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Starting Topics

卒業研究・修士初年で始めやすいテーマ

  • 曲線と曲面の古典微分幾何
  • 極小曲面と平均曲率一定曲面
  • Gauss--Codazzi 方程式と移動標構
  • Lie 群と等質空間の初歩
  • 複素解析と曲面論
  • sine-Gordon 方程式と曲面の幾何
  • 戸田格子と幾何への応用
  • コンピュータを用いた特殊曲面の可視化
  • 情報幾何と統計多様体の基礎

これらのセミナーでは、定義や定理を覚えるだけでなく、数学の文献を丁寧に読み、具体例を計算し、自分の言葉で説明する力を身につけることを目標にします。

最初から高度な内容を知っている必要はありません。学生の興味や準備状況に応じて、調和写像、ループ群、可積分系、タイヒミュラー理論、離散微分幾何、アフィン幾何・情報幾何などへ少しずつ進んでいきます。

Possible Starting Topics for Undergraduate and Master's Seminars

The following are possible starting points for undergraduate and master's seminars.
They are not fixed courses, but examples of topics from which students can enter differential geometry and related areas.

Students usually begin with classical and concrete subjects, such as curves and surfaces, the first and second fundamental forms, the Gauss--Codazzi equations, minimal surfaces, constant mean curvature surfaces, moving frames, Lie groups, complex analysis, and geometric differential equations.

Possible starting topics include:

  • Classical differential geometry of curves and surfaces
  • Minimal surfaces and constant mean curvature surfaces
  • The Gauss--Codazzi equations and moving frames
  • Basic theory of Lie groups and homogeneous spaces
  • Complex analysis and surface theory
  • The sine-Gordon equation and surface geometry
  • The Toda lattice and its geometric applications
  • Computer experiments and visualization of special surfaces
  • Information geometry and statistical manifolds

The aim of these seminars is not only to learn definitions and theorems, but also to develop the ability to read mathematical texts carefully, work out concrete examples, perform computations, and explain geometric ideas in a precise way.

Advanced topics such as harmonic maps, loop groups, integrable systems, Teichmüller theory, discrete differential geometry, and affine or statistical geometry may be introduced gradually, depending on the student's interests and background.

Possible background

  • Linear algebra
  • Calculus and multivariable calculus
  • Ordinary differential equations
  • Complex analysis
  • Basic differential geometry