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Tom Graber
- Virtual Localization
Abstract: I will discuss the formalism of equivariant cohomology and how it leads to the Bott fixed point formula for certain integrals on spaces with circle actions. The focus of these lectures will be on the generalization of this formula to virtual integrals and its application to the numerical invariants of algebraic varieties coming from Gromov-Witten and Donaldson type theories.
Martijn Kool
- Virtual counts on surfaces
Abstract: This series of lectures is divided in three parts. In part 1 I review a sheaf theory of R. Pandharipande and R. P. Thomas known as stable pair theory. Stable pair invariants are closely related to both Gromov-Witten and Donaldson-Thomas invariants. In part 2 we study virtual cycles on Hilbert schemes of curves with points on surfaces. In the absence of points these give rise to the Poincare invariants of M. Durr, A. Kabanov and Ch. Okonek. Poincare invariants are an algebraic version of Seiberg-Witten invariants. In part 3 we relate the virtual cycles of part 1 and part 2 for the total space of the canonical bundle over a surface. We apply the MNOP correspondence in this setting to obtain characterizations of Severi degrees and Seiberg-Witten invariants.
Cristina Manolache
- Virtual fundamental classes and virtual functoriality
Abstract: Virtual classes allow us to do intersection theory on singular spaces and they are a key ingredient in modern enumerative geometry. The lecture series will have two main parts: how to define virtual classes and how to work with them. In the first part I will discuss classical intersection theory and its generalization to certain singular spaces such as the moduli space of stable maps and Hilbert schemes. In the second part I will concentrate on similarities and differences to classical intersection theory. This will be a discussion on functorial properties and open problems in this direction.
Daily schedule:
- Virtual Localization
Abstract: I will discuss the formalism of equivariant cohomology and how it leads to the Bott fixed point formula for certain integrals on spaces with circle actions. The focus of these lectures will be on the generalization of this formula to virtual integrals and its application to the numerical invariants of algebraic varieties coming from Gromov-Witten and Donaldson type theories.
Martijn Kool
- Virtual counts on surfaces
Abstract: This series of lectures is divided in three parts. In part 1 I review a sheaf theory of R. Pandharipande and R. P. Thomas known as stable pair theory. Stable pair invariants are closely related to both Gromov-Witten and Donaldson-Thomas invariants. In part 2 we study virtual cycles on Hilbert schemes of curves with points on surfaces. In the absence of points these give rise to the Poincare invariants of M. Durr, A. Kabanov and Ch. Okonek. Poincare invariants are an algebraic version of Seiberg-Witten invariants. In part 3 we relate the virtual cycles of part 1 and part 2 for the total space of the canonical bundle over a surface. We apply the MNOP correspondence in this setting to obtain characterizations of Severi degrees and Seiberg-Witten invariants.
Cristina Manolache
- Virtual fundamental classes and virtual functoriality
Abstract: Virtual classes allow us to do intersection theory on singular spaces and they are a key ingredient in modern enumerative geometry. The lecture series will have two main parts: how to define virtual classes and how to work with them. In the first part I will discuss classical intersection theory and its generalization to certain singular spaces such as the moduli space of stable maps and Hilbert schemes. In the second part I will concentrate on similarities and differences to classical intersection theory. This will be a discussion on functorial properties and open problems in this direction.
Daily schedule:
Jan 6:
10:30 - 12:00 Manolache
14:00 - 15:30 Graber
16:30 - 18:00 Kool
Jan 7:
10:30 - 12:00 Manolache
14:00 - 15:30 Graber
16:30 - 18:00 Kool
Jan 8:
10:30 - 12:00 Graber
14:00 - 15:30 Kool
16:30 - 18:00 Manolache
Jan 9:
10:30 - 12:00 Kool
14:00 - 15:30 Manolache
16:30 - 18:00 Graber
18:15 - Onwards Workshop Dinner at Outside Location