2026 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics

Special lectures on advanced topics in Mathematics D

Academic unit or major: Graduate major in Mathematics
Instructor(s): Masahiro Futaki / Takefumi Nosaka
Class Format: Lecture
Media-enhanced courses: -
Day of week/Period (Classrooms)
Class: -
Course Code: MTH.E434
Number of credits: 200
Course offered: 2026
Offered quarter: 3Q
Syllabus updated: Mar 5, 2026
Language: English

Syllabus

Course overview and goals

"The subject of this course is Lagrangian intersection Floer cohomology and Fukaya category for toric Fano manifolds and their application to homological mirror symmetry.

Lagrangian intersection Floer cohomology is defined using pseudo-holomorphic disks. We first review basic definitions and analysis of pseudo-holomorphic curves. This leads to a differential-topological construction of the moduli space of pseudo-holomorphic disks, which requires the notion of orbifolds with corners (or more generally, Kuranishi spaces with corners). Using the moduli spaces we are ready to define the Floer A-infinity algebra and its cohomology, following Fukaya-Oh-Ohta-Ono’s works.

Fukaya category, a categorified version of the Floer A-infinity algebra, was introduced to formulate a cohomological version of mirror symmetry. To conclude this lecture, we discuss homological mirror symmetry in the case of toric manifolds, and also discuss its equivariant version if time permits. The aim of this course is to familiarize participants with homological mirror symmetry for toric Fano manifolds and with the concrete aspects of global analysis in that context."

Course description and aims

・Be familiar with pseudo-holomorphic curves and Floer cohomology
・Understand how to calculate Floer cohomology of the moment fibers of toric Fano manifolds
・Understand the mathematical formulation of homological mirror symmetry"

Keywords

Symplectic manifold, holomorphic curve, mirror symmetry, orbifold, equivariant cohomology

Competencies

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

Class flow

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

Course schedule/Objectives

	Course schedule	Objectives
Class 1	"The following topics will be covered: - Overview: Fukaya category and homological mirror symmetry - Definition of pseudo-holomorphic disks - Some analysis: decay estimates - Floer cohomology - Orbifolds and Kuranishi structures - Floer A-infinity algebra - Example: toric case - Equivariant cohomology and its Floer cohomology analogue - Equivariant homological mirror symmetry"	Details will be provided during each class session.

Study advice (preparation and review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.

Textbook(s)

None required

Reference books, course materials, etc.

"McDuff and Salamon, J-homolorphic curves and symplectic topology. AMS Colloquium Publications.
Fukaya, Oh, Ohta and Ono, Lagrangian Floer theory on compact toric manifolds, I. Duke Mathematical Journal (2010)."

Evaluation methods and criteria

Assignments (100%).

Related courses

MTH.B301 ： Geometry I
MTH.B302 ： Geometry II
MTH.B331 ： Geometry III

Prerequisites

Good understanding on the materials in the "related courses" is expected

Other

Not in particular