2019 Special courses on advanced topics in Mathematics I

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Academic unit or major
Mathematics
Instructor(s)
Hayano Kenta 
Course component(s)
Lecture
Day/Period(Room No.)
Intensive ()  
Group
-
Course number
ZUA.E343
Credits
2
Academic year
2019
Offered quarter
1Q
Syllabus updated
2019/3/18
Lecture notes updated
-
Language used
English
Access Index

Course description and aims

It was shown recently that the diffeomorphism type of a closed oriented four-manifold can be described by a combinatorial data, consisting of vanishing cycles of a stable mapping from a four-manifold to a surface (cf. a surface diagram by Williams, and a trisection by Gay-Kirby). In this course, we first introduce basic notions concerning stable mappings to surfaces, such as folds, cusps and vanishing cycles. We then explain how to analyse vanishing cycles of stable mappings on four-manifolds and combinatorial data mentioned above relying on the theory of mapping class groups of surfaces.

The aim of this course is to explain the effects of generic homotopies between stable mappings on their vanishing cycles in terms of mapping class groups of surfaces. We would also like to explain how to determine combinatorial data coming from (explicit examples of) stable mappings on four-manifolds.

Student learning outcomes

・Understand local models of critical points in stable mappings from 4-manifolds to surfaces
・Be familier with surgery homomorphisms on mapping class groups of surfaces
・Understand the effects of homotopies between stable mappings on their vanishing cycles in terms of mapping class groups
・Understand how to obtain diagrams of trisections relying on the theory of mapping class groups

Keywords

stable mapping, fold, cusp, vanishing cycle, mapping class group, surgery homomorphism, monodoromy, parallel transport, trisection

Competencies that will be developed

Intercultural skills Communication skills Specialist 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/Required learning

  Course schedule Required learning
Class 1 The following topics will be covered in this order : -- The definition and examples of stable mappings -- Critical points in stable mappings from 4-manifolds to surfaces -- Mapping class groups of surfaces -- Surgery homomorphisms -- Relation between homotopies of stable mappings and surgery homomorphisms -- Trisections of 4-manifolds Details will be provided during each class.

Textbook(s)

None required

Reference books, course materials, etc.

K. Hayano, Modification rule of monodromies in an R_2 move, AGT, 14(2014), no. 4, 2181-2222.
S. Behrens and K. Hayano, Elimination of cusps in dimension 4 and its applications, PLMS, (3) 113(2016), 674-724.
K. Hayano, On diagrams of simplified trisections and mapping class groups, to appear in OJM.

Assessment criteria and methods

Assignments (100%)

Related courses

  • MTH.E533 : Special lectures on advanced topics in Mathematics I
  • MTH.E639 : Special lectures on current topics in Mathematics I

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

None required

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