現 理工学研究科 教授（数理電子情報部門・数理領域）
[Partial differential equations，Harmonic analysis，Geometric analysis，Geometric computation]
[Strichartz estimates，Restriction conjecture，Brascamp-Lieb type inequalities， Bézier curves，subdivision of curves and surfaces]
■数学科のホームページ→ http://www.rimath.saitama-u.ac.jp/lab.jp/nealbez/Main.html My main research interests lie in partial differential equations, harmonic analysis and geometric analysis. For example, Strichartz estimates for the solution of various evolution equations, including dispersive equations and transport equations. For dispersive equations, such estimates are directly related to questions in the context of the Stein restriction conjecture, which is a central problem in euclidean harmonic analysis, and enjoys further connections to problems in geometric analysis, such as the Kakeya conjecture and, via the multilinear perspective, to various generalisations of the Brascamp-Lieb inequality. Recently, I have been particularly interested in establishing optimal versions of such estimates, seeking sharp constants and information about the nature of maximising input functions when they exist, for example, via heat flow methods. In a different direction, I am also interested in geometric computation, including the analysis of Bézier curves and subdivision of curves and surfaces. 【日本語訳文】私の主な研究分野は、偏微分方程式、調和解析と幾何解析です。例を挙げると､分散型偏微分方程式のStrichartzの評価式、Steinのrestriction予想、掛谷予想、Brascamp-Liebの不等式などです。最近、私はこの不等式の最適な定数に特に 興味があります。例えば、熱流単調性の方法などです。また、Bézier曲線 と曲線 と曲面のサブディビジョンを含む､計算幾何にも興味があります。
A central open problem in modern euclidean harmonic analysis is Stein’s restriction conjecture, formulated in the 1960s, and is concerned with the very delicate problem of whether the Fourier transform of an Lp function in N dimensions can be meaningfully restricted to the N-1 dimensional unit sphere. The problem makes sense in two dimensions and higher, and currently is only known in the two dimensional case.
Stein’s restriction conjecture enjoys many beautiful connections across mathematics. This includes a direct one with the Kakeya conjecture from geometric measure theory, which says that the (Minkowski or Hausdorff) dimension of a Kakeya (or Besicovitch) set must be as large as possible. A Kakeya set is a set in euclidean space containing a line segment of length one in every possible direction. Remarkably, a solution of the restriction conjecture would immediately lead to a solution of the Kakeya conjecture.
Interest in the restriction conjecture is also fuelled by the direct link with space-time estimates for the solution of dispersive and wave-like partial differential equations. These powerful estimates may be combined with iteration arguments to yield highly desired conclusions for the solutions of associated nonlinear partial differential equations.
Bézier curves and surfaces are a class of parametrised (rational) curves and surfaces widely used in computer-aided design and computer graphics. The applicability to engineering was pioneered by Pierre Bézier and Paul de Casteljau in the late 1950s and early 1960s whilst working for different automobile manufacturers. Subdivision is a widely used scheme for generating curves and surfaces, and the mathematical analysis of such schemes has strong connections with geometric analysis, Fourier analysis and wavelets.