariants
1.11 Small quantum cohomology
1.12 Further properties of small quantum product
1.13 Small quantum cohomology of CY manifolds
Part 2: Donaldson-Thomas theory and sheaf theory
2.1 IdealsheavesandDonaldson-Thomastheory
2.2 Deformationtheory
2.3 Primaryinvariants
2.4 Correspondence bewteen DT invariants and GW invariants
2.5 Virtual localizationin DT theory
2.6 Some examples of Torus equivariant localization of DT, for C3 and local P1
2.7 Relative DT Theory(JunLi, BaoseauWu)
2.8 Torsion DT theories and modular forms
Part 3: Physics behind study of torsion sheaves
3.1 Partition function of solutions to SYM theory
3.2 Modularity of torsion sheaf DT theory for K3-fibered 3folds
Part 4: Vafa-Witten theory over algebraic projective surfaces
4.1 Vafa-Witten invariants
4.2 Fixed loci of torsion sheaf moduli space, i.e., C∗-equivariant torsion sheaves
Part 5 4 dimensional DT theories
5.1 Calabi-Yau 4folds and Kapustin-Witten invariants
5.2 Atiyah class
5.3 Universal truncated relative Atiyah class.
5.4 Categorical DT theory for local surfaces
5.5 Work of Yukinobu Toda
5.6 Singular support of coherent sheaves
1.1 Stable curves
1.2 Families and moduli spaces of smooth and stale pointed curves
1.3 Gromov-Witten classes
1.4 Gromov-Witten invariant
1.5 Virtualfundamentalclasses
1.6 Formal construction of virtual fundamental class
1.7 Behrend-Pantechi Construction of virtual fundamental class
1.8 Computation of virtual fundamental class in some examples
1.9 Propertiesof GW classes
1.10 Tree-level classes and invariants
1.11 Small quantum cohomology
1.12 Further properties of small quantum product
1.13 Small quantum cohomology of CY manifolds
Part 2: Donaldson-Thomas theory and sheaf theory
2.1 IdealsheavesandDonaldson-Thomastheory
2.2 Deformationtheory
2.3 Primaryinvariants
2.4 Correspondence bewteen DT invariants and GW invariants
2.5 Virtual localizationin DT theory
2.6 Some examples of Torus equivariant localization of DT, for C3 and local P1
2.7 Relative DT Theory(JunLi, BaoseauWu)
2.8 Torsion DT theories and modular forms
Part 3: Physics behind study of torsion sheaves
3.1 Partition function of solutions to SYM theory
3.2 Modularity of torsion sheaf DT theory for K3-fibered 3folds
Part 4: Vafa-Witten theory over algebraic projective surfaces
4.1 Vafa-Witten invariants
4.2 Fixed loci of torsion sheaf moduli space, i.e., C∗-equivariant torsion sheaves
Part 5 4 dimensional DT theories
5.1 Calabi-Yau 4folds and Kapustin-Witten invariants
5.2 Atiyah class
5.3 Universal truncated relative Atiyah class.
5.4 Categorical DT theory for local surfaces
5.5 Work of Yukinobu Toda
5.6 Singular support of coherent sheaves
Audience
Graduate
Video Public
Yes
Notes Public
No
Language
English
Lecturer Intro
Artan Sheshmani is a Professor of pure Mathematics, specialized in Algebraic geometry, Enumerative and Derived Geometry, and Mathematics of String Theory. He joined BIMSA as a Professor in September 2023. Prior to BIMSA he was a senior personnel (Professor) at Simons Collaboration Program on Homological Mirror Symmetry at Harvard University Center for Mathematical Sciences and Applications (CMSA) for 7 years, during which he was also an Associate Professor of Mathematics at Institut for Mathematik (formerly the Center for Quantum Geometry of Moduli Spaces) at Aarhus University in Denmark (2016-2022). He is working on geometry of moduli spaces of sheaves and curves from enumerative geometry point of view as well as studying their structural properties from derived geometry and geometric representation theory point of view. He has been an invited speaker to ICCM and ICBS and a recipient of several awards. In 2019 he was one of the 30 winners of the IRFD "Research Leader" grant (approx 1M USD) in 2019.