2016 Special courses on advanced topics in Mathematics M

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Naito Satoshi  Maeno Toshiaki 
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Day/Period(Room No.)
Intensive ()  
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Syllabus updated
Lecture notes updated
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Course description and aims

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.

Student learning outcomes

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

Competencies that will be developed

Intercultural skills Communication skills Specialist skills Critical thinking skills Practical and/or problem-solving skills
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Class flow

This is a standard lecture course. There will be some assignments.

Course schedule/Required learning

  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


none required

Reference books, course materials, etc.

Course materials are provided during class.

Assessment criteria and methods

Assignments (100%).

Related courses

  • ZUA.A301 : Algebra I
  • MTH.A302 : Algebra II

Prerequisites (i.e., required knowledge, skills, courses, etc.)

No prerequisites are necessary, but enrollment in the related courses is desirable.

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