TEACHING

卒業研究(埼玉大学の学生へ)

概要
:
現代調和解析およびそれに関連する幾何学と偏微分方程式の話題を研究します。調和解析は現在、純粋数学や応用数学でも非常にアクティブな研究分野です。

例として、Fourier変換Stein’s Restriction予想、幾何学のKakeya予想、物理学の様々な微分方程式 (Schrödinger方程式など)の解析です。意外なことに、このような分野にはお互いに深い関係があります。

調和解析による他分野の数学への応用を研究します。注目すべきことに、数論(Dirichletの定理など)、等周不等式熱流連続かつ微分不可能な関数の存在、などが含まれています。


Topics from modern harmonic analysis, and related topics from geometry and partial differential equations, can be researched. Harmonic analysis is a very active research topic in current mathematics, both pure and applied.

As an example, the study of the Stein's Restriction Conjecture for the Fourier transform, and its surprising connections to various topics, including the Kakeya Conjecture from geometry and analysis of various differential equations from physics (such as Schrödinger equations).

Applications of harmonic analysis to a variety of other fields of mathematics may also be researched. Remarkably, these include number theory (such as Dirichlet's theorem), the isoperimetric inequality, heat-flow, continuous but nowhere differentiable functions, etc.


教科書の候補:
Fourier Analysis: An Introduction, Princeton Lectures in Analysis I, Elias M. Stein and Rami Shakarchi, Princeton University Press, 2003.
Real Analysis: Measure Theory, Integration, & Hilbert Spaces, Princeton Lectures in Analysis III, Elias M. Stein and Rami Shakarchi, Princeton University Press, 2003.
Functional Analysis: Introduction to Further Topics in Analysis, Princeton Lectures in Analysis IV, Elias M. Stein and Rami Shakarchi, Princeton University Press, 2003.
Modern Fourier Analysis (Second Edition), Loukas Grafakos, Springer Graduate Texts in Mathematics, 2008.
Classical and Multilinear Harmonic Analysis,
Camil Muscalu, Wilhelm Schlag, Cambridge Studies in Advanced Mathematics, Volumes I and II, 2013.
Fourier Analysis,
Javier Duoandikoetxea, American Mathematical Society Graduate Studies in Mathematics (Volume 29), 2001.

CURRENT TEACHING

2018-19
前期:
解析学C

後期:
Modern Harmonic Analysis