The main subject of this course is the theory of quantum cohomology. In this course, we start with the moduli space of stable maps to see the outline of the construction of the Gromov-Witten invariants, and then introduce their fundamental properties. After that, we discuss the relationship between the structure of quantum cohomology rings and integrable systems via the J-functions. If time permits, we also see the topics on the quantum K-theory etc.
The developments of the topological field theory in the particle physics at the end of the 20th century provide a new approach to the classical enumerative algebraic geometry, which is now known as the Gromov-Witten theory or the theory of quantum cohomology. The theory of quantum cohomology rings was established in early 90s mainly to understand the Mirror Symmetry phenomenon, and it has been playing important roles in the geometry of the field theory over since. The main aim of this course is to make students understand the relationship between the quantum cohomology ring of the flag variety and the Toda system.
By the end of this course, students will be able to:
1) Understand the construction of Gromov-Witten invariants.
2) Understand the construction of quantum cohomology rings.
3) Be familiar with examples of quantum cohomology rings.
moduli space, intersection theory, quantum cohomology ring, flag variety
|Intercultural skills||Communication skills||Specialist skills||Critical thinking skills||Practical and/or problem-solving skills|
This is a standard lecture course. There will be some assignments.
|Course schedule||Required learning|
|Class 1||Moduli space of stable maps I||Details will be provided during each class session.|
|Class 2||Moduli space of stable maps II|
|Class 3||Moduli space of stable maps III|
|Class 4||Gromov-Witten invariants I|
|Class 5||Gromov-Witten invariants I|
|Class 6||Gromov-Witten invariants III|
|Class 7||WDVV equations and quantum cohomology rings I|
|Class 8||WDVV equations and quantum cohomology rings II|
|Class 9||WDVV equations and quantum cohomology rings III|
|Class 10||J-functions I|
|Class 11||J-functions II|
|Class 12||J-functions III|
|Class 13||Quantum cohomology ring of the flag variety I|
|Class 14||Quantum cohomology ring of the flag variety II|
|Class 15||Quantum cohomology ring of the flag variety III|
Course materials are provided during class.
No prerequisites are necessary, but enrollment in the related courses is desirable.