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Neal Bez

2015.01更新

卒業研究の内容

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変換のStein’s Restriction予想、幾何学のKakeya予想、物理学の様々な微分方程式 (Schroedinger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 Schroedinger equations).

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

教科書の候補

  • Elias M. Stein, Rami Shakarchi, Fourier Analysis: An Introduction, Princeton University Press, 2003.
  • Loukas Grafakos, Modern Fourier Analysis (Second Edition), Springer Graduate Texts in Mathematics, 2008.
  • Camil Muscalu, Wilhelm Schlag, Classical and Multilinear Harmonic Analysis, Cambridge Studies in Advanced Mathematics, Volumes I and II, 2013.
  • Javier Duoandikoetxea, Fourier Analysis, American Mathematical Society Graduate Studies in Mathematics (Volume 29), 2001.