2006年度上智大学シラバス
2006/05/27更新
| ◆解析学特別講義Ⅰ － (後) |
古谷 賢朗 |
| ○科目サブタイトル |
| ベキ零 Lie 群上の熱核の構成 |
| ○講義概要 |
| Since the heat kernel of the sub-Laplacian on Heisenberg group was constructed in an explicit integral form by A. Hulanicki, we have several ways to construct the heat kernel for the sub-Laplacian and the Laplacian on 2-step nilpotent Lie groups. In this course we explain a method effectively employed by Beals-Gaveau-Greiner, so called the complex Hamilton-Jacobi theory, and illustrate the construction of the heat kernel for general 2-step cases. Main contents of the course will be the discussion on the solution of the generalized Hamilton-Jacobi equation and a quantity similar to van Vleck determinant and their roles in the integral expression of the heat kernel. We expect this method will work also for 3-step cases to construct the heat kernel together with the theory of elliptic functions. So as an example, we consider the solution of the generalized Hamilton-Jacobi equation for the lowest dimensional 3-step nilpotent Lie group (Engel group). If we have an enough time, we discuss a hierarchy of heat kernels for the three dimensional Heisenberg group and Heisenberg manifolds as a simple example. |
| ○評価方法 |
| 出席状況(60%)、レポート(40%) |
| ○参考書 |
| Kenro Furutani : Heat Kernels of the sub-Laplacian and the Laplacian on Nilpotent Lie Groups /最初の講義で配ります。 R. Beals, B. Gaveau and P. Greiner "The Green Function of Model Step two Hypoelliptic Operators and the Analysis of Certain Tangential Cauchy Riemann Complexes, Advances in Mathematics Vol.121(1996), 288--345." Academic press 1996年 Volume 121 |
| ○他学部・他学科生の受講 |
| 可 |
| ○授業計画 | ||||||||||||||||||||||||||
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Copyright (C) 2006 Sophia University
By：上智大学 学事センター
By：上智大学 学事センター