Research in harmonic analysis and partial differential equations

埃尔多安将在谐波分析和偏微分方程（PDE）领域进行数学研究。在调和分析中，埃尔多安关注的是欧几里得空间中以勒贝格范数不等式为中心的问题。一个正在进行的研究课题是傅里叶变换的限制估计。近年来，埃尔多安和他的博士生在一些案例中取得了令人满意的结果。他打算继续研究限制估计及其在偏微分方程和几何测度理论中的应用。在偏微分方程中，重点研究色散偏微分方程的动力学性质。一个正在进行的研究课题是薛定谔方程（SE）的色散衰减和平滑估计。另一个主题是非线性色散偏微分方程的正则性，如Korteweg-de Vries （KdV）方程。埃尔多安和他的合作者最近获得了KdV的平滑边界，并发现了解决方案的长期行为的应用。在这项工作中使用的技术适用于PDE的广泛范围。埃尔多安将继续朝这个方向调查。他也将在光纤通信系统的数值研究中研究非线性SE的数学问题。自从傅立叶在热扩散理论方面的开创性工作以来，谐波分析在纯科学和应用科学中发挥了重要作用，并继续在量子力学中使用薛定谔方程。这个数学研究项目部分涉及来自现实世界应用的问题。例如，x射线变换，一个积分算子，应用于病人身体的密度函数，本质上是通过磁共振成像获得的数据。色散管理孤子在商业上用于跨大西洋光纤互联网电缆。更好地理解潜在的数学结构可能有助于改进这些系统。色散衰减和平滑估计在模拟各种物理现象的非线性偏微分方程中有着广泛的应用。特别是，非线性薛定谔方程模拟了光纤通信系统中的数据传输，Korteweg-de Vries方程模拟了冷等离子体中的表面水波和离子声波。

Erdogan will undertake mathematics research in the areas of harmonic analysis and partial differential equations (PDE). In harmonic analysis, Erdogan focuses on problems in Euclidean spaces centered around Lebesgue norm inequalities. One subject of on-going research are the restriction estimates for the Fourier transform. In recent years Erdogan and his Ph.D. student obtained satisfactory results in some cases. He proposes to continue his investigations on restriction estimates, and on their applications in PDE and geometric measure theory. In PDE, the focus is on the dynamical properties of dispersive PDE. One subject of on-going research is dispersive decay and smoothing estimates for the Schrodinger equation (SE). Another topic is the regularity of nonlinear dispersive PDE such as the Korteweg-de Vries (KdV) equation. Erdogan and his collaborators recently obtained smoothing bounds for KdV, and found applications to long-term behavior of the solutions. The techniques employed in this work are applicable to a wide range of PDE. Erdogan will continue his investigations in this direction. He will also work on mathematical problems on non-linear SE motivated by the numerical studies in fiber optic communication systems.Harmonic analysis has played major roles in pure and applied sciences since Fourier's seminal work on the theory of heat diffusion, continuing on with Schrodinger's equation in quantum mechanics. This mathematics research project is partly concerned with problems coming from real-world applications. For example, the X-ray transform, an integral operator, applied to the density function of a patient's body is essentially the data obtained by magnetic resonance imaging. The dispersion managed solitons are used commercially in transatlantic fiber optic internet cables. A better understanding of the underlying mathematical structure may be useful in improving these systems. The dispersive decay and smoothing estimates have a wide range of applications in nonlinear partial differential equations modeling diverse physical phenomena. In particular, the nonlinear Schrodinger equation models the transmission of data in fiber optic communication systems, and the Korteweg-de Vries equation models surface water waves as well as ion-acoustic waves in cold plasma.