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2023　Special lectures on advanced topics in Mathematics J

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
  Masai Hidetoshi 
Class Format
Lecture     
Media-enhanced courses
Day/Period(Room No.)
-
Group
-
Course number
MTH.E534
Credits
2
Academic year
2023
Offered quarter
2Q
Syllabus updated
2023/3/20
Lecture notes updated
-
Language used
Japanese
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Course description and aims

The main subject of this course is the topology and the geometry of 3-manifolds. In the first half of the course, we begin by explaining a relation between the torus and the Farey tessellation, and then explain the topological classification theorem of (once-punctured) torus bundles over the circle and that of 2-bridge links from the viewpoint of the Farey tessellation. In the latter half of the course, we first explain basic facts in hyperbolic geometry and then explain concrete constructions of the hyperbolic structures and the canonical decompositions of once-punctured torus bundles and 2-bridge link complements. We intend to explain an intimate relation between the topology and the geometry of 3-manifolds through once-punctured torus bundles and 2-bridge link complements, which form special but important families of 3-manifolds.

Student learning outcomes

・Be familiar with the Farey tessellation and its relation with the torus
・Understand the classification theorems of 2-bridge links and punctured torus bundles
・Be familiar with basic facts in hyperbolic geometry
・Understand intimate relation between the topology and the geometry in dimension three through 2-bridge links and punctured torus bundles

Keywords

knot theory, hyperbolic geometry, 2-bridge links, once punctured torus bundles, Farey tessalations

Competencies that will be developed

✔ 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/Required learning

  Course schedule Required learning
Class 1 The following topics will be covered in this order: - the (once-punctured) torus and the Faray tessellations - the classification theorem of (once-punctured) tori from the viewpoint of the Farey tessellation - the classification theorem of 2-bridge links from the viewpoint of the Farey tessellation - Jorgensen -Floyd-Hatcher decompositions of once-punctured torus bundles and their analogies for 2-bridge link complements - basic facts in hyperbolic geometry - the Epstein-Penner decompositions of cusped hyperbolic manifolds - Jorgensen's work on once-punctured Kleinian groups and their extension Details will be provided during each class session.

Out-of-Class Study Time (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.
They should do so by referring to textbooks and other course material.

Textbook(s)

None required.

Reference books, course materials, etc.

J. S. Purcell, Hyperbolic knot theory, Graduate Studies in Mathematics 209, American Mathematical Society, Providence, RI, 2020. H. Akiyoshi, M. Sakuma, M. Wada, and Y. Yamashita, Punctured torus groups and 2-bridge knot groups (I), Lecture Notes in Mathematics 1909, Springer, Berlin, 2007.

Assessment criteria and methods

Assignments (100%).

Related courses

  • MTH.B301 ： Geometry I
  • MTH.B302 ： Geometry II
  • MTH.B331 ： Geometry III
  • MTH.B341 ： Topology

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

To have basic knowledge in the theory of differentiable manifolds

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